# Blocks of Ariki–Koike algebras as a superlevel set
for the weight function

Salim Rostam Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

###### Abstract

We study a natural generalisation of Fayers’s definition of the weight of a
multipartition. In level one we prove that the set of all blocks, where the
size of the partition varies, can be seen as a $0$-superlevel set for the
generalised weight function. In higher levels, we use the notion of core
block, as introduced by Fayers, to prove that the set of blocks contains a
superlevel set (with infinite cardinality) for the generalised weight
function. We apply the result in level one to study a shift operation on
partitions.

##  1 Introduction

Studying the representation theory of a finite group $G$ over a field $k$ of
characteristic zero reduces to find the irreducible representations of $G$. If
$G=\mathfrak{S}_{n}$ is the symmetric group on $n$ letters, the irreducible
representations are indexed by the _partitions_ of $n$, that is, non-
increasing sequences of positive integers
$\lambda=(\lambda_{1}\geq\dots\geq\lambda_{h}>0)$ with sum
$\lvert\lambda\rvert=n$. If the field $k$ is of positive characteristic $p$,
some representations may not be written as a direct sum of irreducible ones.
Hence, we are also interested in the _blocks_ of the group algebra, that is,
indecomposable two-sided ideals. These blocks are parametrised by the _$p$
-cores_ of the partitions of $n$, in particular, any block is uniquely
determined by its $p$-core and its _$p$ -weight_. Note that the set of
irreducible representations can be parametrised by the _$p$ -regular_
partitions of $n$.

More generally, we can replace the symmetric group $\mathfrak{S}_{n}$ by a
_complex reflection group_. The set of _irreducible_ complex reflection groups
consists of an infinite family $\\{G(r,p,n)\\}_{r,p,n}$ where $r,p,n$ are
positive integers with $p\mid r$, and also a finite number of exceptions (see
[ShTo]). The complex reflection group $G(r,1,n)$ is isomorphic to
$(\mathbb{Z}/r\mathbb{Z})\wr\mathfrak{S}_{n}\simeq(\mathbb{Z}/r\mathbb{Z})^{n}\rtimes\mathfrak{S}_{n}$
and can be seen as the set of $n\times n$ monomial matrices with non-zero
entries in the set of complex $r$-roots of unity, while $G(r,p,n)$ is a
certain subgroup of $G(r,1,n)$ of index $p$. We can then study the
representation theory of a _Hecke algebra_
$\mathcal{H}_{n}^{{\boldsymbol{\mathsf{s}}}}(q)$ of $G(r,1,n)$, where
${\boldsymbol{\mathsf{s}}}=(\mathsf{s}_{1},\dots,\mathsf{s}_{r})\in\mathbb{Z}^{r}$
is a multicharge ([ArKo, BrMa, BMR]). The algebra
$\mathcal{H}_{n}^{{\boldsymbol{\mathsf{s}}}}(q)$ is a particular deformation
of the group algebra of $G(r,1,n)$ and $q\in k$ is the deformation parameter.
Assume that $q\in k\setminus\\{0\\}$ and let $e\geq 0$ be its multiplicative
order (with $e\coloneqq 0$ if $q$ is of infinite order). The representation
theories of $\mathcal{H}_{n}^{{\boldsymbol{\mathsf{s}}}}(q)$ and $G(r,1,n)$
are deeply linked. For instance, if $r=1$ then
$G(1,1,n)\simeq\mathfrak{S}_{n}$ and the situation is the following: if
$\mathcal{H}_{n}^{{\boldsymbol{\mathsf{s}}}}(q)$ is semisimple then its
irreducible representations are indexed by the partitions of $n$, otherwise
they are parametrised by the $e$-regular partitions of $n$ and the blocks are
parametrised by the $e$-cores of partitions of $n$. In the general case $r\geq
1$, if $\mathcal{H}_{n}^{{\boldsymbol{\mathsf{s}}}}(q)$ is semisimple then its
irreducible representations are indexed by the _$r$ -partitions_ of $n$, that
is, $r$-tuples
$\boldsymbol{\lambda}=\bigl{(}\lambda^{(1)},\dots,\lambda^{(r)}\bigr{)}$ with
$\lvert\boldsymbol{\lambda}\rvert=\lvert\lambda^{(1)}\rvert+\dots+\lvert\lambda^{(r)}\rvert=n$.
On the contrary, if $\mathcal{H}_{n}^{{\boldsymbol{\mathsf{s}}}}(q)$ is non-
semisimple then the situation is more complex. First, its irreducible
representations can be indexed by a non-trivial generalisation of $e$-regular
partitions, known as _Kleshchev_ $r$-partitions (see [Ar01, ArMa]). Similarly,
the naive generalisation of $e$-cores to $r$-partitions, the _$e$
-multicores_, do not parametrise in general the blocks of
$\mathcal{H}_{n}^{{\boldsymbol{\mathsf{s}}}}(q)$. Namely, Lyle and Mathas
[LyMa] proved that the blocks of
$\mathcal{H}_{n}^{{\boldsymbol{\mathsf{s}}}}(q)$ are parametrised by the
multisets of ${\boldsymbol{\mathsf{s}}}$-residues modulo $e$ of the
$r$-partitions of $n$. To make things more explicit, let
$Q=\oplus_{i\in\mathbb{Z}/e\mathbb{Z}}\mathbb{Z}\alpha_{i}$ be a free abelian
group. If $\boldsymbol{\lambda}$ is a multipartition, for any
$i\in\mathbb{Z}/e\mathbb{Z}$ we denote by
$c^{{\boldsymbol{\mathsf{s}}}}_{i}(\boldsymbol{\lambda})$ the number of
$i$-nodes of $\boldsymbol{\lambda}$. Then the block _corresponding_ to
$\boldsymbol{\lambda}$ is

$\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})\coloneqq\sum_{i\in\mathbb{Z}/e\mathbb{Z}}c^{{\boldsymbol{\mathsf{s}}}}_{i}(\boldsymbol{\lambda})\alpha_{i}\in
Q,$

and we say that $\boldsymbol{\lambda}$ _lies_ in
$\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})$. During their proof,
Lyle and Mathas used a generalisation of the $e$-weight to $r$-partitions,
introduced by Fayers [Fa06]. If
$\boldsymbol{\lambda}=\bigl{(}\lambda^{(1)},\dots,\lambda^{(r)}\bigr{)}$ is an
$r$-partition, its _${\boldsymbol{\mathsf{s}}}$ -weight_ is given by

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\coloneqq\sum_{j=1}^{r}c^{{\boldsymbol{\mathsf{s}}}}_{\mathsf{s}_{j}}(\boldsymbol{\lambda})-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(c^{{\boldsymbol{\mathsf{s}}}}_{i}(\boldsymbol{\lambda})-c^{{\boldsymbol{\mathsf{s}}}}_{i+1}(\boldsymbol{\lambda})\right)^{2}.$

In particular, two $r$-partitions that lie in the same block (i.e.
$\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})=\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\mu})$)
have the same ${\boldsymbol{\mathsf{s}}}$-weight. Fayers proved that if $r=1$
then this definition of $\mathsf{s}$-weight coincides with the usual notion of
$e$-weight for partitions. For instance, we have

$c_{0}(\lambda)=\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(c_{i}(\lambda)-c_{i+1}(\lambda)\right)^{2},$

if the partition $\lambda$ is an $e$-core (in level $1$ we may omit to write
the multicharge, assuming, without loss of generality, that
$\mathsf{s}=\mathsf{0}$). Moreover, while the fact that the $e$-weight of a
partition is non-negative follows from the definition, this is a non-trivial
statement for the ${\boldsymbol{\mathsf{s}}}$-weight in higher levels (see
[Fa06]).

In level one, a partition is an $e$-core if and only if its $e$-weight is
zero. In higher levels, we still have an implication: if an $r$-partition has
${\boldsymbol{\mathsf{s}}}$-weight at most $r-1$ then it is an $e$-multicore.
However, we can find $e$-multicores of arbitrary large
${\boldsymbol{\mathsf{s}}}$-weight. Moreover, we can have two $r$-partitions
$\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$ lying in the same block, thus
having the same ${\boldsymbol{\mathsf{s}}}$-weight, but with
$\boldsymbol{\lambda}$ (respectively $\boldsymbol{\mu}$) being (resp. not
being) an $e$-multicore. In order to obtain a more satisfying notion of core
of a multipartition, Fayers [Fa07a] introduced the notion of _core block_ and
_reduced $e$-multicore_ (the latter expression is due to [LyMa]): an
$e$-multicore $\boldsymbol{\lambda}$ is _${\boldsymbol{\mathsf{s}}}$ -reduced_
if any multipartition $\boldsymbol{\mu}$ such that
$\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\mu})=\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})$
is an $e$-multicore, in which case the block associated with
$\boldsymbol{\lambda}$ is a _core block_. If $r=1$ then every $e$-core is
$\mathsf{s}$-reduced and every block is a core block. Now if $r>1$, to any
multipartition we can still associate a unique core block, however there is no
canonical choice for a reduced $e$-multicore inside this core block. Note that
Jacon–Lecouvey [JaLe] managed to define what they have called the
_$(e,{\boldsymbol{\mathsf{s}}})$ -core_ of a multipartition (they also use the
notion of _reduced $(e,{\boldsymbol{\mathsf{s}}})$-core_, which is different
from the notion of reduced $e$-multicore that we use here). The
$(e,{\boldsymbol{\mathsf{s}}})$-core of an $r$-partition is again an
$r$-partition, and the situation is then entirely similar to the level one
case, namely, for the combinatorics of blocks. However, the
$(e,{\boldsymbol{\mathsf{s}}})$-core of a multipartition is associated with a
possibly different multicharge, which depends on the multipartition.

Now let $p\mid r$. We can use Clifford theory to study the representation
theory of $G(r,p,n)$, and this involves the following natural _shift_
operation of order $p$ on $r$-partitions:

$\boldsymbol{\lambda}=\bigl{(}\lambda^{(1)},\dots,\lambda^{(r)}\bigr{)}\longmapsto\prescript{\sigma}{}{\boldsymbol{\lambda}}\coloneqq\bigl{(}\lambda^{(r-d+1)},\dots,\lambda^{(r)},\lambda^{(1)},\dots,\lambda^{(r-d)}\bigr{)},$

where $d\coloneqq\frac{r}{p}$ (see, for instance, [Ar95, ChJa, GeJa, HuMa]).
Now take $e\geq 2$ and assume that $p$ also divides $e$. There is a map
$\sigma:Q\to Q$ of order $p$ such that if
${\boldsymbol{\mathsf{s}}}\in\mathbb{Z}^{r}$ is a multicharge satisfying some
compatibility conditions, we have the relation

$\alpha^{{\boldsymbol{\mathsf{s}}}}\bigl{(}\prescript{\sigma}{}{\boldsymbol{\lambda}})=\sigma\cdot\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda}).$
(1.1)

In [Ro19a], the author proved that if $r\geq 2$ then a block $\alpha\in
Q^{\boldsymbol{\mathsf{s}}}$ that is _stuttering_ , that is
$\sigma\cdot\alpha=\alpha$, always corresponds to a _stuttering_
multipartition, that is, a multipartition $\boldsymbol{\lambda}$ satisfying
$\prescript{\sigma}{}{\boldsymbol{\lambda}}=\boldsymbol{\lambda}$.

The aim of this paper is to study the natural generalisation of Fayers’s
weight function to
$Q=\oplus_{i\in\mathbb{Z}/e\mathbb{Z}}\mathbb{Z}\alpha_{i}$, and then apply
this result in level one to define a shift operation on partitions so that a
relation such as (1.1) holds. More precisely, let
${\boldsymbol{\mathsf{s}}}\in\mathbb{Z}^{r}$ be a multicharge and for any
$\alpha=\sum_{i\in\mathbb{Z}/e\mathbb{Z}}c_{i}\alpha_{i}\in Q$ define

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\coloneqq\sum_{j=1}^{r}c_{\mathsf{s}_{j}}-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(c_{i}-c_{i+1}\right)^{2}\in\mathbb{Z},$

so that
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}\bigl{(}\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})\bigr{)}=\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})$
for any $r$-partition $\boldsymbol{\lambda}$. We also define the set

$Q^{\boldsymbol{\mathsf{s}}}\coloneqq\bigl{\\{}\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda}):\boldsymbol{\lambda}\text{
is an }r\text{-partition}\bigr{\\}}\subseteq Q.$

Since $\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\geq 0$ for
any $r$-partition $\boldsymbol{\lambda}$, we have an inclusion
$Q^{\boldsymbol{\mathsf{s}}}\subseteq\\{\alpha\in
Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\geq 0\\}$. The paper is
mainly concerned in studying a reverse inclusion. More precisely, we prove the
following results.

######  Theorem A (Corollaries 3.7 and 3.24).

Let $r\geq 2$ and $e>0$. We have

$Q^{\boldsymbol{\mathsf{s}}}\supseteq\bigl{\\{}\alpha\in
Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)>N^{\prime}_{r,e}-r\bigr{\\}},$

where
$N^{\prime}_{r,e}\coloneqq\left\lfloor\frac{r^{2}}{2e}\left\lfloor\frac{e^{2}}{4}\right\rfloor\right\rfloor$.

######  Theorem B (Propositions 2.37 and 3.9).

Let $r\geq 1$ and $e\geq 0$. Assume that $r=1$ or
$(r,e)\in\bigl{\\{}(2,2),(2,3),(3,2)\bigr{\\}}$. Then

$Q^{\boldsymbol{\mathsf{s}}}=\bigl{\\{}\alpha\in
Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\geq 0\bigr{\\}}.$

Note that Theorem A does not hold when $e=0$, whereas Theorem B does (when
$r=1$). If $e>0$, a set of the form $\\{\alpha\in
Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\geq C\bigr{\\}}$ for $C\geq
0$ is never empty (and even always infinite), so that the inclusion of Theorem
A is not trivially true. The main idea of the proof of Theorem A is to prove
that the weight function is bounded above by a constant $N_{r,e}$ on the set
of _reduced_ $e$-multicores. Note that this assertion is wrong if we only
consider $e$-multicores, which can have arbitrarily large weight. With some
calculations involving binary matrices, we then give a sharp estimation of the
bound $N_{r,e}$.

Now assume that $r=1$, take $p$ dividing $e$ and let $\lambda$ be a partition.
Shifting the components of the _$e$ -quotient_ of $\lambda$, we define another
partition $\sigma\lambda$. Using Theorem B, we prove the following analogue of
(1.1) in level one.

######  Theorem C (Corollary 4.10).

Let $\lambda$ be a partition. With $e^{\prime}\coloneqq\frac{e}{p}$, we have

$\alpha(\sigma\lambda)=\sigma\cdot\alpha(\lambda)\iff\lvert\sigma\lambda\rvert=\lvert\lambda\rvert\iff
c_{0}(\lambda)=c_{e^{\prime}}(\lambda).$

In particular, if $\lambda$ is a partition of $n$ then $\sigma\lambda$ is a
partition of $m$ with possibly $m\neq n$. Finally, we complete the study of
stuttering blocks initiated in [Ro19b], by giving its analogue in level one.
It turns out that a similar equivalence holds, now with an additional
condition on the weight of the block.

######  Theorem D (Lemma 4.15 and Corollary 4.17).

Assume that $r=1$ and let $\alpha\in Q^{\mathsf{0}}$. The block $\alpha$
corresponds to a partition $\lambda$ satisfying $\sigma\lambda=\lambda$ if and
only if $\sigma\cdot\alpha=\alpha$ and $p\mid\mathsf{w}(\alpha)$.

We now give a brief overview of the paper. Section 2 introduces the necessary
material to define core blocks and give their proprieties as stated in
[Fa07a]. In particular, in §2.1 we define the weight of any element of
$Q=\oplus_{i\in\mathbb{Z}/e\mathbb{Z}}\mathbb{Z}\alpha_{i}$ and in §2.2 we
define the abaci representations of a partition and recall how to recover the
block associated with a partition from its abacus. We use these results to
prove that an element $\alpha\in Q$ corresponds to an $e$-core if and only if
$\alpha$ has weight $0$ (Lemma 2.36) and we deduce the case $r=1$ of Theorem B
(Proposition 2.37). In §2.4 we recall Fayers’s definition of weight for
multipartitions, and we reprove that the weight of a multipartition is non-
negative using abaci. We also recall an important result from [Fa06]
(Proposition 2.48), which expresses the weight of a multipartition as the
minimum of the cardinalities of two sets. Finally, in §2.5 we recall from
[Fa07a] the notion of core block and reduced multicore, and we prove that any
element of $Q$ is canonically associated with a core block (Lemma 2.59). This
leads to the definition of the _${\boldsymbol{\mathsf{s}}}$ -core_ of any
element of $Q$ (Definition 2.62). We conclude this section by recalling a key
result from [Fa07a] (Proposition 2.65), which characterises the abaci of
reduced $e$-multicores.

Section 3 is the heart of the paper. In §3.1 we show that the weights of
reduced $e$-multicores can be obtained via a simple functional on binary
matrices (Lemma 3.2), where the columns are seen as characteristic vectors of
subsets of $\\{1,\dots,e\\}$. In §3.2 we prove that for fixed $r$ and $e$ the
weight of a reduced $e$-multicore is bounded above by a constant $N_{r,e}$
(Theorem 3.6). We then deduce the first part of Theorem A, stating that
$Q^{\boldsymbol{\mathsf{s}}}$ contains an (infinite) superlevel set for the
weight function on $Q$ (Corollary 3.7). We also give the second part of
Theorem B, using some results of the next subsection (Proposition 3.9). In
§3.3 the aim is to give sharp bounds for the constant $N_{r,e}$. We first
compute $N_{r,e}$ for $r=2$ and $e=2$ (Propositions 3.10 and 3.11), and we
give a lower bound for $N_{r,e}$ using the superadditivity of the sequence
$(N_{r,e})_{e\geq 1}$ (Corollary 3.13). The computation of a sharp upper bound
is more elaborate. We use the fact that the computation of $N_{r,e}$ reduces
to maximising a certain quadratic form with integer coefficients on the
$(2^{e}-1)$-sphere for the $1$-norm (Proposition 3.14). Namely, using a
calculation taken from graph theory, we compute the eigenvalues of the matrix
$A_{e,k}=(\lvert E\cap F\rvert)_{E,F}$, where $E,F$ run over the subsets of
$\\{1,\dots,e\\}$ of cardinality $k$ (Lemma 3.21). Note that the matrix
$A_{e,k}$ often appears in the literature (see, for instance, [Ry81, Ry82]).
We then deduce an upper bound for $N_{r,e}$ (Corollary 3.24), and we compute
other values of $N_{r,e}$ for small $r$ or $e$, proving that
$N_{r,e}=N^{\prime}_{r,e}$ (where $N^{\prime}_{r,e}$ is the constant appearing
in Theorem A) for (at least) $r\in\\{2,4\\}$ and $e\in\\{2,\dots,6\\}$
(Proposition 3.43). We conclude the section by a quick study of the asymptotic
behaviour of $N_{r,e}$, for $e\to\infty$ (see (3.46)) and $r\to\infty$
(Corollary 3.49).

Finally, Section 4 is devoted to an application of Theorem B in level one. In
§4.1 we define a shift operation $\alpha\mapsto\sigma\cdot\alpha$ on $Q$ of
order $p$ where $p$ divides $e$, and we characterise the blocks $\alpha\in Q$
such that both $\alpha$ and $\sigma\cdot\alpha$ correspond to a partition
(Corollary 4.4). In §4.2 we propose a definition of a shift operation
$\lambda\mapsto\sigma\lambda$ on the set of partitions, uniquely defined on
the set of $e$-cores by the following requirement: if $\lambda$ is an $e$-core
then $\alpha(\sigma\lambda)$ is the core of $\sigma\cdot\alpha(\lambda)$. In
particular, if $\sigma\cdot\alpha(\lambda)=\alpha(\mu)$ for some partition
$\mu$ then $\sigma\lambda$ is the $e$-core of $\mu$. Moreover, we prove that
the two shift operations, on blocks and on partitions, are compatible under
some conditions, giving an analogue of (1.1) in level one (Corollary 4.10).
Finally, in §4.3 we give some properties of the map
$\alpha(\lambda)\mapsto\sigma\cdot\alpha(\lambda)$. We first give some
properties involving $e$-cores and $e^{\prime}$-cores, where
$e^{\prime}\coloneqq\frac{e}{p}$ (Propositions 4.13 and 4.14). Then, as in
[Ro19b] we study the existence of a stuttering partition inside a stuttering
block. More precisely, given a partition $\lambda$ such that
$\sigma\cdot\alpha(\lambda)=\alpha(\lambda)$, we prove that there exists a
partition $\mu$ satisfying $\sigma\mu=\mu$ and $\alpha(\mu)=\alpha(\lambda)$
if and only if $p$ divides the $e$-weight of $\lambda$ (Lemma 4.15 and
Corollary 4.17).

##  2 Background on core blocks

We recall here the combinatorics of blocks of Ariki–Koike algebras. We will
write $\mathbb{N}$ for $\mathbb{Z}_{\geq 0}$. Let $e\in\mathbb{N}$. If $e>0$
we identify $\mathbb{Z}/e\mathbb{Z}$ and $\\{0,\dots,e-1\\}$, and if $e=0$
then $\mathbb{Z}/e\mathbb{Z}\simeq\mathbb{Z}$.

###  2.1 Partitions

Let $n\in\mathbb{N}$. Let $\lambda$ be a _partition_ of $n$, that is, a non-
increasing sequence of positive integers
$\lambda=(\lambda_{1}\geq\dots\geq\lambda_{h}>0)$ with sum $n$. We will write
$\lvert\lambda\rvert\coloneqq n$ and $h(\lambda)\coloneqq h$. We denote by
$\emptyset$ the empty partition, which satisfies
$\lvert\emptyset\rvert=h(\emptyset)=0$. The Young diagram associated with
$\lambda$ is the region of $\mathbb{N}^{2}$ given by

$\mathcal{Y}(\lambda)\coloneqq\left\\{(a,b)\in\mathbb{N}^{2}:1\leq a\leq
h(\lambda)\text{ and }1\leq b\leq\lambda_{a}\right\\}.$

An element of $\mathcal{Y}(\lambda)$ is a _node of $\lambda$_. More generally,
we will call _node_ any element of $\mathbb{N}^{2}$. A node
$\gamma\notin\mathcal{Y}(\lambda)$ is _addable_ (respectively,
$\gamma\in\mathcal{Y}(\lambda)$ is _removable_) if
$\mathcal{Y}(\lambda)\cup\\{\gamma\\}$ (resp.
$\mathcal{Y}\setminus\\{\gamma\\}$) is the Young diagram of a partition. A
_rim hook_ of $\lambda$ is a subset of $\mathcal{Y}(\lambda)$ of the following
form:

$r^{\lambda}_{(a,b)}\coloneqq\left\\{(a^{\prime},b^{\prime})\in\mathcal{Y}(\lambda):a^{\prime}\geq
a,\,b^{\prime}\geq b\text{ and
}(a^{\prime}+1,b^{\prime}+1)\notin\mathcal{Y}(\lambda)\right\\},$

where $(a,b)\in\mathcal{Y}(\lambda)$. We say that $r^{\lambda}_{(a,b)}$ is an
_$l$ -rim hook_ if it has cardinality $l$. Note that $1$-rim hooks are exactly
removable nodes. The set $\mathcal{Y}(\lambda)\setminus r^{\lambda}_{(a,b)}$
is the Young diagram of a certain partition $\mu$, obtained by _unwrapping_ or
_removing_ the rim hook $r^{\lambda}_{(a,b)}$ from $\lambda$. Conversely, we
say that $\lambda$ is obtained from $\mu$ by _wrapping on_ or _adding_ the rim
hook $r^{\lambda}_{(a,b)}$. We say that $\lambda$ is an _$e$ -core_ if
$\lambda$ has no $e$-rim hook. Note that if $e=0$ then every partition is an
$e$-core, and if $e=1$ then the empty partition $\emptyset$ is the only
$e$-core. In particular, the combinatorics of $1$-cores is very easy, and all
the future statements in the paper can easily be proven for $e=1$.

######  Example 2.1.

We consider the partition $\lambda\coloneqq(3,2,2,1)$. An example of a $3$-rim
hook is

$r^{\lambda}_{(3,1)}=\ytableau{}&{}{}\\\ {}{}\\\ \times\times\\\ \times\,,$

and a $4$-rim hook is for instance

$r^{\lambda}_{(2,1)}=\ytableau{}&{}{}\\\ {}\times\\\ \times\times\\\
\times\,.$

We can check that $\lambda$ has no $5$-rim hooks so it is a $5$-core. We will
see in §2.2 how to use abaci to easily know whether a partition is an
$e$-core.

More generally, we can successively remove $e$-rim hooks to the partition
$\lambda$ until we reach an $e$-core partition. This $e$-core is uniquely
defined from $\lambda$, in particular it does not depend on the order we chose
to remove the $e$-rim hooks (see Lemma 2.17). We denote by
$\overline{\lambda}$ the $e$-core of $\lambda$. The number of $e$-rim hooks
that we remove to obtain $\overline{\lambda}$ from $\lambda$ is the _$e$
-weight_ of $\lambda$, denoted by $\mathsf{w}_{e}(\lambda)$. Note that
$\lvert\lambda\rvert=\lvert\overline{\lambda}\rvert+\mathsf{w}_{e}(\lambda)e$
if $e>0$ and $\mathsf{w}_{0}(\lambda)=0$.

Now let $\mathsf{s}\in\mathbb{Z}$ be a _charge_. Given
$i\in\mathbb{Z}/e\mathbb{Z}$, an _$(i,\mathsf{s})$ -node_ or simply _$i$
-node_ is a node of $\mathsf{s}$-residue $i$, where the _$\mathsf{s}$
-residue_ of the node $(a,b)\in\mathbb{N}^{2}$ is
$\mathrm{res}^{\mathsf{s}}(a,b)\coloneqq b-a+\mathsf{s}\pmod{e}$. More
generally, if $i\in\mathbb{Z}$ then an $i$-node will be a node
$\gamma\in\mathbb{N}^{2}$ such that
$\mathrm{res}^{\mathsf{s}}(\gamma)=i\pmod{e}$. We denote by
$c^{\mathsf{s}}_{i}(\lambda)$ the number of $(i,\mathsf{s})$-nodes of
$\lambda$. Note the following simple equality, where $\mathsf{0}$ denotes the
charge $0\in\mathbb{Z}$:

$c^{\mathsf{s}}_{i}(\lambda)=c^{\mathsf{0}}_{i-\mathsf{s}}(\lambda).$ (2.2)

######  Remark 2.3.

Let $i\in\mathbb{Z}$. The integer $c^{\mathsf{s}}_{i}(\lambda)$ does not
depend on $e$ if $e\geq\max\bigl{(}\lvert\lambda\rvert,\lvert
i-\mathsf{s}\rvert\bigr{)}$, in which case the value of
$c^{\mathsf{s}}_{i}(\lambda)$ is the one for $e=0$. In particular, we will
sometimes deal with the case $e=0$ by taking $e\gg 0$. Besides, we have
$c^{\mathsf{s}}_{i}(\lambda)=0$ as soon as $e\geq\lvert
i-\mathsf{s}\rvert\geq\lvert\lambda\rvert$ (as soon as $\lvert
i-\mathsf{s}\rvert\geq\lvert\lambda\rvert$ if $e=0$).

A simple consequence of the definition of a rim hook is the following.

######  Lemma 2.4.

Let $\lambda$ be a partition. Any $e$-rim hook of $\lambda$ has exactly one
node of residue $i$ for each $i\in\mathbb{Z}/e\mathbb{Z}$.

Fayers [Fa06] introduced another weight function on partitions: the
_$\mathsf{s}$ -weight_ of $\lambda$, given by

$\mathsf{w}^{\mathsf{s}}(\lambda)\coloneqq
c^{\mathsf{s}}_{\mathsf{s}}(\lambda)-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(c^{\mathsf{s}}_{i}(\lambda)-c^{\mathsf{s}}_{i+1}(\lambda)\right)^{2}.$

It is well-defined for $e=0$ by Remark 2.3. We will see that the
$\mathsf{s}$-weight coincide with the $e$-weight, however these weights will
have different generalisations to multipartitions (see §2.4).

Let $Q$ be a free $\mathbb{Z}$-module with a basis indexed by
$\mathbb{Z}/e\mathbb{Z}$ that we denote by
$\\{\alpha_{i}\\}_{i\in\mathbb{Z}/e\mathbb{Z}}$. We have
$Q=\oplus_{i\in\mathbb{Z}/e\mathbb{Z}}\mathbb{Z}\alpha_{i}$ and we define
$Q_{+}\coloneqq\oplus_{i\in\mathbb{Z}/e\mathbb{Z}}\mathbb{N}\alpha_{i}$. We
will also use the element $\mathbf{1}\in Q_{+}$ defined by

$\mathbf{1}=\begin{cases}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\alpha_{i},&\text{if
}e>0,\\\ 0,&\text{if }e=0.\end{cases}$

For any $\alpha\in Q$, we denote by $\lvert\alpha\rvert\in\mathbb{Z}$ the sum
of its coordinates in the basis
$\\{\alpha_{i}\\}_{i\in\mathbb{Z}/e\mathbb{Z}}$. For any partition $\lambda$,
we define

$\alpha^{\mathsf{s}}(\lambda)\coloneqq\sum_{\gamma\in\mathcal{Y}(\lambda)}\alpha_{\mathrm{res}^{\mathsf{s}}(\gamma)}=\sum_{i\in\mathbb{Z}/e\mathbb{Z}}c^{\mathsf{s}}_{i}(\lambda)\alpha_{i}\in
Q_{+}.$

Note that $\lvert\alpha^{\mathsf{s}}(\lambda)\rvert=\lvert\lambda\rvert$. We
will say that the partition $\lambda$ _lies_ in $\alpha\in Q$ if
$\alpha^{\mathsf{s}}(\lambda)=\alpha$. Lemma 2.4 can be reformulated as
follows.

######  Lemma 2.5.

Let $\lambda$ and $\mu$ be two partitions such that $\mu$ is obtained from
$\lambda$ by adding an $e$-rim hook. Then
$\alpha^{\mathsf{s}}(\mu)=\alpha^{\mathsf{s}}(\lambda)+\mathbf{1}$.

We extend the definition of the $\mathsf{s}$-weight to $Q$ by setting

$\mathsf{w}^{\mathsf{s}}(\alpha)\coloneqq
c_{\mathsf{s}}-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(c_{i}-c_{i+1}\right)^{2}\in\mathbb{Z},$

for any $\alpha=\sum_{i\in\mathbb{Z}/e\mathbb{Z}}c_{i}\alpha_{i}\in Q$. Note
that $\mathsf{w}^{\mathsf{s}}(\alpha)\in\mathbb{Z}$ indeed, since
$\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\bigl{(}c_{i}-c_{i+1}\bigr{)}=0$ and thus
$\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\bigl{(}c_{i}-c_{i+1}\bigr{)}^{2}$ is even.
Moreover, we have

$\mathsf{w}^{\mathsf{s}}(\lambda)=\mathsf{w}^{\mathsf{s}}\bigl{(}\alpha^{\mathsf{s}}(\lambda)\bigr{)},$
(2.6)

and if $e>0$ then

$\mathsf{w}^{\mathsf{s}}(\alpha+h\mathbf{1})=\mathsf{w}^{\mathsf{s}}(\alpha)+h,$
(2.7)

for any $h\in\mathbb{Z}$. Finally, we define

$Q^{\mathsf{s}}\coloneqq\left\\{\alpha^{\mathsf{s}}(\lambda):\lambda\text{
partition of }n\text{ for some }n\in\mathbb{N}\right\\}.$

In §2.2 we will give a parametrisation of $Q^{\mathsf{s}}$ using abaci, while
in §2.3 we will give an implicit description of $Q^{\mathsf{s}}$, using a
superlevel set for the $\mathsf{s}$-weight function.

###  2.2 Abaci

Let $\lambda$ be a partition. The _charged beta-number_ associated with the
partition $\lambda$ and the charge $\mathsf{s}$ is the sequence
$\beta^{\mathsf{s}}(\lambda)=\bigl{(}\beta^{\mathsf{s}}(\lambda)_{a}\bigr{)}_{a\geq
1}$ defined by

$\beta^{\mathsf{s}}(\lambda)_{a}\coloneqq\mathsf{s}+\lambda_{a}-a\in\mathbb{Z},$

for any $a\geq 1$. It is a (strictly) decreasing sequence of integers. For
$a>h(\lambda)$ we have $\beta^{\mathsf{s}}(\lambda)_{a}=\mathsf{s}-a$, and
conversely if $\beta=(\beta_{a})_{a\geq 1}$ is a decreasing sequence of
integers such that $\beta_{a}=\mathsf{s}-a$ for $a\gg 0$ then
$\beta=\beta^{\mathsf{s}}(\lambda)$ for some partition $\lambda$. Representing
$\beta^{\mathsf{s}}(\lambda)$ by a copy of $\mathbb{Z}$ where we put a bead on
position $i\in\mathbb{Z}$ if $i\in\beta^{\mathsf{s}}(\lambda)$ and a gap
otherwise gives the _$\mathsf{s}$ -abacus_ associated with $\lambda$.

######  Example 2.8.

The $\mathsf{0}$-beta-number associated with the empty partition is
$(-1,-2,-3,\dots)$ and the associated charged abacus is

$\cdots$

.

######  Example 2.9.

The $\mathsf{1}$-beta-number associated with the partition $(4,3,3,1)$ is
$(4,2,1,-2,-4,-5,\dots)$ and the associated charged abacus is

$\cdots$

.

The next result is well-known, see for instance [Ol, Proposition (1.8)].

######  Lemma 2.10.

Let $l\in\mathbb{N}$. The partition $\lambda$ has an $l$-rim hook if and only
if there is an element $b\in\beta(\lambda)$ such that
$b-l\notin\beta(\lambda)$. In that case, if $\mu$ is the partition that we
obtain by removing this $l$-rim hook, then $\beta(\mu)$ is obtained by
replacing $b$ by $b-l$ in $\beta(\lambda)$ and then sorting in decreasing
order.

Unless mentioned otherwise, we now assume that $e>0$. For each
$i\in\\{0,\dots,e-1\\}$ we can define
$\mathfrak{b}_{i}^{\mathsf{s}}(\lambda)\in\mathbb{Z}$ to be the largest
element of $\beta^{\mathsf{s}}(\lambda)$ congruent to $i$ modulo $e$. By
[Fa07a, Lemma 3.2], if $\lambda$ is an $e$-core we have the following
relation:

$\mathsf{s}=\frac{e+1}{2}+\frac{1}{e}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\mathfrak{b}^{\mathsf{s}}_{i}(\lambda).$
(2.11)

Setting
$y^{\mathsf{s}}_{i}(\lambda)\coloneqq\frac{1}{e}\bigl{(}\mathfrak{b}^{\mathsf{s}}_{i}(\lambda)-i\bigr{)}+1\in\mathbb{Z}$
for each $i\in\\{0,\dots,e-1\\}$, the relation (2.11) becomes

$\mathsf{s}=\sum_{i=0}^{e-1}y^{\mathsf{s}}_{i}(\lambda).$ (2.12)

######  Proposition 2.13.

Let $y_{0},\dots,y_{e-1}\in\mathbb{Z}$. There exists an $e$-core $\lambda$
such that $y_{i}=y_{i}^{\mathsf{s}}(\lambda)$ for all $i\in\\{0,\dots,e-1\\}$
if and only if $y_{0}+\dots+y_{e-1}=\mathsf{s}$.

###### Proof.

The direct implication follows from (2.12). Now assume that
$y_{0}+\dots+y_{e-1}=\mathsf{s}$ and define $\mathfrak{b}_{i}\coloneqq
e(y_{i}-1)+i$ for all $i\in\\{0,\dots,e-1\\}$. The (unique) decreasing
sequence of integers $\beta=(\beta_{a})_{a\geq 1}$ defined by:

  * •

for all $i\in\\{1,\dots,e-1\\}$, the integer $\mathfrak{b}_{i}$ is the largest
element of $\beta$ congruent to $i$ modulo $e$;

  * •

for all $h\in\beta$ we have $h-e\in\beta$;

is the charged beta-number associated with some partition $\lambda$ and charge
$\mathsf{s}^{\prime}\in\mathbb{Z}$. By Lemma 2.10 the partition $\lambda$ is
an $e$-core and by (2.12) we have $\mathsf{s}^{\prime}=\mathsf{s}$. This
concludes the proof. ∎

The abacus representation of a partition that we now use has been first
introduced by James [Ja]; we use here the setting of [LyMa]. We dispose the
elements of $\beta^{\mathsf{s}}(\lambda)$ on an abacus with now $e$ runners,
the _charged $e$-abacus of $\lambda$_, each runner being a horizontal copy of
$\mathbb{Z}$ and displayed in the following way: the $0$-th runner is on the
bottom and the origins of each copy of $\mathbb{Z}$ are aligned with respect
to a vertical line. We record the elements of $\beta^{\mathsf{s}}(\lambda)$ on
this abacus according to the following rule: there is a bead at position
$j\in\mathbb{Z}$ on the runner $i\in\\{0,\dots,e-1\\}$ if and only if there
exists $a\geq 1$ such that $\beta^{\mathsf{s}}(\lambda)_{a}=i+je$. For each
$i\in\\{0,\dots,e-1\\}$, the $i$-th runner corresponds to the charged abacus
(as defined at the beginning of §2.2) of a certain partition $\lambda^{[i]}$.
The $e$-tuple $\bigl{(}\lambda^{[0]},\dots,\lambda^{[e-1]}\bigr{)}$ is the
_$e$ -quotient_ of $\lambda$. In particular, any partition is uniquely
determined by its $e$-core and its $e$-quotient.

######  Remark 2.14.

The $e$-quotient of $\lambda$ depends on the charge $\mathsf{s}$ only up to a
shift: if $\bigl{(}\lambda^{[0]},\dots,\lambda^{[e-1]}\bigr{)}$ is the
$e$-quotient of $\lambda$ computed on the $\mathsf{s}$-charged $e$-abacus then
$\bigl{(}\lambda^{[e-1]},\lambda^{[0]},\dots,\lambda^{[e-2]}\bigr{)}$ is the
$e$-quotient computed on the $(\mathsf{s}+1)$-charged $e$-abacus.

######  Example 2.15.

We take the charge $\mathsf{s}=\mathsf{1}$ and we consider the partition
$\lambda\coloneqq(4,3,3,1)$ as in Example 2.9. The associated charged
$3$-abacus is

$\dots$$\dots$$\dots$

,

and the associated $6$-abacus is

$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$

.

Counting the number of gaps down each bead (continuing counting on the left
starting from the top runner when reaching the bottom one) recovers the
underlying partition. The $3$-quotient of $\lambda$ is
$\bigl{(}\emptyset,\emptyset,(1)\bigr{)}$ and its $6$-quotient has only empty
partitions.

The $e$-abacus representation is particularly adapted to the addition and
deletion of rim hooks: we give two particular cases, as corollaries of Lemma
2.10.

######  Lemma 2.16.

Let $\lambda$ be a partition and $i\in\\{0,\dots,e-1\\}$.

  * •

The partition $\lambda$ has a removable $i$-node if and only if we can move a
bead from runner $i$ to runner $i-1$ (to runner $e-1$ if $i=0$), keeping the
same position $j\in\mathbb{Z}$ (from position $j$ to $j-1$ if $i=0$).

  * •

The partition $\lambda$ has an addable $i$-node if and only if we can move a
bead from runner $i$ to runner $i+1$ (to runner $0$ if $i=e-1$), keeping the
same position $j\in\mathbb{Z}$ (from position $j$ to $j+1$ if $i=e-1$).

######  Lemma 2.17.

Let $\lambda$ be a partition.

  * •

The partition $\lambda$ has an $e$-rim hook if and only if on some runner we
can slide a bead at some position $j\in\mathbb{Z}$ to the previously free
position $j-1$. Hence, the partition $\lambda$ is an $e$-core if and only if
its associated $e$-abacus has no gaps, that is, there are no gaps between two
beads on a same runner.

  * •

The partition $\lambda$ has an addable $e$-rim hook if and only if on some
runner we can slide a bead at some position $j\in\mathbb{Z}$ to the previously
free position $j+1$. In particular, to any partition we can add at least $e$
different $e$-rim hooks.

######  Remark 2.18.

Let $\lambda$ be an $e$-core. For any $i\in\\{0,\dots,e-1\\}$, the integer
$y^{\mathsf{s}}_{i}(\lambda)$ corresponds to the position of the first gap on
runner $i$ (pictured by
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\blacklozenge}$
in the abacus below). For instance, by Lemma 2.17 the partition
$\lambda=(4,3,3,1)$ of Example 2.15 is a $6$-core and we have
$y^{\mathsf{s}}(\lambda)=(0,1,1,-1,1,-1)$. In particular, note that (2.12) is
satisfied indeed.

$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$$\blacklozenge$

Note that Lemma 2.17 implies

$\mathsf{w}_{e}(\lambda)=\sum_{i=0}^{e-1}\left\lvert\lambda^{[i]}\right\rvert.$
(2.19)

The next result follows from Lemma 2.16 (see, for instance, [Ro19b, Lemma
2.11]).

######  Proposition 2.20.

Assume that $\lambda$ is an $e$-core. For any $i\in\\{0,\dots,e-1\\}$ we have:

$y^{\mathsf{s}}_{i}(\lambda)=c^{\mathsf{s}}_{i}(\lambda)-c^{\mathsf{s}}_{i+1}(\lambda)+y^{\mathsf{s}}_{i}(\emptyset).$

Note that we can compute the value of $y^{\mathsf{s}}_{i}(\emptyset)$.

######  Lemma 2.21.

Write $\mathsf{s}=be+\mathsf{s}^{\prime}$, where $b\in\mathbb{Z}$ and
$\mathsf{s}^{\prime}\in\\{0,\dots,e-1\\}$. For any $i\in\\{0,\dots,e-1\\}$ we
have:

$y^{\mathsf{s}}_{i}(\emptyset)=\begin{cases}b+1,&\text{if
}i\in\\{0,\dots,\mathsf{s}^{\prime}-1\\},\\\ b,&\text{otherwise}.\end{cases}$

###### Proof.

We have $\beta^{\mathsf{s}}(\emptyset)_{a}=\mathsf{s}-a$ for all $a\geq 1$.
Thus, for any $a\in\\{1,\dots,e\\}$, if $i\in\\{0,\dots,e-1\\}$ is the residue
modulo $e$ of $\mathsf{s}-a$ then
$\mathfrak{b}^{\mathsf{s}}_{i}(\emptyset)=\mathsf{s}-a$. Thus, if $a$
describes $\\{1,\dots,\mathsf{s}^{\prime}\\}$ then
$i\coloneqq\mathsf{s}^{\prime}-a$ describes
$\\{0,\dots,\mathsf{s}^{\prime}-1\\}$, moreover $i=\mathsf{s}-a\pmod{e}$ thus
$\mathfrak{b}^{\mathsf{s}}_{i}(\emptyset)=\mathsf{s}-a=be+i$ and
$y^{\mathsf{s}}_{i}(\emptyset)=\frac{1}{e}\bigl{(}\mathfrak{b}^{\mathsf{s}}_{i}(\emptyset)-i\bigr{)}+1=b+1$.
Similarly, if $a\in\\{\mathsf{s}^{\prime}+1,\dots,e\\}$ then with
$i\coloneqq\mathsf{s}^{\prime}-a+e\in\\{\mathsf{s}^{\prime},\dots,e-1\\}$ we
have $i=\mathsf{s}-a\pmod{e}$ thus
$\mathfrak{b}^{\mathsf{s}}_{i}(\emptyset)=be+i-e$ and
$y^{\mathsf{s}}_{i}(\emptyset)=b$. ∎

For any $i\in\mathbb{Z}/e\mathbb{Z}$, define

$x^{\mathsf{s}}_{i}(\lambda)\coloneqq
y^{\mathsf{s}}_{i}(\lambda)-y^{\mathsf{s}}_{i}(\emptyset).$

By Proposition 2.20 (and (2.12)), for an $e$-core $\lambda$ we have

$\displaystyle x^{\mathsf{s}}_{i}(\lambda)$
$\displaystyle=c^{\mathsf{s}}_{i}(\lambda)-c^{\mathsf{s}}_{i+1}(\lambda),$
$\displaystyle\text{for all }i\in\mathbb{Z}/e\mathbb{Z},$ (2.22)
$\displaystyle\sum_{i\in\mathbb{Z}/e\mathbb{Z}}x^{\mathsf{s}}_{i}(\lambda)$
$\displaystyle=0.$ (2.23)

As for the $e$-quotient, the family
$x^{\mathsf{s}}(\lambda)=\bigl{(}x^{\mathsf{s}}_{i}(\lambda)\bigr{)}_{i\in\mathbb{Z}/e\mathbb{Z}}$
depends on the charge $\mathsf{s}$ only up to a shift (by (2.2) and (2.22)).
The following proposition is a direct consequence of (2.12) and Proposition
2.13.

######  Proposition 2.24.

Let $\mathsf{s}\in\mathbb{Z}$ and $x_{0},\dots,x_{e-1}\in\mathbb{Z}$. There
exists an $e$-core $\lambda$ such that $x_{i}=x_{i}^{\mathsf{s}}(\lambda)$ for
all $i\in\\{0,\dots,e-1\\}$ if and only if $x_{0}+\dots+x_{e-1}=0$.

For completeness, we also give the version of Proposition 2.24 in the case
$e=0$. We recall that every partition $\lambda$ is a $0$-core, and we define
the integers $x^{\mathsf{s}}_{i}(\lambda)$ for any $i\in\mathbb{Z}$ using
(2.22). By Remark 2.3 the family
$\bigl{(}x^{\mathsf{s}}_{i}(\lambda)\bigr{)}_{i\in\mathbb{Z}}$ has finite
support, equality (2.23) is still satisfied and by (2.22) we have

$x^{\mathsf{s}}_{i}(\lambda)\in\begin{cases}\\{0,1\\},&\text{if
}i\geq\mathsf{s},\\\ \\{0,-1\\},&\text{if }i<\mathsf{s}.\end{cases}$ (2.25)

######  Proposition 2.26.

Let $(x_{i})_{i\in\mathbb{Z}}\in\mathbb{Z}^{\mathbb{Z}}$ with finite support.
There exists a partition $\lambda$ such that
$x_{i}=x^{\mathsf{s}}_{i}(\lambda)$ for all $i\in\mathbb{Z}$ if and only if

$\displaystyle x_{i}$ $\displaystyle\in\begin{cases}\\{0,1\\},&\text{if
}i\geq\mathsf{s},\\\ \\{0,-1\\},&\text{if }i<\mathsf{s},\end{cases}$ (2.27)
for all $i\in\mathbb{Z}$, and $\displaystyle\sum_{i\in\mathbb{Z}}x_{i}$
$\displaystyle=0.$

###### Proof.

We have just seen that these conditions are necessary. Now let
$(x_{i})_{i\in\mathbb{Z}}\in\mathbb{Z}^{\mathbb{Z}}$ with finite support
satisfy the conditions of Proposition 2.26 and let $N>0$ such that $x_{i}=0$
if $\lvert i\rvert\geq N$. We define a family $(c_{i})_{i\in\mathbb{Z}}$ by
setting $c_{N}\coloneqq 0$ and

$c_{i}\coloneqq\begin{cases}c_{i+1}+x_{i},&\text{if }i<N,\\\
c_{i-1}-x_{i-1},&\text{if }i>N.\end{cases}$

By assumption we have $c_{i}=0$ if $\lvert i\rvert\geq N$, moreover
$c_{i}-c_{i+1}=x_{i}$ for all $i\in\mathbb{Z}$. By (2.27), it is clear that we
can construct a partition $\lambda$ such that
$c_{i}=c^{\mathsf{s}}_{i}(\lambda)$ for all $i\in\mathbb{Z}$. ∎

The following result shows that the two notions of weight that we have
introduced coincide.

######  Proposition 2.28 ([Fa06, Proposition 2.1]).

Let $e\in\mathbb{N}$. We have
$\mathsf{w}_{e}(\lambda)=\mathsf{w}^{\mathsf{s}}(\lambda)$.

We are thus able to recover the integers $c^{\mathsf{s}}_{i}(\lambda)$ from
$x^{\mathsf{s}}(\lambda)$.

######  Corollary 2.29.

Let $e\in\mathbb{N}$.

  * •

Assume that $e>0$. If $\lambda$ is an $e$-core, then

$\displaystyle c^{\mathsf{s}}_{\mathsf{s}}(\lambda)$
$\displaystyle=\frac{1}{2}\lVert
x^{\mathsf{s}}(\lambda)\rVert^{2}=\frac{1}{2}\sum_{i=0}^{e-1}x^{\mathsf{s}}_{i}(\lambda)^{2},$
$\displaystyle c^{\mathsf{s}}_{\mathsf{s}+i}(\lambda)$
$\displaystyle=\frac{1}{2}\lVert
x^{\mathsf{s}}(\lambda)\rVert^{2}-x^{\mathsf{s}}_{\mathsf{s}}(\lambda)-\dots-x^{\mathsf{s}}_{\mathsf{s}+i-1}(\lambda),$

for all $i\in\\{1,\dots,e-1\\}$.

  * •

Assume that $e=0$. We have

$\displaystyle c^{\mathsf{s}}_{\mathsf{s}}(\lambda)$
$\displaystyle=\frac{1}{2}\lVert
x^{\mathsf{s}}(\lambda)\rVert^{2}=\frac{1}{2}\sum_{i\in\mathbb{Z}}x^{\mathsf{s}}_{i}(\lambda)^{2},$
$\displaystyle c^{\mathsf{s}}_{\mathsf{s}+i}(\lambda)$
$\displaystyle=\frac{1}{2}\lVert
x^{\mathsf{s}}(\lambda)\rVert^{2}-x^{\mathsf{s}}_{\mathsf{s}}(\lambda)-\dots-x^{\mathsf{s}}_{\mathsf{s}+i-1}(\lambda),$
$\displaystyle c^{\mathsf{s}}_{\mathsf{s}-i}(\lambda)$
$\displaystyle=\frac{1}{2}\lVert
x^{\mathsf{s}}(\lambda)\rVert^{2}+x^{\mathsf{s}}_{\mathsf{s}-1}(\lambda)+\dots+x^{\mathsf{s}}_{\mathsf{s}-i+1}(\lambda),$

for all $i\in\mathbb{Z}_{>0}$.

###  2.3 Blocks for partitions

Let $e\in\mathbb{N}$. We first recall some standard facts concerning blocks
associated with partitions.

######  Lemma 2.30.

If $\lambda$ and $\mu$ are two $e$-cores with
$\alpha^{\mathsf{s}}(\lambda)=\alpha^{\mathsf{s}}(\mu)$ then $\lambda=\mu$.

###### Proof.

The statement is clear if $e=0$. If $e>0$, by (2.22) we have
$x^{\mathsf{s}}(\lambda)=x^{\mathsf{s}}(\mu)$. We thus conclude that
$y^{\mathsf{s}}_{i}(\lambda)=y^{\mathsf{s}}_{i}(\mu)$ and thus
$\mathfrak{b}^{\mathsf{s}}_{i}(\lambda)=\mathfrak{b}_{i}^{\mathsf{s}}(\mu)$
for all $i\in\\{0,\dots,e-1\\}$ whence $\lambda=\mu$. ∎

Together with Lemma 2.5, (2.6) and Proposition 2.28, we deduce the next lemma,
which can be seen as a combinatorial version of the so-called “Nakayama’s
Conjecture”.

######  Lemma 2.31 ([JaKe, Theorem 2.7.41]).

Two partitions that lie in the same block share the same $e$-core.

In particular, the elements $\alpha^{\mathsf{s}}(\lambda)$ encode the blocks
of the associated _Iwahori–Hecke algebra_ , see for instance [Ma, 5.38
Corollary]. Finally, by (2.6), Proposition 2.28 and Lemma 2.31, we have the
following result.

######  Lemma 2.32.

Let $\alpha\in Q^{\mathsf{s}}$ with $\mathsf{w}^{\mathsf{s}}(\alpha)=0$. Then
there is a unique partition lying in $\alpha$, and this partition is an
$e$-core.

######  Remark 2.33.

We will see in Lemma 2.36 that the statement of Lemma 2.32 remains true if we
replace $\alpha\in Q^{\mathsf{s}}$ with $\alpha\in Q$.

We can now give a $1{:}1$-parametrisation of the set
$Q^{\mathsf{s}}=\\{\alpha^{\mathsf{s}}(\lambda):\lambda$ partition of $n$ for
some $n\in\mathbb{N}\\}$ by $\mathbb{Z}^{e-1}\times\mathbb{N}$, if $e>0$. By
Lemmas 2.30 and 2.32, for any $\alpha\in Q^{\mathsf{s}}$ we can denote by
$\lambda_{\alpha}$ the (unique) common $e$-core of the partitions lying in
$\alpha$.

######  Proposition 2.34.

Assume that $e>0$. The map

$Q^{\mathsf{s}}\longrightarrow\bigl{\\{}x=(x_{0},\dots,x_{e-1})\in\mathbb{Z}^{e}:x_{0}+\dots+x_{e-1}=0\bigr{\\}}\times\mathbb{N},$

given by
$\alpha\mapsto\bigl{(}x^{\mathsf{s}}(\lambda_{\alpha}),\mathsf{w}^{\mathsf{s}}(\alpha)\bigr{)}$,
is a bijection. Its inverse is given by

$(x,w)\longmapsto\left(\frac{1}{2}\lVert
x\rVert^{2}+w\right)\sum_{i=0}^{e-1}\alpha_{i}-\sum_{i=1}^{e-1}(x_{\mathsf{s}}+\dots+x_{\mathsf{s}+i-1})\alpha_{\mathsf{s}+i},$

where the indices of $x=(x_{0},\dots,x_{e-1})$ are taken modulo $e$.

######  Remark 2.35.

Using Proposition 2.26 and Corollary 2.29, we can also give a version of
Proposition 2.34 in the case $e=0$, using the map $\alpha\mapsto
x^{\mathsf{s}}(\lambda_{\alpha})$. Note that in this case, an element
$\alpha=\sum_{i\in\mathbb{Z}}c_{i}\alpha_{i}\in Q$ is in $Q^{\mathsf{s}}$ if
and only if

$c_{i}-c_{i+1}\in\begin{cases}\\{0,1\\},&\text{if }i\geq\mathsf{s},\\\
\\{0,-1\\},&\text{if }i<\mathsf{s}.\end{cases}$

We now want to give an implicit description of $Q^{\mathsf{s}}$. Note that in
the following results we consider the $\mathsf{s}$-weight inside $Q$ and not
$Q_{+}$.

######  Lemma 2.36.

We have $\bigl{\\{}\alpha^{\mathsf{s}}(\lambda):\lambda\text{ is an
}e\text{-core}\bigr{\\}}=\\{\alpha\in Q:\mathsf{w}^{\mathsf{s}}(\alpha)=0\\}$.

###### Proof.

We first assume that $e>0$. The inclusion $\subseteq$ follows from (2.6) and
Proposition 2.28. Now let
$\alpha=\sum_{i\in\mathbb{Z}/e\mathbb{Z}}c_{i}\alpha_{i}\in Q$ such that
$\mathsf{w}^{\mathsf{s}}(\alpha)=0$ and define $x_{i}\coloneqq c_{i}-c_{i+1}$
for all $i\in\mathbb{Z}/e\mathbb{Z}$. By Proposition 2.24 we know that
$x_{i}=x^{\mathsf{s}}_{i}(\lambda)$ for some $e$-core $\lambda$. By Corollary
2.29 and since $\mathsf{w}^{\mathsf{s}}(\alpha)=0$ we have

$\displaystyle c^{\mathsf{s}}_{\mathsf{s}}(\lambda)$
$\displaystyle=\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}x^{\mathsf{s}}_{i}(\lambda)^{2}$
$\displaystyle=\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}x_{i}^{2}$
$\displaystyle=c_{\mathsf{s}},$

and

$\displaystyle c^{\mathsf{s}}_{\mathsf{s}+i}(\lambda)$
$\displaystyle=c^{\mathsf{s}}_{\mathsf{s}}(\lambda)-x^{\mathsf{s}}_{\mathsf{s}}(\lambda)-\dots-x^{\mathsf{s}}_{\mathsf{s}+i-1}(\lambda)$
$\displaystyle=c^{\mathsf{s}}_{\mathsf{s}}-x^{\mathsf{s}}_{\mathsf{s}}-\dots-x^{\mathsf{s}}_{\mathsf{s}+i-1}$
$\displaystyle=c_{\mathsf{s}+i},$

thus $\alpha^{\mathsf{s}}(\lambda)=\alpha$.

We now assume that $e=0$. Again $\subseteq$ follows from (2.6) and Proposition
2.28, thus let $\alpha=\sum_{i\in\mathbb{Z}}c_{i}\alpha_{i}\in Q$ such that
$\mathsf{w}^{\mathsf{s}}(\alpha)=0$. We can repeat the proof of the case
$e>0$, using Proposition 2.26 instead of Proposition 2.24. However, we have to
ensure that $c_{i}-c_{i+1}\in\begin{cases}\\{0,1\\},&\text{if
}i\geq\mathsf{s},\\\ \\{0,-1\\},&\text{if }i<\mathsf{s}.\end{cases}$ Since
$\mathsf{w}^{\mathsf{s}}(\alpha)=0$ we have

$\displaystyle 2c_{\mathsf{s}}$
$\displaystyle=\sum_{i\in\mathbb{Z}}(c_{i}-c_{i+1})^{2}$
$\displaystyle=\sum_{i\geq\mathsf{s}}(c_{i}-c_{i+1})^{2}+\sum_{i<\mathsf{s}}(c_{i}-c_{i+1})^{2}$
$\displaystyle\geq\sum_{i\geq\mathsf{s}}(c_{i}-c_{i+1})+\sum_{i<\mathsf{s}}(c_{i+1}-c_{i})$
$\displaystyle=c_{\mathsf{s}}+c_{\mathsf{s}},$

thus we deduce that

$(c_{i}-c_{i+1})^{2}=\begin{cases}c_{i}-c_{i+1},&\text{if }i\geq\mathsf{s},\\\
c_{i+1}-c_{i},&\text{if }i<\mathsf{s},\end{cases}$

for all $i\in\mathbb{Z}$, thus we obtain the desired result. Another way to
see the result for $e=0$ is to consider $e>0$ large enough so that we can
apply the previous case (using Remark 2.3). ∎

######  Proposition 2.37.

We have $Q^{\mathsf{s}}=\\{\alpha\in Q:\mathsf{w}^{\mathsf{s}}(\alpha)\geq
0\\}$.

###### Proof.

Assume first that $e>0$. Again $\subseteq$ follows from (2.6) and Proposition
2.28. Let $\alpha=\sum_{i\in\mathbb{Z}/e\mathbb{Z}}c_{i}\alpha_{i}\in Q$ such
that $\mathsf{w}^{\mathsf{s}}(\alpha)\geq 0$. By (2.7), setting
$\hat{\alpha}\coloneqq\alpha-\mathsf{w}^{\mathsf{s}}(\alpha)\mathbf{1}\in Q$
we have $\mathsf{w}^{\mathsf{s}}(\hat{\alpha})=0$ thus by Lemma 2.36 we have
$\hat{\alpha}=\alpha^{\mathsf{s}}(\lambda)$ for some $e$-core $\lambda$. Now
if $\mu$ is any partition obtained from $\lambda$ by adding
$\mathsf{w}^{\mathsf{s}}(\alpha)$ times an $e$-rim hook we have
$\alpha^{\mathsf{s}}(\mu)=\alpha^{\mathsf{s}}(\lambda)+\mathsf{w}^{\mathsf{s}}(\alpha)\mathbf{1}=\hat{\alpha}+\mathsf{w}^{\mathsf{s}}(\alpha)\mathbf{1}=\alpha$
which concludes the proof in the case $e>0$.

Now if $e=0$, if $\alpha=\sum_{i\in\mathbb{Z}}c_{i}\alpha_{i}\in Q$ is such
that $\mathsf{w}^{\mathsf{s}}(\alpha)\geq 0$ then as in the proof of Lemma
2.36 we have

$2c_{\mathsf{s}}\geq\sum_{i\in\mathbb{Z}}(c_{i}-c_{i+1})^{2}\geq\sum_{i\geq\mathsf{s}}(c_{i}-c_{i+1})+\sum_{i<\mathsf{s}}(c_{i+1}-c_{i})=2c_{\mathsf{s}},$

thus we have in fact $\mathsf{w}^{\mathsf{s}}(\alpha)=0$ and we apply Lemma
2.36. ∎

The aim of Section 3 is to study an analogue of Proposition 2.37 in higher
levels. By (2.7) we obtain the following corollary of Proposition 2.37.

######  Corollary 2.38.

Assume $e>0$. For any $\alpha\in Q$ and any
$h\geq-\mathsf{w}^{\mathsf{s}}(\alpha)$ we have

$\alpha+h\mathbf{1}\in Q^{\mathsf{s}}.$

######  Example 2.39.

Assume $e\geq 2$ and take $\mathsf{s}=\mathsf{0}$. Let $h\in\mathbb{N}$ and
define $\alpha\coloneqq h\alpha_{0}\in Q_{+}$. If $h=0$ then
$\alpha=0=\alpha^{\mathsf{0}}(\emptyset)$ and if $h=1$ then
$\alpha=\alpha_{0}=\alpha^{\mathsf{0}}\bigl{(}(1)\bigr{)}$. Thus, we now
assume that $h\geq 2$. We have $\mathsf{w}^{\mathsf{0}}(\alpha)=h-h^{2}<0$,
and

$\mathsf{w}^{\mathsf{0}}\bigl{(}\alpha+(h^{2}-h)\mathbf{1}\bigr{)}=0.$

Thus, by Lemma 2.36 there is an $e$-core $\lambda$ such that

$\alpha^{\mathsf{0}}(\lambda)=h^{2}\alpha_{0}+(h^{2}-h)\sum_{i=1}^{e-1}\alpha_{i},$

in other words $c^{\mathsf{0}}_{0}(\lambda)=h^{2}$ and
$c^{\mathsf{0}}_{i}(\lambda)=h^{2}-h$ for all $i\in\\{1,\dots,e-1\\}$. Note
that, by (2.22), this $e$-core satisfies
$x^{\mathsf{0}}_{0}(\lambda)=-x^{\mathsf{0}}_{e-1}(\lambda)=h$, the other
integers $x^{\mathsf{0}}_{i}(\lambda)$ being $0$.

######  Example 2.40.

Assume $e\geq 2$ and take $\mathsf{s}=\mathsf{0}$. Let $h\in\mathbb{N}$ and
define $\alpha\coloneqq h\alpha_{i_{0}}\in Q_{+}$ for
$i_{0}\in\\{1,\dots,e-1\\}$. If $h=0$ then
$\alpha=0=\alpha^{\mathsf{0}}(\emptyset)$, thus we now assume that $h\geq 1$.
We have $\mathsf{w}^{\mathsf{0}}(\alpha)=-h^{2}<0$ and

$\mathsf{w}^{\mathsf{0}}\bigl{(}\alpha+h^{2}\mathbf{1}\bigr{)}=0,$

thus there is an $e$-core $\lambda$ such that

$\alpha^{\mathsf{0}}(\lambda)=\bigl{(}h^{2}+h\bigr{)}\alpha_{i_{0}}+h^{2}\sum_{\begin{subarray}{c}i\in\mathbb{Z}/e\mathbb{Z}\\\
i\neq i_{0}\end{subarray}}\alpha_{i},$

in other words $c^{\mathsf{0}}_{i_{0}}(\lambda)=h^{2}+h$ and
$c^{\mathsf{0}}_{i}(\lambda)=h^{2}$ for all $i\neq i_{0}$. This $e$-core
satisfies
$x^{\mathsf{0}}_{i_{0}}(\lambda)=-x^{\mathsf{0}}_{i_{0}-1}(\lambda)=h$, the
other integers $x^{\mathsf{0}}_{i}(\lambda)$ being $0$.

###  2.4 Multipartitions

Let $r\geq 1$ and $e\in\mathbb{N}$. Let $\boldsymbol{\lambda}$ be an _$r$
-partition_ (or _multipartition_) of $n$, that is, an $r$-tuple
$\boldsymbol{\lambda}=\bigl{(}\lambda^{(1)},\dots,\lambda^{(r)}\bigr{)}$ of
partitions such that
$\bigl{\lvert}\lambda^{(1)}\bigr{\rvert}+\dots+\bigl{\lvert}\lambda^{(r)}\bigr{\rvert}=n$.
The Young diagram of the $r$-partition $\boldsymbol{\lambda}$ is the subset of
$\mathbb{N}^{2}\times\\{1,\dots,r\\}$ defined by

$\mathcal{Y}(\boldsymbol{\lambda})\coloneqq\bigcup_{c=1}^{r}\mathcal{Y}\bigl{(}\lambda^{(c)}\bigr{)}\times\\{c\\}.$

An element of $\mathcal{Y}(\boldsymbol{\lambda})$ is a node of
$\boldsymbol{\lambda}$, more generally any element of
$\mathbb{N}^{2}\times\\{1,\dots,r\\}$ is a node. An $e$-rim hook of
$\boldsymbol{\lambda}$ is an $e$-rim hook of $\lambda^{(j)}$ for some
$j\in\\{1,\dots,r\\}$. We say that $\boldsymbol{\lambda}$ is an _$e$
-multicore_ if $\lambda^{(j)}$ is an $e$-core for all $j\in\\{1,\dots,r\\}$,
and the _$e$ -multicore_ of $\boldsymbol{\lambda}$ is the $r$-partition
$\overline{\boldsymbol{\lambda}}$ given by
$\overline{\boldsymbol{\lambda}}\coloneqq\bigl{(}\overline{\lambda^{(1)}},\dots,\overline{\lambda^{(r)}}\bigr{)}$.

Now let
${\boldsymbol{\mathsf{s}}}=(\mathsf{s}_{1},\dots,\mathsf{s}_{r})\in\mathbb{Z}^{r}$
be a _multicharge_. The ${\boldsymbol{\mathsf{s}}}$-residue
$\mathrm{res}^{{\boldsymbol{\mathsf{s}}}}(\gamma)$ of a node $\gamma=(a,b,c)$
is the $\mathsf{s}_{c}$-residue of the node $(a,b)$, in other words

$\mathrm{res}^{{\boldsymbol{\mathsf{s}}}}(a,b,c)=\mathrm{res}^{\mathsf{s}_{c}}(a,b)=b-a+\mathsf{s}_{c}\pmod{e}.$

Again, given $i\in\mathbb{Z}/e\mathbb{Z}$ an
$(i,{\boldsymbol{\mathsf{s}}})$-node (or simply $i$-node) is a node of
${\boldsymbol{\mathsf{s}}}$-residue $i$. We denote by
$c^{{\boldsymbol{\mathsf{s}}}}_{i}(\boldsymbol{\lambda})$ the number of
$(i,{\boldsymbol{\mathsf{s}}})$-nodes of $\boldsymbol{\lambda}$. We define

$\alpha^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\coloneqq\sum_{\gamma\in\mathcal{Y}(\boldsymbol{\lambda})}\alpha_{\mathrm{res}^{{\boldsymbol{\mathsf{s}}}}(\gamma)}=\sum_{i\in\mathbb{Z}/e\mathbb{Z}}c^{{\boldsymbol{\mathsf{s}}}}_{i}(\boldsymbol{\lambda})\alpha_{i}\in
Q_{+}.$

Following [Fa06], the _${\boldsymbol{\mathsf{s}}}$ -weight_ (or simply
_weight_) of an $r$-partition $\boldsymbol{\lambda}$ is

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})=\sum_{j=1}^{r}c^{{\boldsymbol{\mathsf{s}}}}_{\mathsf{s}_{j}}(\boldsymbol{\lambda})-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(c^{{\boldsymbol{\mathsf{s}}}}_{i}(\boldsymbol{\lambda})-c^{{\boldsymbol{\mathsf{s}}}}_{i+1}(\boldsymbol{\lambda})\right)^{2}.$

Note that we recover the corresponding definition given at §2.1 when $r=1$. We
extend the definition of the ${\boldsymbol{\mathsf{s}}}$-weight to $Q$ by
setting

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\coloneqq\sum_{j=1}^{r}c_{\mathsf{s}_{j}}-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\bigl{(}c_{i}-c_{i+1}\bigr{)}^{2}\in\mathbb{Z},$

for any $\alpha=\sum_{i\in\mathbb{Z}/e\mathbb{Z}}c_{i}\alpha_{i}\in Q$. Note
that $\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\in\mathbb{Z}$ for the
same reason as in level one, and

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})=\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}\bigl{(}\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})\bigr{)}.$

We now recall some results from [Fa06]. The first one follows from Lemma 2.4.

######  Lemma 2.41 ([Fa06, Corollary 3.4]).

Assume that $e>0$ and let $\boldsymbol{\lambda}$ be an $r$-partition obtained
from $\boldsymbol{\mu}$ by adding an $e$-rim hook. We have
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})=\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\mu})+r$.

More generally, if $e>0$ then for any $\alpha\in Q$ and any $h\in\mathbb{Z}$
we have

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha+h\mathbf{1})=\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)+rh.$
(2.42)

If $\boldsymbol{\lambda}$ is an $e$-multicore, for any
$i\in\mathbb{Z}/e\mathbb{Z}$ and $j\in\\{1,\dots,r\\}$ we define

$\displaystyle\mathfrak{b}^{\boldsymbol{\mathsf{s}}}_{ij}(\boldsymbol{\lambda})$
$\displaystyle\coloneqq\mathfrak{b}^{\mathsf{s}_{j}}_{i}\bigl{(}\lambda^{(j)}\bigr{)},$
$\displaystyle y^{\boldsymbol{\mathsf{s}}}_{ij}(\boldsymbol{\lambda})$
$\displaystyle\coloneqq y^{\mathsf{s}_{j}}_{i}\bigl{(}\lambda^{(j)}\bigr{)},$
$\displaystyle x^{\boldsymbol{\mathsf{s}}}_{ij}(\boldsymbol{\lambda})$
$\displaystyle\coloneqq x^{\mathsf{s}_{j}}_{i}\bigl{(}\lambda^{(j)}\bigr{)}.$

Moreover, if $e>0$ then for any $j,k\in\\{1,\dots,r\\}$ we define

$u_{j,k}(\boldsymbol{\lambda})\coloneqq-x^{\boldsymbol{\mathsf{s}}}_{\mathsf{s}_{j},j}(\boldsymbol{\lambda})-\dots-x^{\boldsymbol{\mathsf{s}}}_{\mathsf{s}_{j}+i-1,j}(\boldsymbol{\lambda}),$

if $\mathsf{s}_{j}\neq\mathsf{s}_{k}$, where $i\geq 1$ is any integer such
that $i=\mathsf{s}_{k}-\mathsf{s}_{j}\pmod{e}$ (recall (2.23)). We define
$u_{j,k}(\boldsymbol{\lambda})\coloneqq 0$ if $\mathsf{s}_{j}=\mathsf{s}_{k}$.
The point of this definition is that

$c^{\mathsf{s}_{j}}_{\mathsf{s}_{k}}(\lambda^{(j)})=\frac{1}{2}\bigl{\lVert}x^{\mathsf{s}_{j}}\bigl{(}\lambda^{(j)}\bigr{)}\bigr{\rVert}^{2}+u_{j,k}(\boldsymbol{\lambda}),$
(2.43)

for all $j,k\in\\{1,\dots,r\\}$, by Corollary 2.29

######  Lemma 2.44 ([Fa06, Proposition 3.5]).

Write
$\boldsymbol{\lambda}=\bigl{(}\lambda^{(1)},\dots,\lambda^{(r)}\bigr{)}$. If
$\boldsymbol{\lambda}$ is a multicore then

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})=\sum_{1\leq
j<k\leq
r}\mathsf{w}^{(\mathsf{s}_{j},\mathsf{s}_{k})}\bigl{(}\lambda^{(j)},\lambda^{(k)}\bigr{)}.$

Note that Lemma 2.44 can be proved by a direct calculation using the elements
$x^{\boldsymbol{\mathsf{s}}}_{ij}(\boldsymbol{\lambda})$ together with (2.22)
and (2.43). Using the same idea, we will give a proof of the next proposition.

######  Proposition 2.45 ([Fa06, Corollary 3.9]).

For any $r$-partition $\boldsymbol{\lambda}$ we have
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\geq 0$.

###### Proof.

By Lemma 2.41 it suffices to consider the case where $\boldsymbol{\lambda}$ is
a multicore, moreover by Lemma 2.44 it suffices to consider the case $r=2$.
Besides, by Remark 2.3 we can assume $e>0$. By (2.2) and since
$\mathsf{w}^{(\mathsf{s}_{1},\mathsf{s}_{2})}(\lambda^{(1)},\lambda^{(2)})=\mathsf{w}^{(\mathsf{s}_{2},\mathsf{s}_{1})}(\lambda^{(2)},\lambda^{(1)})$,
we can further assume that ${\boldsymbol{\mathsf{s}}}=(0,\mathsf{s})$ with
$\mathsf{s}\in\\{0,\dots,e-1\\}$. We have, using (2.22) and (2.43),

$\displaystyle\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})$
$\displaystyle=c^{{\boldsymbol{\mathsf{s}}}}_{0}(\boldsymbol{\lambda})+c^{{\boldsymbol{\mathsf{s}}}}_{\mathsf{s}}(\boldsymbol{\lambda})-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(c^{{\boldsymbol{\mathsf{s}}}}_{i}(\boldsymbol{\lambda})-c^{{\boldsymbol{\mathsf{s}}}}_{i+1}(\boldsymbol{\lambda})\right)^{2}$
$\displaystyle=\sum_{j=1}^{2}\left(c^{\mathsf{s}_{j}}_{0}(\lambda^{(j)})+c^{\mathsf{s}_{j}}_{\mathsf{s}}(\lambda^{(j)})\right)-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(\sum_{j=1}^{2}\left(c^{\mathsf{s}_{j}}_{i}(\lambda^{(j)})-c^{\mathsf{s}_{j}}_{i+1}(\lambda^{(j)})\right)\right)^{2}$
$\displaystyle=\left(\bigl{\lVert}x^{\mathsf{0}}(\lambda^{(1)})\bigr{\rVert}^{2}+u_{1,2}(\boldsymbol{\lambda})\right)+\left(\bigl{\lVert}x^{\mathsf{s}}(\lambda^{(2)})\bigr{\rVert}^{2}+u_{2,1}(\boldsymbol{\lambda})\right)-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(x^{\boldsymbol{\mathsf{s}}}_{i1}(\boldsymbol{\lambda})+x^{\boldsymbol{\mathsf{s}}}_{i2}(\boldsymbol{\lambda})\right)^{2}$
$\displaystyle=\bigl{\lVert}x^{\mathsf{0}}(\lambda^{(1)})\bigr{\rVert}^{2}+\bigl{\lVert}x^{\mathsf{s}}(\lambda^{(2)})\bigr{\rVert}^{2}+u_{1,2}(\boldsymbol{\lambda})+u_{2,1}(\boldsymbol{\lambda})-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(x^{\boldsymbol{\mathsf{s}}}_{i1}(\boldsymbol{\lambda})^{2}+x^{\boldsymbol{\mathsf{s}}}_{i2}(\boldsymbol{\lambda})^{2}+2x^{\boldsymbol{\mathsf{s}}}_{i1}(\boldsymbol{\lambda})x^{\boldsymbol{\mathsf{s}}}_{i2}(\boldsymbol{\lambda})\right)$
$\displaystyle=\frac{1}{2}\left(\lVert
x^{\mathsf{0}}(\lambda^{(1)})\rVert^{2}+\lVert
x^{\mathsf{s}}(\lambda^{(2)})\rVert^{2}\right)+u_{1,2}(\boldsymbol{\lambda})+u_{2,1}(\boldsymbol{\lambda})-\sum_{i\in\mathbb{Z}/e\mathbb{Z}}x^{\boldsymbol{\mathsf{s}}}_{i1}(\boldsymbol{\lambda})x^{\boldsymbol{\mathsf{s}}}_{i2}(\boldsymbol{\lambda})$
$\displaystyle=\frac{1}{2}\left(\lVert
x^{\mathsf{0}}(\lambda^{(1)})\rVert^{2}+\lVert
x^{\mathsf{s}}(\lambda^{(2)})\rVert^{2}\right)+u_{1,2}(\boldsymbol{\lambda})+u_{2,1}(\boldsymbol{\lambda})-\sum_{i\in\mathbb{Z}/e\mathbb{Z}}x^{\mathsf{0}}_{i}(\lambda^{(1)})x^{\mathsf{s}}_{i}(\lambda^{(2)})$
$\displaystyle=\frac{1}{2}\lVert
x^{\mathsf{0}}(\lambda^{(1)})-x^{\mathsf{s}}(\lambda^{(2)})\rVert^{2}+u_{1,2}(\boldsymbol{\lambda})+u_{2,1}(\boldsymbol{\lambda}).$

If $\mathsf{s}=0$ we have
$u_{1,2}(\boldsymbol{\lambda})=u_{2,1}(\boldsymbol{\lambda})=0$ so that
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\geq 0$ as
desired, otherwise we have $\mathsf{s}\in\\{1,\dots,e-1\\}$ so that

$\displaystyle u_{1,2}(\boldsymbol{\lambda})$
$\displaystyle=-x^{\boldsymbol{\mathsf{s}}}_{0,1}(\boldsymbol{\lambda})-\dots-x^{\boldsymbol{\mathsf{s}}}_{\mathsf{s}-1,1}(\boldsymbol{\lambda}),$
$\displaystyle u_{2,1}(\boldsymbol{\lambda})$
$\displaystyle=-x^{\boldsymbol{\mathsf{s}}}_{\mathsf{s},2}(\boldsymbol{\lambda})-\dots-x^{\boldsymbol{\mathsf{s}}}_{e-1,2}(\boldsymbol{\lambda}).$

Now by (2.23) we have

$u_{2,1}(\boldsymbol{\lambda})=x^{\boldsymbol{\mathsf{s}}}_{0,2}(\boldsymbol{\lambda})+\dots+x^{\boldsymbol{\mathsf{s}}}_{\mathsf{s}-1,2}(\boldsymbol{\lambda}).$

Hence, setting $z\coloneqq
x^{\mathsf{0}}(\lambda^{(1)})-x^{\mathsf{s}}(\lambda^{(2)})$ we obtain

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})=\frac{1}{2}\lVert
z\rVert^{2}-z_{0}-\dots-z_{\mathsf{s}-1},$

thus we conclude by the below Lemma 2.46, recalling (2.23). ∎

######  Lemma 2.46.

Assume that $e>0$. Let $z=(z_{1},\dots,z_{e})\in\mathbb{Z}^{e}$ such that
$z_{1}+\dots+z_{e}=0$. Then $\lVert z\rVert^{2}\geq
2(z_{1}+\dots+z_{\mathsf{s}})$ for any $\mathsf{s}\in\\{1,\dots,e\\}$.

###### Proof.

First, note that for any integer $t\in\mathbb{Z}$ we have $t^{2}\geq 2t$
unless if $t=1$. We define the following two sets:

$\displaystyle I$
$\displaystyle\coloneqq\left\\{i\in\\{1,\dots,\mathsf{s}\\}:z_{i}=1\right\\},$
$\displaystyle J$ $\displaystyle\coloneqq\left\\{i\in\\{1,\dots,e\\}:z_{i}\leq
0\right\\},$

in particular $I\cap J=\emptyset$. We have

$0=z_{1}+\dots+z_{e}\geq\sum_{i\in J}z_{i}+\sum_{i\in I}z_{i},$

so that $\lvert I\rvert+\sum_{i\in J}z_{i}\leq 0$. For any $i\in J$ we have
$z_{i}\in\mathbb{Z}_{\leq 0}$ thus

$z_{i}^{2}-2z_{i}\geq z_{i}^{2}\geq-z_{i},$

hence we obtain

$\displaystyle\lVert z\rVert^{2}-2(z_{1}+\dots+z_{\mathsf{s}})$
$\displaystyle\geq\sum_{i\in I}(z_{i}^{2}-2z_{i})+\sum_{i\in J}(-z_{i})$
$\displaystyle=\sum_{i\in I}(-1)-\sum_{i\in J}z_{i}$ $\displaystyle=-\lvert
I\rvert-\sum_{i\in J}z_{i}$ $\displaystyle\geq 0,$

as desired. ∎

Note that $\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\geq
r\sum_{j=1}^{r}\mathsf{w}^{\mathsf{s}_{j}}(\lambda^{(j)})$. Indeed, by
Propositions 2.28 and 2.45 the inequality holds when $\boldsymbol{\lambda}$ is
an $e$-multicore, and we conclude by Lemma 2.41.

Finally, if $\boldsymbol{\lambda}$ is an $e$-multicore, for any
$i\in\mathbb{Z}/e\mathbb{Z}$ and $j,k\in\\{1,\dots,r\\}$, as in [Fa06, Fa07a]
we define

$\gamma_{i,jk}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\coloneqq\frac{\mathfrak{b}^{{\boldsymbol{\mathsf{s}}}}_{ij}(\boldsymbol{\lambda})-\mathfrak{b}^{{\boldsymbol{\mathsf{s}}}}_{ik}(\boldsymbol{\lambda})}{e}=y^{\boldsymbol{\mathsf{s}}}_{ij}(\boldsymbol{\lambda})-y^{\boldsymbol{\mathsf{s}}}_{ik}(\boldsymbol{\lambda})\in\mathbb{Z}.$

These integers depend on the multicharge ${\boldsymbol{\mathsf{s}}}$, however
for any $i,l\in\mathbb{Z}/e\mathbb{Z}$ and $j,k\in\\{1,\dots,r\\}$ the set
defined by the integers

$\gamma_{il,jk}(\boldsymbol{\lambda})\coloneqq\gamma_{i,jk}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})-\gamma_{l,jk}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda}),$
(2.47)

does not.

######  Proposition 2.48 ([Fa06, Proposition 3.8]).

Assume that $r=2$ and let $\boldsymbol{\lambda}$ be a bicore. Assume that
$\gamma_{il,12}(\boldsymbol{\lambda})\leq 2$ for all
$i,l\in\mathbb{Z}/e\mathbb{Z}$. Then
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})$ is the smaller
of the two integers

$\\#\bigl{\\{}i\in\mathbb{Z}/e\mathbb{Z}:\gamma_{il,12}(\boldsymbol{\lambda})=2\text{
for some }l\in\mathbb{Z}/e\mathbb{Z}\bigr{\\}},$

and

$\\#\bigl{\\{}l\in\mathbb{Z}/e\mathbb{Z}:\gamma_{il,12}(\boldsymbol{\lambda})=2\text{
for some }i\in\mathbb{Z}/e\mathbb{Z}\bigr{\\}}.$

Proposition 2.48 immediately implies that if $\boldsymbol{\lambda}$ is a
bicore satisfying $\gamma_{il,12}(\boldsymbol{\lambda})\leq 2$ for all
$i,l\in\mathbb{Z}/e\mathbb{Z}$ then
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\leq e$. We will
see that we can straighten this inequality for a certain class of
multipartitions. One on the aim of this paper is to study the consequences of
such an inequality, namely, on the blocks of multipartitions.

######  Remark 2.49.

By [Fa06, Lemma 3.7], the ${\boldsymbol{\mathsf{s}}}$-weight function on
$e$-multicores is not bounded. For instance, if $e>0$ assume that $r=2$ and
${\boldsymbol{\mathsf{s}}}=(0,0)$, and let $\lambda$ and $\mu$ be two bicores.
By (2.22) and Corollary 2.29 we have

$\displaystyle\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\lambda,\mu)$
$\displaystyle=2c^{0}_{0}(\lambda,\mu)-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(c^{0}_{i}(\lambda,\mu)-c^{0}_{i+1}(\lambda,\mu)\right)^{2}$
$\displaystyle=2c^{0}_{0}(\lambda)+2c^{0}_{0}(\mu)-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(c^{0}_{i}(\lambda)-c^{0}_{i+1}(\lambda)+c^{0}_{i}(\mu)-c^{0}_{i+1}(\mu)\right)^{2}$
$\displaystyle=\bigl{\lVert}x^{0}(\lambda)\bigr{\rVert}^{2}+\bigl{\lVert}x^{0}(\mu)\bigr{\rVert}^{2}-\frac{1}{2}\sum_{i\in\mathbb{Z}/e\mathbb{Z}}\left(x^{0}_{i}(\lambda)+x^{0}_{i}(\mu)\right)^{2}$
$\displaystyle=\frac{1}{2}\bigl{\lVert}x^{0}(\lambda)-x^{0}(\mu)\bigr{\rVert}^{2}.$

Thus, if we chose the $e$-core $\mu$ so that $x^{0}(\mu)=-x^{0}(\lambda)$
(which satisfies the conditions of Proposition 2.24 indeed) we obtain

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\lambda,\mu)=2\bigl{\lVert}x^{0}(\lambda)\bigr{\rVert}^{2}=c^{0}_{0}(\lambda),$

which is not bounded since we can find $e$-cores with arbitrary large number
of $0$-nodes (see, for instance, Example 2.39). If $e=0$ one can simply
consider the partition $\lambda^{[h]}\coloneqq(h,\dots,h)$ where $h\geq 1$ is
repeated $h$ times, and see that
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\lambda^{[h]},\emptyset)=\mathsf{w}^{0}(\lambda^{[h]})+c^{0}_{0}(\lambda^{[h]})=0+h=h$.

###  2.5 Blocks for multipartitions

The combinatorics between an $r$-partition $\boldsymbol{\lambda}$ and
$\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})$ is more intricate
than in the case $r=1$. For instance, the natural notion of $e$-multicore no
more suffices to distinguish the blocks between two $r$-partitions. However,
as in the level $1$ case the quantities
$\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})$ determine the blocks
of the associated _Ariki–Koike algebra_ (see [LyMa]).

######  Example 2.50.

Let $r=2$, $e=2$ and ${\boldsymbol{\mathsf{s}}}=(0,1)$. The bipartitions
$\boldsymbol{\lambda}=\bigl{(}(1),(1)\bigr{)}$ and
$\boldsymbol{\mu}=\bigl{(}(2),\emptyset\bigr{)}$ lie in the same block
$\alpha_{0}+\alpha_{1}$, however $\boldsymbol{\lambda}$ is a bicore whereas
$\boldsymbol{\mu}$ is not.

Again, we define

$Q^{\boldsymbol{\mathsf{s}}}\coloneqq\left\\{\alpha^{\mathsf{s}}(\boldsymbol{\lambda}):\boldsymbol{\lambda}\text{
is an }r\text{-partition of }n\text{ for some }n\in\mathbb{N}\right\\}.$

######  Remark 2.51.

As we mentioned in §2.1, the case $e=1$ is particularly easy to deal with, and
indeed we have here $Q^{\boldsymbol{\mathsf{s}}}=\mathbb{N}\alpha_{0}=Q_{+}$.

Since
$\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})=\sum_{j=1}^{r}\alpha^{\mathsf{s}_{j}}(\lambda^{(j)})$
we have $Q^{\boldsymbol{\mathsf{s}}}=\sum_{j=1}^{r}Q^{\mathsf{s}_{j}}$,
however the sum is not direct. In particular, using the
$1{:}1$-parametrisation of §2.3 for each $Q^{\mathsf{s}_{j}}$, we can give a
naive parametrisation of $Q^{\boldsymbol{\mathsf{s}}}$ but this
parametrisation will not be $1{:}1$. We will rather aim at a generalisation of
Lemma 2.36 and Proposition 2.37, looking for an implicit description of
$Q^{\boldsymbol{\mathsf{s}}}$. It turns out that the following simple result,
which comes from Lemmas 2.5 and 2.17, is the key of what follows.

######  Proposition 2.52.

Let $\alpha\in Q^{\boldsymbol{\mathsf{s}}}$ and $h\in\mathbb{N}$. Then
$\alpha+h\mathbf{1}\in Q^{\boldsymbol{\mathsf{s}}}$.

We have the following generalisation of Corollary 2.38 (see also [Fa07b,
Theorem 4.7]).

######  Proposition 2.53.

Assume $e>0$. For any $\alpha\in Q$ and any
$h\geq-\max\bigl{\\{}\mathsf{w}^{\mathsf{s}_{j}}(\alpha):j\in\\{1,\dots,r\\}\bigr{\\}}$
we have

$\alpha+h\mathbf{1}\in Q^{\boldsymbol{\mathsf{s}}}.$

###### Proof.

Let $j\in\\{1,\dots,r\\}$. If $h\geq-\mathsf{w}^{\mathsf{s}_{j}}(\alpha)$ then
by Corollary 2.38 we have $\alpha+h\mathbf{1}\in Q^{\mathsf{s}_{j}}$. We
conclude the proof since $Q^{\mathsf{s}_{j}}\subseteq
Q^{\boldsymbol{\mathsf{s}}}=\sum_{k=1}^{r}Q^{\mathsf{s}_{k}}$. ∎

As we saw in Example 2.50, the raw notion of $e$-multicores is not adapted to
the study of blocks.

######  Definition 2.54 ([Fa07a]).

Let $\alpha\in Q$. We say that $\alpha$ is a _core block_ if $\alpha\in
Q^{\boldsymbol{\mathsf{s}}}$ and either:

  * •

$e=0$;

  * •

$e>0$ and $\alpha-\mathbf{1}\notin Q^{\boldsymbol{\mathsf{s}}}$.

By Lemmas 2.5 and 2.17, a block $\alpha\in Q^{\boldsymbol{\mathsf{s}}}$ is a
core block if and only if every $r$-partition $\boldsymbol{\lambda}$ such that
$\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})=\alpha$ is an
$e$-multicore.

######  Remark 2.55.

In particular, if $r=1$ then $\alpha^{\mathsf{s}}(\lambda)$ is a core block if
and only if $\lambda$ is an $e$-core, in which case $\lambda$ is the only
$e$-core (and partition) that lies in $\alpha^{\mathsf{s}}(\lambda)$ (see
Lemma 2.32).

######  Remark 2.56.

If $e=1$ then the only $e$-multicore is the empty multipartition, thus there
is only one core block, which is $0\in Q^{\boldsymbol{\mathsf{s}}}$.

######  Definition 2.57 ([LyMa]).

A multipartition $\boldsymbol{\lambda}$ is an _${\boldsymbol{\mathsf{s}}}$
-reduced $e$-multicore_ if
$\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})$ is a core block.

Note that the term _core multipartition_ has a different meaning, see [Fa19,
JaLe]. Any ${\boldsymbol{\mathsf{s}}}$-reduced $e$-multicore is an
$e$-multicore, and the converse holds when $r=1$ by Remark 2.55.

######  Lemma 2.58.

Let $\boldsymbol{\lambda}=\bigl{(}\lambda^{(1)},\dots,\lambda^{(r)}\bigr{)}$
be an ${\boldsymbol{\mathsf{s}}}$-reduced multicore. Then for each $1\leq
j<k\leq r$, the bipartition $\bigl{(}\lambda^{(j)},\lambda^{(k)}\bigr{)}$ is
an $(\mathsf{s}_{j},\mathsf{s}_{k})$-reduced bicore.

###### Proof.

We prove the contraposition. Assume that
$\bigl{(}\lambda^{(j)},\lambda^{(k)}\bigr{)}$ is not an
$(\mathsf{s}_{j},\mathsf{s}_{k})$-reduced bicore. Then we can find a
bipartition $\bigl{(}\mu^{(j)},\mu^{(k)}\bigr{)}$ in its block which is not a
bicore, in particular we can remove an $e$-rim hook to $\mu^{(j)}$ or
$\mu^{(k)}$. But then the multipartition

$\bigl{(}\lambda^{(1)},\dots,\lambda^{(j-1)},\mu^{(j)},\lambda^{(j+1)},\dots,\lambda^{(k-1)},\mu^{(k)},\lambda^{(k+1)},\dots,\lambda^{(r)}\bigr{)}$

lie in the same block as $\boldsymbol{\lambda}$ and is not an $e$-multicore,
thus $\boldsymbol{\lambda}$ is not a reduced
${\boldsymbol{\mathsf{s}}}$-multicore. ∎

######  Lemma 2.59.

Assume that $e>0$ and let $\alpha\in Q$. There is a unique integer
$h\in\mathbb{Z}$ such that $\alpha-h\mathbf{1}$ is a core block and we have
$h=\max\bigl{\\{}k\in\mathbb{Z}:\alpha-k\mathbf{1}\in
Q^{\mathsf{s}}\bigr{\\}}$. Moreover, we have $\alpha\in
Q^{\boldsymbol{\mathsf{s}}}\iff h\geq 0$.

###### Proof.

Write $H\coloneqq\bigl{\\{}k\in\mathbb{Z}:\alpha-k\mathbf{1}\in
Q^{\mathsf{s}}\bigr{\\}}$. The set $H$ is non-empty by Proposition 2.53 and
bounded above by (2.42) and Proposition 2.45. If $h\coloneqq\max H$, then by
definition we have $\alpha-h\mathbf{1}\in Q^{\boldsymbol{\mathsf{s}}}$ and
$\alpha-(h+1)\mathbf{1}\notin Q^{\boldsymbol{\mathsf{s}}}$ thus
$\alpha-h\mathbf{1}$ is a core block. We now prove the unicity. If
$h^{\prime}\in\mathbb{Z}$ is such that $\alpha-h^{\prime}\mathbf{1}$ is a core
block then in particular $\alpha-h^{\prime}\mathbf{1}\in
Q^{\boldsymbol{\mathsf{s}}}$ thus $h^{\prime}\leq h$ by maximality of $h$. Now
if $h^{\prime}<h$ then

$\alpha-(h^{\prime}+1)\mathbf{1}=(\alpha-h\mathbf{1})+(h-h^{\prime}-1)\mathbf{1},$

but the left-hand side is not in $Q^{\boldsymbol{\mathsf{s}}}$ since
$\alpha-h^{\prime}\mathbf{1}$ is a core block while the right-hand side is in
$Q^{\boldsymbol{\mathsf{s}}}$ by Proposition 2.52, thus $h^{\prime}=h$.
Finally, we have $\alpha\in Q^{\boldsymbol{\mathsf{s}}}\iff 0\in H\iff h\geq
0$, which concludes the proof. ∎

######  Remark 2.60.

If $e>0$, Lemma 2.59 ensures that
$Q/\langle\mathbf{1}\rangle\simeq\mathbb{Z}^{e-1}$ is in
$1{:}1$-correspondence with the set of core blocks. In particular, if $e\geq
2$ then given a multicharge there is an infinite number of core blocks.

######  Remark 2.61.

Let $\alpha\in Q$ and let $h\in\mathbb{Z}$ such that
$\hat{\alpha}\coloneqq\alpha-h\mathbf{1}$ is a core block. By (2.7) we have
$h=\frac{\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)-\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\hat{\alpha})}{r}$.
Note that the quantity $\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\hat{\alpha})$
can be computed from $\alpha$, see [Fa07b]. However, the formula given in
[Fa07b] does not seem suitable for the purpose of this paper.

######  Definition 2.62.

Assume that $e>0$ and let $\alpha\in Q$. We say that the unique core block of
the form $\alpha-h\mathbf{1}$ for $h\in\mathbb{Z}$ as in Lemma 2.59 is the
_${\boldsymbol{\mathsf{s}}}$ -core_ of $\alpha$.

We will simply use _core_ instead of ${\boldsymbol{\mathsf{s}}}$-core when the
multicharge ${\boldsymbol{\mathsf{s}}}$ is understood from the context. Given
a multipartition $\boldsymbol{\lambda}$, by Lemma 2.5 to compute the core of
$\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})$ we can proceed as
follows. We first look at the $e$-multicore $\overline{\boldsymbol{\lambda}}$
of $\boldsymbol{\lambda}$. If every $r$-partition lying in the same block as
$\overline{\boldsymbol{\lambda}}$ is an $e$-multicore then we are done (in
which case $\alpha^{\boldsymbol{\mathsf{s}}}(\overline{\boldsymbol{\lambda}})$
is the core of $\alpha^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})$),
otherwise we repeat the procedure with an $r$-partition in the block of
$\overline{\boldsymbol{\lambda}}$ that is not an $e$-multicore.

######  Remark 2.63.

The terminology fits with the usual notion of $e$-core when $r=1$: if
$\lambda$ and $\mu$ are two partitions then $\alpha^{\mathsf{s}}(\lambda)$ is
the $\mathsf{s}$-core of $\alpha^{\mathsf{s}}(\mu)$ if and only if $\lambda$
is the $e$-core of $\mu$ (by Remark 2.55 together with Lemmas 2.5 and 2.30).

By Lemma 2.41 and Proposition 2.45, we have the following particular case of
core block. The aim of this paper is to give a (weak) converse.

######  Proposition 2.64.

Let $\alpha\in Q^{\boldsymbol{\mathsf{s}}}$ with
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\leq r-1$. Then $\alpha$ is a
core block.

We say that a multicharge
${\boldsymbol{\mathsf{s}}}^{\prime}\in\mathbb{Z}^{e}$ is _compatible_ with
${\boldsymbol{\mathsf{s}}}$ (or simply _compatible_ if
${\boldsymbol{\mathsf{s}}}$ is clear from the context) if
${\boldsymbol{\mathsf{s}}}^{\prime}-{\boldsymbol{\mathsf{s}}}\in e\mathbb{Z}$.
In particular, note that if ${\boldsymbol{\mathsf{s}}}^{\prime}$ is a
compatible multicharge then a multicore $\boldsymbol{\lambda}$ is
${\boldsymbol{\mathsf{s}}}$-reduced if and only if it is
${\boldsymbol{\mathsf{s}}}^{\prime}$-reduced. We have the following
characterisation of core blocks in terms of abaci.

######  Proposition 2.65 ([Fa07a, Theorem 3.1]).

Assume that $e>0$ and let $\boldsymbol{\lambda}$ be an $e$-multicore. The
following assertions are equivalent.

  1. $(i)$

The $e$-multicore $\boldsymbol{\lambda}$ is
${\boldsymbol{\mathsf{s}}}$-reduced.

  2. $(ii)$

There exist a compatible multicharge ${\boldsymbol{\mathsf{s}}}^{\prime}$ and
integers $\mathfrak{b}_{1},\dots,\mathfrak{b}_{e}\in\mathbb{Z}$ such that

$\mathfrak{b}^{{\boldsymbol{\mathsf{s}}}^{\prime}}_{ij}(\boldsymbol{\lambda})\in\\{\mathfrak{b}_{i},\mathfrak{b}_{i}+e\\},$

for all $i\in\mathbb{Z}/e\mathbb{Z}$ and $j\in\\{1,\dots,r\\}$.

  3. $(iii)$

There exist a compatible multicharge ${\boldsymbol{\mathsf{s}}}^{\prime}$ and
integers $\sigma_{1},\dots,\sigma_{r}\in\mathbb{Z}$ such that

$\gamma_{i,jk}^{{\boldsymbol{\mathsf{s}}}^{\prime}}(\boldsymbol{\lambda})\leq\sigma_{j}-\sigma_{k}+1,$

for all $i\in\mathbb{Z}/e\mathbb{Z}$ and $j,k\in\\{1,\dots,r\\}$.

  4. $(iv)$

For any compatible multicharge ${\boldsymbol{\mathsf{s}}}^{\prime}$, there
exist integers $\sigma_{1},\dots,\sigma_{r}\in\mathbb{Z}$ such that

$\gamma_{i,jk}^{{\boldsymbol{\mathsf{s}}}^{\prime}}(\boldsymbol{\lambda})\leq\sigma_{j}-\sigma_{k}+1,$

for all $i\in\mathbb{Z}/e\mathbb{Z}$ and $j,k\in\\{1,\dots,r\\}$.

######  Corollary 2.66.

Assume that $e>0$. An $e$-multicore $\boldsymbol{\lambda}$ is
${\boldsymbol{\mathsf{s}}}$-reduced if and only if there exists a compatible
multicharge ${\boldsymbol{\mathsf{s}}}^{\prime}$ such that

$\gamma^{{\boldsymbol{\mathsf{s}}}^{\prime}}_{i,jk}(\boldsymbol{\lambda})\leq
1,$

for all $i\in\mathbb{Z}/e\mathbb{Z}$ and $j,k\in\\{1,\dots,r\\}$.

###### Proof.

If the $e$-multicore $\boldsymbol{\lambda}$ is
${\boldsymbol{\mathsf{s}}}$-reduced then by Proposition 2.65ii we can write
$\mathfrak{b}^{{\boldsymbol{\mathsf{s}}}^{\prime}}_{ij}(\boldsymbol{\lambda})=\mathfrak{b}_{i}+\epsilon_{ij}e$
for all $i\in\mathbb{Z}/e\mathbb{Z}$ and $j\in\\{1,\dots,r\\}$, where
$\epsilon_{ij}\in\\{0,1\\}$ and ${\boldsymbol{\mathsf{s}}}^{\prime}$ is a
compatible multicharge. Thus, we obtain

$\gamma_{i,jk}^{{\boldsymbol{\mathsf{s}}}^{\prime}}(\boldsymbol{\lambda})=\frac{\mathfrak{b}^{{\boldsymbol{\mathsf{s}}}^{\prime}}_{ij}(\boldsymbol{\lambda})-\mathfrak{b}^{{\boldsymbol{\mathsf{s}}}^{\prime}}_{ik}(\boldsymbol{\lambda})}{e}=\epsilon_{ij}-\epsilon_{ik}\leq
1,$

for all $i\in\mathbb{Z}/e\mathbb{Z}$ and $j,k\in\\{1,\dots,r\\}$. Conversely,
if there exists a compatible multicharge ${\boldsymbol{\mathsf{s}}}^{\prime}$
such that
$\gamma^{{\boldsymbol{\mathsf{s}}}^{\prime}}_{i,jk}(\boldsymbol{\lambda})\leq
1$ for all $i\in\mathbb{Z}/e\mathbb{Z}$ and $j,k\in\\{1,\dots,r\\}$ then item
iii of Proposition 2.65 is satisfied with $\sigma_{1}=\dots=\sigma_{r}$ thus
the $e$-multicore $\boldsymbol{\lambda}$ is
${\boldsymbol{\mathsf{s}}}$-reduced. ∎

##  3 Implicit description of the blocks

We will give in this section the central results of the paper. We will first
give a formula to compute the weight as a map from binary matrices. Let $r\geq
2$ and $e\geq 1$.

###  3.1 Weight of binary matrices

Recall from Proposition 2.65ii that to any ${\boldsymbol{\mathsf{s}}}$-reduced
$e$-multicore we can associate an $e\times r$ binary matrix. Conversely, given
an $e\times r$ binary matrix $\mathcal{E}=(\epsilon_{ij})$ and integers
$\mathfrak{b}_{1},\dots,\mathfrak{b}_{e}\in\mathbb{Z}$ such that
$\mathfrak{b}_{i}=i\pmod{e}$ for each $i\in\\{1,\dots,e\\}$, we can construct
an ${\boldsymbol{\mathsf{s}}}$-reduced $e$-multicore
$\boldsymbol{\lambda}=\boldsymbol{\lambda}(\mathcal{E},(\mathfrak{b}_{i}))$,
where the multicharge ${\boldsymbol{\mathsf{s}}}$ is determined by (2.11).
Note that the integers

$\gamma_{i,jk}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})=\epsilon_{ij}-\epsilon_{ik},$
(3.1)

do not depend on the choice of $(\mathfrak{b}_{i})$ so that the weight
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\eqqcolon\mathsf{w}(\mathcal{E})$
only depends on $\mathcal{E}$ (cf. Proposition 2.48).

##### Notation

Given an $e\times r$ binary matrix $\mathcal{E}$, we denote by
$E_{1},\dots,E_{r}$ the subsets of $\\{1,\dots,e\\}$ such that for any
$j\in\\{1,\dots,r\\}$ the $j$-th column of $\mathcal{E}$ is exactly the
characteristic vector $\mathbf{1}_{j}$ of $E_{j}$. Conversely, if
$E_{1},\dots,E_{r}$ are subsets of $\\{1,\dots,e\\}$ then we denote by
$\mathcal{E}$ the $e\times r$ binary matrix such that $\mathbf{1}_{j}$ is the
$j$-th column of $\mathcal{E}$ for any $j\in\\{1,\dots,r\\}$.

######  Lemma 3.2.

Assume that $r=2$ and let $E_{1},E_{2}\subseteq\\{1,\dots,e\\}$. We have:

$\mathsf{w}(\mathcal{E})=\min(\lvert E_{1}\rvert,\lvert E_{2}\rvert)-\lvert
E_{1}\cap E_{2}\rvert.$

###### Proof.

Let $\mathfrak{b}_{1},\dots,\mathfrak{b}_{e}\in\mathbb{Z}$ with
$\mathfrak{b}_{i}=i\pmod{e}$, let ${\boldsymbol{\mathsf{s}}}$ be the
multicharge associated with $(\mathfrak{b}_{i})$ and let
$\boldsymbol{\lambda}$ be the ${\boldsymbol{\mathsf{s}}}$-reduced
$e$-multicore associated with $(\mathfrak{b}_{i})$ and the $e\times 2$ binary
matrix $\mathcal{E}$. By (3.1), for any $i\in\mathbb{Z}/e\mathbb{Z}$ we have

$\gamma_{i,12}^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})=\mathbf{1}_{1}(i)-\mathbf{1}_{2}(i),$

so that $\gamma_{i,12}^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})=1$ if
and only if $i\in E_{1}\setminus E_{2}$. Hence, recalling (2.47),

$\displaystyle\gamma_{il,12}(\boldsymbol{\lambda})=2$
$\displaystyle\iff\gamma_{i,12}^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})=1\text{
and }\gamma_{l,12}^{\boldsymbol{\mathsf{s}}}(\boldsymbol{\lambda})=-1$
$\displaystyle\iff i\in E_{1}\setminus E_{2}\text{ and }l\in E_{2}\setminus
E_{1},$

so that

$A\coloneqq\\#\left\\{i\in\mathbb{Z}/e\mathbb{Z}:\gamma_{il,12}(\boldsymbol{\lambda})=2\text{
for some }l\in\mathbb{Z}/e\mathbb{Z}\right\\}=\begin{cases}\lvert
E_{1}\setminus E_{2}\rvert,&\text{if }\lvert E_{2}\setminus E_{1}\rvert\neq
0,\\\ 0,&\text{otherwise},\end{cases}$

and similarly

$B\coloneqq\\#\left\\{l\in\mathbb{Z}/e\mathbb{Z}:\gamma_{il,12}(\boldsymbol{\lambda})=2\text{
for some }i\in\mathbb{Z}/e\mathbb{Z}\right\\}=\begin{cases}\lvert
E_{2}\setminus E_{1}\rvert,&\text{if }\lvert E_{1}\setminus E_{2}\rvert\neq
0,\\\ 0,&\text{otherwise},\end{cases}$

Recall from Proposition 2.48 that $\mathsf{w}(\mathcal{E})=\min(A,B)$. First
assume that $\lvert E_{1}\setminus E_{2}\rvert\neq 0$ and $\lvert
E_{2}\setminus E_{1}\rvert\neq 0$. We obtain that $A=\lvert E_{1}\setminus
E_{2}\rvert$ and $B=\lvert E_{2}\setminus E_{1}\rvert$, thus

$\displaystyle\mathsf{w}(\mathcal{E})$ $\displaystyle=\min\Bigl{(}\lvert
E_{1}\rvert-\lvert E_{1}\cap E_{2}\rvert,\>\lvert E_{2}\rvert-\lvert E_{1}\cap
E_{2}\rvert\Bigr{)}$ $\displaystyle=\min(\lvert E_{1}\rvert,\lvert
E_{2}\rvert)-\lvert E_{1}\cap E_{2}\rvert,$

as announced. Now assume that $\lvert E_{1}\setminus E_{2}\rvert=0$ or $\lvert
E_{2}\setminus E_{1}\rvert=0$. Without loss of generality, we can assume that
we are in the first case. On the one hand, we have $A=B=0$ thus
$\mathsf{w}(\mathcal{E})=0$. On the other hand, the assumption implies that
$E_{1}\subseteq E_{2}$ thus $\min(\lvert E_{1}\rvert,\lvert
E_{2}\rvert)=\lvert E_{1}\rvert$ and $\lvert E_{1}\cap E_{2}\rvert=\lvert
E_{1}\rvert$. Hence, we have $\min(\lvert E_{1}\rvert,\lvert
E_{2}\rvert)-\lvert E_{1}\cap E_{2}\rvert=0=\mathsf{w}(\mathcal{E})$ and this
concludes the proof. ∎

Note that with Lemma 2.44 we have

$\mathsf{w}(\mathcal{E})=\sum_{1\leq j<k\leq r}\Bigl{[}\min(\lvert
E_{j}\rvert,\lvert E_{k}\rvert)-\lvert E_{j}\cap E_{k}\rvert\Bigr{]}.$ (3.3)

We conclude this subsection by the following particular case of Lemma 3.2.

######  Lemma 3.4.

For any $E\subseteq\\{1,\dots,e\\}$ we have:

$\mathsf{w}(E,E)=\mathsf{w}(E,\emptyset)=\mathsf{w}(E,\\{1,\dots,e\\})=0.$

######  Remark 3.5.

Recalling Remark 2.56, if $e=1$ then we recover the fact that
$\mathsf{w}(\mathcal{E})=0$ for any $e\times 1$ binary matrix $\mathcal{E}$.

###  3.2 Bound for the weight of core blocks

Recall from Remark 2.60 that there is an infinite number of reduced
$e$-multicores.

######  Theorem 3.6.

Let $r\geq 2$ and $e>0$. There exists a constant $C\geq 0$ such that for all
multicharge ${\boldsymbol{\mathsf{s}}}$ and all $r$-partition
$\boldsymbol{\lambda}$ that is an ${\boldsymbol{\mathsf{s}}}$-reduced
$e$-multicore we have

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\leq C.$

Note that the statement is wrong if $e=0$ (see Remark 2.49 or the coming
Remark 3.8).

###### Proof.

We first prove the case $r=2$, which is clear since by Proposition 2.48 and
Corollary 2.66 we have
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\leq e$ (we could
also have use Lemma 3.2). We now assume that $r\geq 3$. Since
$\boldsymbol{\lambda}$ is a multicore, by Lemma 2.44 we have

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})=\sum_{1\leq
j<k\leq
r}\mathsf{w}^{(\mathsf{s}_{j},\mathsf{s}_{k})}\bigl{(}\lambda^{(j)},\lambda^{(k)}\bigr{)}.$

By Lemma 2.58 we know that each $\bigl{(}\lambda^{(j)},\lambda^{(k)}\bigr{)}$
is a reduced $(\mathsf{s}_{j},\mathsf{s}_{k})$-bicore, thus using the case
$r=2$ we find

$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda})\leq
e\binom{r}{2},$

which concludes the proof. ∎

We denote by $N_{r,e}$ the smallest possible constant $C$ as in Theorem 3.6,
in particular

$N_{r,e}=\mathrm{sup}\bigl{\\{}\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\boldsymbol{\lambda}):{\boldsymbol{\mathsf{s}}}\in\mathbb{Z}^{r}\text{
a multicharge and }\boldsymbol{\lambda}\text{ an
}{\boldsymbol{\mathsf{s}}}\text{-reduced
}e\text{-multicore}\bigr{\\}}<\infty.$

######  Corollary 3.7.

Assume that $e>0$. We have

$Q^{\boldsymbol{\mathsf{s}}}\supseteq\bigl{\\{}\alpha\in
Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)>N_{r,e}-r\bigr{\\}}.$

###### Proof.

Let $\alpha\in Q$ and assume that $\alpha\notin Q^{\boldsymbol{\mathsf{s}}}$.
We will prove that $\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\leq
N_{r,e}-r$. By Lemma 2.59, we can find $h\in\mathbb{Z}$ such that
$\alpha-h\mathbf{1}$ is a core block, and since $\alpha\notin
Q^{\boldsymbol{\mathsf{s}}}$ we have $h<0$. Now by Theorem 3.6 we have
$\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha-h\mathbf{1})\leq N_{r,e}$ thus
by (2.42) we have $\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\leq
N_{r,e}+rh\leq N_{r,e}-r$. This concludes the proof. ∎

Note that the inclusion of Corollary 3.7 is not trivial, that is, the set
$\bigl{\\{}\alpha\in
Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)>N_{r,e}-r\bigr{\\}}$ is not
empty, and in fact it is even infinite (by (2.42)).

######  Remark 3.8.

The result of Corollary 3.7 does not hold if $e=0$. For instance, assume that
$r=2$ and take ${\boldsymbol{\mathsf{s}}}=(0,\mathsf{s})$ for $\mathsf{s}\geq
0$. For any $h\geq 1$, as in Remark 2.49 we consider the partition
$\lambda^{[h]}=(h,\dots,h)$ where $h$ is repeated $h$ times and we define

$\alpha^{[h]}\coloneqq\alpha^{0}(\lambda^{[h]})+\alpha_{h+\mathsf{s}+1}=\sum_{\lvert
i\rvert<h}(h-\lvert i\rvert)\alpha_{i}+\alpha_{h+\mathsf{s}+1}.$

Let $(c_{i})_{i\in\mathbb{Z}}\in\mathbb{Z}^{\mathbb{Z}}$ so that
$\alpha^{[h]}=\sum_{i\in\mathbb{Z}}c_{i}\alpha_{i}$. Since
$c_{h+\mathsf{s}}=0<c_{h+\mathsf{s}+1}$ and
${\boldsymbol{\mathsf{s}}}=(0,\mathsf{s})$, we have $\alpha^{[h]}\notin
Q^{\boldsymbol{\mathsf{s}}}$ (by (2.22) and (2.25)), however

$\displaystyle\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha^{[h]})$
$\displaystyle=2c_{0}-\frac{1}{2}\sum_{i\in\mathbb{Z}}(c_{i}-c_{i+1})^{2}$
$\displaystyle=2c_{0}-\frac{1}{2}\left(\sum_{i=-h}^{h-1}(c_{i}-c_{i+1})^{2}+(c_{h+\mathsf{s}}-c_{h+\mathsf{s}+1})^{2}+(c_{h+\mathsf{s}+1}-c_{h+\mathsf{s}+2})^{2}\right)$
$\displaystyle=2h-\frac{1}{2}\left(\sum_{i=-h}^{h-1}1^{2}+1^{2}+1^{2}\right)$
$\displaystyle=2h-\frac{1}{2}(2h+2)$ $\displaystyle=h-1.$

Note that this calculation remains valid for $e\geq 2(h+1)+\mathsf{s}$. This
proves that we can find elements in $Q\setminus Q^{\boldsymbol{\mathsf{s}}}$
of arbitrarily large weight.

We will give in §3.3 more information about the constant $N_{r,e}$. In
particular, in the following particular cases the situation is the same as in
level one (cf. Proposition 2.37).

######  Proposition 3.9.

Assume that $(r,e)\in\bigl{\\{}(2,2),(2,3),(3,2)\bigr{\\}}$. Then
$Q^{\boldsymbol{\mathsf{s}}}=\bigl{\\{}\alpha\in
Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\geq 0\bigr{\\}}$.

###### Proof.

We will see in Proposition 3.10 (respectively, Proposition 3.29) that
$N_{r,e}=1$ (resp. $N_{r,e}=2$) if $r=2$ and $e\in\\{2,3\\}$ (resp. $r=3$ and
$e=2$). In these three cases we have $N_{r,e}\leq r-1$, thus

$\left\\{\alpha\in Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\geq
0\right\\}\subseteq\left\\{\alpha\in
Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\geq N_{r,e}-r+1\right\\}.$

We conclude the proof by Proposition 2.45 and Corollary 3.7 since we then find

$Q^{\boldsymbol{\mathsf{s}}}\subseteq\\{\alpha\in
Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)\geq
0\\}\subseteq\left\\{\alpha\in
Q:\mathsf{w}^{{\boldsymbol{\mathsf{s}}}}(\alpha)>N_{r,e}-r\right\\}\subseteq
Q^{\boldsymbol{\mathsf{s}}}.$

∎

###  3.3 Optimality

The aim of this subsection is to give sharp bounds for $N_{r,e}$. Note that we
have seen during the proof of Theorem 3.6 that $N_{r,e}\leq\frac{er(r-1)}{2}$.
In fact, we will see that this first bound is roughly the value of $N_{r,e}$
multiplied by a factor of $4$ (see Corollary 3.26). We will also compute
$N_{r,e}$ for small values of $e$ or $r$ (see §3.3.1 and §3.3.3).

####  3.3.1 Values for small parameters

We will here compute $N_{r,e}$ when $r=2$ or $e=2$.

######  Proposition 3.10 (Case $r=2$).

We have $N_{2,e}=\lfloor\frac{e}{2}\rfloor$.

###### Proof.

By Lemma 3.2, it suffices to prove that for any $E,F\subseteq\\{1,\dots,e\\}$
we have $\min(\lvert E\rvert,\lvert F\rvert)-\lvert E\cap
F\rvert\leq\lfloor\frac{e}{2}\rfloor$ and that equality happen for some $E,F$.
Writing $E=(E\cap F)\sqcup E^{\prime}$ and $F=(E\cap F)\sqcup F^{\prime}$, we
have

$\min(\lvert E\rvert,\lvert F\rvert)-\lvert E\cap F\rvert=\min(\lvert
E^{\prime}\rvert,\lvert
F^{\prime}\rvert)\leq\left\lfloor\frac{e}{2}\right\rfloor,$

since $E^{\prime}\cap F^{\prime}=\emptyset$. We conclude the proof since
equality holds when $\lvert E\rvert=\lfloor\frac{e}{2}\rfloor$ and $F=E^{c}$.
∎

######  Proposition 3.11 (Case $e=2$).

We have $N_{r,2}=\left\lfloor\frac{r^{2}}{4}\right\rfloor$.

###### Proof.

Let $E_{1},\dots,E_{r}\subseteq\\{1,2\\}$. By Lemma 3.4, if $E_{j}=\emptyset$
or $E_{j}=\\{1,2\\}$ for some $j\in\\{1,\dots,r\\}$ then
$\mathsf{w}(\mathcal{E})$ does not decrease if we replace $E_{j}$ by
$\\{1\\}$. Hence, we can assume that for all $j\in\\{1,\dots,r\\}$ we have
$E_{j}=\\{1\\}$ or $E_{j}=\\{2\\}$. Let $s\in\\{0,\dots,r\\}$ so that, after
reordering, we have $E_{1}=\dots=E_{s}=\\{1\\}$ and
$E_{s+1}=\dots=E_{r}=\\{2\\}$. Then by Lemmas 2.44 and 3.4 we have

$\mathsf{w}(\mathcal{E})=s(r-s)\,\mathsf{w}(\\{1\\},\\{2\\})=s(r-s).$

This quantity is maximal for $s\in\\{0,\dots,r\\}$ at
$s=\lfloor\frac{r}{2}\rfloor$ thus

$N_{r,2}=\left\lfloor\frac{r}{2}\right\rfloor\left(r-\left\lfloor\frac{r}{2}\right\rfloor\right).$

If $r$ is even then we obtain $N_{r,2}=\lfloor\frac{r^{2}}{4}\rfloor$ as
announced, and if $r$ is odd we obtain
$N_{r,2}=\frac{r-1}{2}\frac{r+1}{2}=\frac{r^{2}-1}{4}=\lfloor\frac{r^{2}}{4}\rfloor$
as well. ∎

####  3.3.2 Bounds

Except in Lemma 3.12, we always assume that $e\geq 2$ (recall from Remark 3.5
that $N_{r,1}=0$). We will first give a lower bound for $N_{r,e}$.

######  Lemma 3.12.

Let $r\geq 2$. The sequence $(N_{r,e})_{e\geq 1}$ is superadditive, in other
words:

$N_{r,e+e^{\prime}}\geq N_{r,e}+N_{r,e^{\prime}},$

for any $e,e^{\prime}\geq 1$.

###### Proof.

Let $E_{1},\dots,E_{r}\subseteq\\{1,\dots,e\\}$ and
$F_{1},\dots,F_{r}\subseteq\\{1,\dots,e^{\prime}\\}$.For any
$j\in\\{1,\dots,r\\}$ we define $F^{\prime}_{j}\coloneqq\\{i+e:i\in
F_{j}\\}\subseteq\\{e+1,\dots,e+e^{\prime}\\}$. Note that $E_{j}\cap
F_{k}=\emptyset$ for all $1\leq j<k\leq r$. For any $j\in\\{1,\dots,r\\}$, we
define $G_{j}\coloneqq E_{j}\sqcup
F^{\prime}_{j}\subseteq\\{1,\dots,e+e^{\prime}\\}$ (the corresponding binary
matrix $\mathcal{G}$ is obtained by putting $\mathcal{F}$ below
$\mathcal{E}$). For any $1\leq j<k\leq r$ we have

$\displaystyle\min(\lvert G_{j}\rvert,\lvert G_{k}\rvert)$
$\displaystyle=\min(\lvert E_{j}\rvert+\lvert F_{j}\rvert,\lvert
E_{k}\rvert+\lvert F_{k}\rvert)$ $\displaystyle\geq\min(\lvert
E_{j}\rvert,\lvert E_{k}\rvert)+\min(\lvert F_{j}\rvert,\lvert F_{k}\rvert),$

and, recalling that $E_{j}\cap F^{\prime}_{k}=\emptyset$,

$\displaystyle G_{j}\cap G_{k}$ $\displaystyle=(E_{j}\sqcup
F^{\prime}_{j})\cap(E_{k}\sqcup F^{\prime}_{k})$ $\displaystyle=(E_{j}\cap
E_{k})\sqcup(F^{\prime}_{j}\cap F^{\prime}_{k}),$

so that
$\mathsf{w}(G_{j},G_{k})\geq\mathsf{w}(E_{j},E_{k})+\mathsf{w}(F_{j},F_{k})$.
By Lemma 2.44 we obtain
$\mathsf{w}(\mathcal{G})\geq\mathsf{w}(\mathcal{E})+\mathsf{w}(\mathcal{F})$
and this conclude the proof since
$N_{r,e+e^{\prime}}\geq\mathsf{w}(\mathcal{G})$ and by taking the maximum on
$\mathcal{E}$ and $\mathcal{F}$. ∎

######  Corollary 3.13.

For any $e\geq 2$ we have
$N_{r,e}\geq\bigl{\lfloor}\frac{e}{2}\bigr{\rfloor}\bigl{\lfloor}\frac{r^{2}}{4}\bigr{\rfloor}$.

###### Proof.

Since $N_{r,1}=0$ we have $N_{r,e}\geq
N_{r,2\lfloor\frac{e}{2}\rfloor}\geq\lfloor\frac{e}{2}\rfloor N_{r,2}$ by
Lemma 3.12 and we conclude by Proposition 3.11. ∎

We will now give an upper bound for $N_{r,e}$. The next proposition is the key
of the next results. We write
${\mathcal{P}_{e}}\coloneqq\mathcal{P}(\\{1,\dots,e\\})$ for the powerset of
$\\{1,\dots,e\\}$, the set of subsets of $\\{1,\dots,e\\}$. We denote by
$\lVert\cdot\rVert_{1}$ the $1$-norm on $\mathbb{R}^{{\mathcal{P}_{e}}}$,
given by $\lVert x\rVert_{1}=\sum_{E\in{\mathcal{P}_{e}}}\lvert x_{E}\rvert$
for all $x\in\mathbb{R}^{{\mathcal{P}_{e}}}$. For any
$x,y\in\mathbb{R}^{{\mathcal{P}_{e}}}$ we write $\langle x,y\rangle\coloneqq
x^{\intercal}y$ for the canonical scalar product. We have $\langle
x,x\rangle=\lVert x\rVert^{2}$, where $\lVert\cdot\rVert$ is the Euclidean
norm as in Section 2. Let $A_{e}=(a_{E,F})_{E,F\in{\mathcal{P}_{e}}}$ be the
matrix given by $a_{E,F}\coloneqq\mathsf{w}(E,F)$ for all
$E,F\in{\mathcal{P}_{e}}$. The matrix $A_{e}$ is symmetric of size $2^{e}$
with all its diagonal entries being $0$ and only non-negative entries. We
denote by $q_{e}$ the quadratic form given by
$q_{e}(x)\coloneqq\frac{1}{2}\langle x,A_{e}x\rangle$ for all
$x\in\mathbb{R}^{{\mathcal{P}_{e}}}$.

######  Proposition 3.14.

We have

$N_{r,e}=\max_{\begin{subarray}{c}x\in\mathbb{N}^{{\mathcal{P}_{e}}}\\\ \lVert
x\rVert_{1}=r\end{subarray}}q_{e}(x).$

###### Proof.

We have a one-to-one correspondence between $r$-tuples
$E_{1},\dots,E_{r}\subseteq\\{1,\dots,e\\}$ of subsets of $\\{1,\dots,e\\}$
and elements
$x=(x_{E})_{E\in{\mathcal{P}_{e}}}\in\mathbb{N}^{{\mathcal{P}_{e}}}$ with
$\lVert x\rVert_{1}=r$ given as follows:

$x_{E}=\\#\bigl{\\{}j\in\\{1,\dots,r\\}:E_{j}=E\\}.$

For such elements, by Lemmas 2.44 and 3.4 we have

$\displaystyle\mathsf{w}(\mathcal{E})$ $\displaystyle=\sum_{1\leq j<k\leq
r}\mathsf{w}(E_{j},E_{k})$ $\displaystyle=\sum_{1\leq j<k\leq
r}a_{E_{j},E_{k}}$ $\displaystyle=\frac{1}{2}\sum_{1\leq j,k\leq
r}a_{E_{j},E_{k}}$
$\displaystyle=\frac{1}{2}\sum_{E,F\in{\mathcal{P}_{e}}}a_{E,F}x_{E}x_{F}$
$\displaystyle=q_{e}(x),$

which concludes the proof. ∎

We will now study the quadratic form $q_{e}$. Our first aim is to prove that
it suffices to study the restrictions of $q_{e}$ to the subspaces of
$\mathbb{R}^{{\mathcal{P}_{e}}}$ corresponding to subsets of $\\{1,\dots,e\\}$
of same cardinalities.

######  Lemma 3.15.

Let $E,F\subseteq\\{1,\dots,e\\}$. We have
$\mathsf{w}(E,F)=\mathsf{w}(E^{c},F^{c})$.

###### Proof.

A simple calculation gives

$\displaystyle\mathsf{w}(E^{c},F^{c})$ $\displaystyle=\min(\lvert
E^{c}\rvert,\lvert F^{c}\rvert)-\lvert E^{c}\cap F^{c}\rvert$
$\displaystyle=\min(e-\lvert E\rvert,e-\lvert F\rvert)-\lvert(E\cup
F)^{c}\rvert$ $\displaystyle=e-\max(\lvert E\rvert,\lvert F\rvert)-e+\lvert
E\cup F\rvert$ $\displaystyle=-\max(\lvert E\rvert,\lvert F\rvert)+\lvert
E\rvert+\lvert F\rvert-\lvert E\cap F\rvert$ $\displaystyle=\min(\lvert
E\rvert,\lvert F\rvert)-\lvert E\cap F\rvert$ $\displaystyle=\mathsf{w}(E,F).$

∎

######  Lemma 3.16.

Let $E_{1},\dots,E_{r}\subseteq\\{1,\dots,e\\}$. Let $m\coloneqq\min_{1\leq
j\leq r}\lvert E_{j}\rvert$ and assume that $m<e$. Let $j_{0}$ such that
$\lvert E_{j_{0}}\rvert=m$ and let $x\in\\{1,\dots,e\\}\setminus E_{j_{0}}$.
For $j\in\\{1,\dots,r\\}$, we define

$\widetilde{E}_{j}\coloneqq\begin{cases}E_{j},&\text{if }j\neq j_{0},\\\
E_{j_{0}}\cup\\{x\\},&\text{if }j=j_{0}.\end{cases}$

Then

$\mathsf{w}(\widetilde{\mathcal{E}})=\mathsf{w}(\mathcal{E})+\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert
E_{j}\rvert>m\text{ and }x\notin E_{j}\bigr{\\}}\\\
-\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert E_{j}\rvert=m\text{ and }x\in
E_{j}\bigr{\\}}.$

###### Proof.

Let $j\neq k\in\\{1,\dots,e\\}$. If $j,k\neq j_{0}$ then
$\mathsf{w}(\widetilde{E}_{j},\widetilde{E}_{k})=\mathsf{w}(E_{j},E_{k})$. If
$k=j_{0}$ then, recalling that $\mathbf{1}_{j}$ denotes the characteristic
vector of $E_{j}$,

$\displaystyle\mathsf{w}(\widetilde{E}_{j},\widetilde{E}_{j_{0}})$
$\displaystyle=\mathsf{w}(E_{j},E_{j_{0}}\sqcup\\{x\\})$
$\displaystyle=\min(\lvert E_{j}\rvert,m+1)-\lvert E_{j}\cap
E_{j_{0}}\rvert-\mathbf{1}_{j}(x)$
$\displaystyle=\begin{cases}\mathsf{w}(E_{j},E_{j_{0}})-\mathbf{1}_{j}(x),&\text{if
}\lvert E_{j}\rvert=m,\\\
\mathsf{w}(E_{j},E_{j_{0}})+1-\mathbf{1}_{j}(x),&\text{if }\lvert
E_{j}\rvert>m.\end{cases}$

Thus, by Lemma 2.44 we have

$\displaystyle\mathsf{w}(\widetilde{\mathcal{E}})-\mathsf{w}(\mathcal{E})$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq j\leq r\\\ j\neq
j_{0}\end{subarray}}\bigl{[}\mathsf{w}(\widetilde{E}_{j},\widetilde{E}_{j_{0}})-\mathsf{w}(E_{j},E_{j_{0}})\bigr{]}$
$\displaystyle=\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert
E_{j}\rvert>m\bigr{\\}}-\\#\bigl{\\{}j\in\\{1,\dots,r\\}:x\in
E_{j}\bigr{\\}}.$

Writing

$\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert
E_{j}\rvert>m\bigr{\\}}=\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert
E_{j}\rvert>m\text{ and }x\in E_{j}\bigr{\\}}\\\
\sqcup\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert E_{j}\rvert>m\text{ and }x\notin
E_{j}\bigr{\\}},$

and

$\bigl{\\{}j\in\\{1,\dots,r\\}:x\in
E_{j}\bigr{\\}}=\bigl{\\{}j\in\\{1,\dots,r\\}:x\in E_{j}\text{ and }\lvert
E_{j}\rvert>m\bigr{\\}}\\\ \sqcup\bigl{\\{}j\in\\{1,\dots,r\\}:x\in
E_{j}\text{ and }\lvert E_{j}\rvert=m\bigr{\\}},$

gives

$\mathsf{w}(\widetilde{\mathcal{E}})-\mathsf{w}(\mathcal{E})=\begin{multlined}\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert
E_{j}\rvert>m\text{ and }x\notin E_{j}\bigr{\\}}\\\
-\\#\bigl{\\{}j\in\\{1,\dots,r\\}:x\in E_{j}\text{ and }\lvert
E_{j}\rvert=m\bigr{\\}},\end{multlined}\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert
E_{j}\rvert>m\text{ and }x\notin E_{j}\bigr{\\}}\\\
-\\#\bigl{\\{}j\in\\{1,\dots,r\\}:x\in E_{j}\text{ and }\lvert
E_{j}\rvert=m\bigr{\\}},$

as announced. ∎

######  Proposition 3.17.

There exist $E_{1},\dots,E_{r}\subseteq\\{1,\dots,e\\}$ with $\lvert
E_{1}\rvert=\dots=\lvert E_{r}\rvert$ such that
$\mathsf{w}(\mathcal{E})=N_{r,e}$.

###### Proof.

Let $E_{1},\dots,E_{r}\subseteq\\{1,\dots,e\\}$. It suffices to prove that we
can find $\widetilde{E}_{1},\dots,\widetilde{E}_{r}\subseteq\\{1,\dots,e\\}$
satisfying $\lvert\widetilde{E}_{1}\rvert=\dots=\lvert\widetilde{E}_{r}\rvert$
such that $\mathsf{w}(\widetilde{\mathcal{E}})\geq\mathsf{w}(\mathcal{E})$.

Let $m\coloneqq\min_{1\leq j\leq r}\lvert E_{j}\rvert$ and
$M\coloneqq\max_{1\leq j\leq r}\lvert E_{j}\rvert$, and define

$\displaystyle N_{m}$
$\displaystyle\coloneqq\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert
E_{j}\rvert=m\\},$ $\displaystyle N_{M}$
$\displaystyle\coloneqq\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert
E_{j}\rvert=M\\}.$

We will use induction on $(M-m,N_{m}+N_{M})\in\mathbb{N}^{2}$. If $m=M$ then
we are done, thus assume that $m<M$. Let $j_{m},j_{M}\in\\{1,\dots,e\\}$ such
that $\lvert E_{j_{m}}\rvert=m$ and $\lvert E_{j_{M}}\rvert=M$. We have
$e-m+M>e$ by assumption, thus $E_{j_{m}}^{c}\cap E_{j_{M}}\neq\emptyset$, in
particular we can pick $x\in E_{j_{m}}^{c}\cap E_{j_{M}}$. If

$\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert E_{j}\rvert>m\text{ and }x\notin
E_{j}\bigr{\\}}\geq\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert E_{j}\rvert=m\text{
and }x\in E_{j}\bigr{\\}},$ (3.18)

then by Lemma 3.16 applied with the family $E_{1},\dots,E_{r}$ and $x\notin
E_{j_{m}}$, we can construct a family
$\widetilde{E_{1}},\dots,\widetilde{E}_{r}\subseteq\\{1,\dots,e\\}$ satisfying
$\mathsf{w}(\widetilde{\mathcal{E}})\geq\mathsf{w}(\mathcal{E})$ such that
either

$\displaystyle\max_{1\leq j\leq r}\lvert\widetilde{E}_{j}\rvert$
$\displaystyle=M,$ $\displaystyle\min_{1\leq j\leq
r}\lvert\widetilde{E}_{j}\rvert$ $\displaystyle=m+1,$ or
$\displaystyle\max_{1\leq j\leq r}\lvert\widetilde{E}_{j}\rvert$
$\displaystyle=M,$ $\displaystyle\min_{1\leq j\leq
r}\lvert\widetilde{E}_{j}\rvert$ $\displaystyle=m,$
$\displaystyle\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert\widetilde{E}_{j}\rvert=m\bigr{\\}}$
$\displaystyle=N_{m}-1,$
$\displaystyle\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert\widetilde{E}_{j}\rvert=M\bigr{\\}}$
$\displaystyle=N_{M},$

thus in both cases we conclude by induction. Thus, we now assume that (3.18)
fails, that is,

$\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert E_{j}\rvert>m\text{ and }x\notin
E_{j}\bigr{\\}}<\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert E_{j}\rvert=m\text{
and }x\in E_{j}\bigr{\\}}.$ (3.19)

Defining $F_{j}\coloneqq E_{j}^{c}$ for all $j\in\\{1,\dots,e\\}$, we have

$\displaystyle\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert F_{j}\rvert>e-M\text{
and }x\notin F_{j}\bigr{\\}}$
$\displaystyle=\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert E_{j}\rvert<M\text{ and
}x\in E_{j}\bigr{\\}}$
$\displaystyle\geq\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert E_{j}\rvert=m\text{
and }x\in E_{j}\bigr{\\}}$
$\displaystyle>\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert E_{j}\rvert>m\text{ and
}x\notin E_{j}\bigr{\\}}\text{ (by\leavevmode\nobreak\
\eqref{equation:induction_nremax_same_size_converse})}$
$\displaystyle\geq\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert E_{j}\rvert=M\text{
and }x\notin E_{j}\bigr{\\}}$
$\displaystyle=\\#\bigl{\\{}j\in\\{1,\dots,r\\}:\lvert F_{j}\rvert=e-M\text{
and }x\in F_{j}\bigr{\\}}.$

Since we have chosen $x$ such that $x\notin F_{j_{M}}$ with $\lvert
F_{j_{M}}\rvert=e-M=\min_{1\leq j\leq r}\lvert F_{j}\rvert$, we can apply
Lemma 3.16 to the family $F_{1},\dots,F_{r}$ and $x\notin F_{j_{M}}$. We thus
find a family
$\widetilde{F}_{1},\dots,\widetilde{F}_{r}\subseteq\\{1,\dots,e\\}$ satisfying
$\mathsf{w}(\widetilde{\mathcal{F}})>\mathsf{w}(\mathcal{F})$, and this
concludes the proof since $\mathsf{w}(\mathcal{F})=\mathsf{w}(\mathcal{E})$ by
Lemmas 2.44 and 3.15. ∎

For any $k\in\\{0,\dots,e\\}$, let $q_{e,k}$ be the restriction of $q_{e}$ to
the subspace $\mathbb{R}^{\mathcal{P}_{e,k}}$, where $\mathcal{P}_{e,k}$ is
the subset of ${\mathcal{P}_{e}}$ given by the subsets
$E\subseteq\\{1,\dots,e\\}$ of size $k$. We have
$q_{e,k}(x)=\frac{1}{2}\langle x,A_{e,k}x\rangle$ for all
$x\in\mathbb{R}^{\mathcal{P}_{e,k}}$, where
$A_{e,k}=(a_{E,F})_{E,F\in\mathcal{P}_{e,k}}$ is the symmetric matrix of size
$\binom{e}{k}$ given by

$a_{E,F}=\mathsf{w}(E,F)=k-\lvert E\cap F\rvert,$

for all $E,F\in\mathcal{P}_{e,k}$. In particular, as for $A_{e}$, the matrix
$A_{e,k}$ has only $0$ entries in the diagonal. By Lemmas 3.4 and 3.15 and
Proposition 3.17, we have

$N_{r,e}=\max_{1\leq
k\leq\lfloor\frac{e}{2}\rfloor}\max_{\begin{subarray}{c}x\in\mathbb{N}^{\mathcal{P}_{e,k}}\\\
\lVert x\rVert_{1}=r\end{subarray}}q_{e,k}(x).$ (3.20)

The matrix $A_{e,k}$ appears at some places in the literature (see, for
instance, [Ry81, Ry82]). The result of Lemma 3.21 is maybe well-known; we
provide a proof for convenience.

######  Lemma 3.21.

Recall that $e\geq 2$ and let $k\in\\{1,\dots,e-1\\}$. The eigenvalues of
$A_{e,k}$ are the following:

$\displaystyle k\binom{e-1}{k}$ , $\displaystyle\text{ with multiplicity }1,$
$\displaystyle 0$ , $\displaystyle\text{ with multiplicity }\binom{e}{k}-e,$
$\displaystyle-\binom{e-2}{k-1}$ , $\displaystyle\text{ with multiplicity
}e-1.$

Moreover, the constant vector $\mathbf{1}\in\mathbb{R}^{\mathcal{P}_{e,k}}$ is
an eigenvector for the eigenvalue $k\binom{e-1}{k}$.

###### Proof.

We follow the computation of the eigenvalues for adjacency matrices of
strongly regular graphs (see, for instance, [GoRo]). For any $m,n\geq 1$, we
denote by $J_{m,n}$ the $m\times n$ matrix filled with ones, and we define
$J_{n}\coloneqq J_{n,n}$. We also write $I_{n}$ for the $n\times n$ identity
matrix.

We first note that for any $E\in\mathcal{P}_{e,k}$ we have,

$\displaystyle\sum_{F\in\mathcal{P}_{e,k}}a_{E,F}$
$\displaystyle=\sum_{F\in\mathcal{P}_{e,k}}(k-\lvert E\cap F\rvert)$
$\displaystyle=\sum_{\ell=0}^{k-1}(k-\ell)\binom{k}{\ell}\binom{e-k}{k-\ell}$
$\displaystyle=\sum_{\ell=1}^{k}\ell\binom{k}{\ell}\binom{e-k}{\ell}$
$\displaystyle=\sum_{\ell=1}^{k}k\binom{k-1}{\ell-1}\binom{e-k}{e-k-\ell},$

thus, using Vandermonde’s identity we obtain,

$\displaystyle\sum_{F\in\mathcal{P}_{e,k}}a_{E,F}$
$\displaystyle=k\binom{e-1}{e-k-1}$ $\displaystyle=k\binom{e-1}{k},$

so that the constant vector $\mathbf{1}$ is an eigenvector of $A_{e,k}$ with
eigenvalue $k\binom{e-1}{k}$.

Now let $B=(b_{iE})_{\begin{subarray}{c}1\leq i\leq e\\\
E\in\mathcal{P}_{e,k}\end{subarray}}$ be the $e\times\binom{e}{k}$ incidence
matrix associated with the elements of $\\{1,\dots,e\\}$ and
$\mathcal{P}_{e,k}$. In other words, for any $i\in\\{1,\dots,e\\}$ and
$E\in\mathcal{P}_{e,k}$ we have $b_{iE}=\mathbf{1}_{E}(i)$, where
$\mathbf{1}_{E}$ is the characteristic vector of $E\subseteq\\{1,\dots,e\\}$.
The matrix $B^{\intercal}B$ is square with rows and columns indexed by
$\mathcal{P}_{e,k}$ and for any $E,F\in\mathcal{P}_{e,k}$ we have

$(B^{\intercal}B)_{EF}=\sum_{i=1}^{e}\mathbf{1}_{E}(i)\mathbf{1}_{F}(i)=\lvert
E\cap F\rvert,$

so that $A_{e,k}=kJ_{\binom{e}{k}}-B^{\intercal}B$. We will now compute
$A_{e,k}^{2}$. First, note that the matrix $BB^{\intercal}$ is square of size
$e$ and for any $i,j\in\\{1,\dots,e\\}$ we have

$(BB^{\intercal})_{ij}=\sum_{E\in\mathcal{P}_{e,k}}\mathbf{1}_{E}(i)\mathbf{1}_{E}(j)=\begin{dcases}\binom{e-1}{k-1},&\text{if
}i=j,\\\ \binom{e-2}{k-2},&\text{if }i\neq j\end{dcases}$

(with the convention $\binom{e-2}{k-2}=0$ if $k=1$), so that

$BB^{\intercal}=\binom{e-2}{k-2}J_{e}+\left(\binom{e-1}{k-1}-\binom{e-2}{k-2}\right)I_{e}=\binom{e-2}{k-2}J_{e}+\binom{e-2}{k-1}I_{e}.$

It is clear that $B^{\intercal}J_{e}$ is a scalar multiple of
$J_{\binom{e}{k},e}$, similarly

$J_{\binom{e}{k}}^{2},\quad J_{\binom{e}{k}}B^{\intercal}B\text{ and
}J_{\binom{e}{k},e}B,$

are scalar multiples of $J_{\binom{e}{k}}$. We deduce that

$\displaystyle A_{e,k}^{2}$
$\displaystyle=\left(kJ_{\binom{e}{k}}-B^{\intercal}B\right)^{2},$
$\displaystyle=\alpha J_{\binom{e}{k}}+B^{\intercal}(BB^{\intercal})B$
$\displaystyle=\beta J_{\binom{e}{k}}+\binom{e-2}{k-1}B^{\intercal}B$
$\displaystyle=\gamma J_{\binom{e}{k}}-\binom{e-2}{k-1}A_{e,k},$ (3.22)

for some scalars $\alpha,\beta,\gamma$. Now let $x$ be any eigenvector of
$A_{e,k}$ orthogonal to $\mathbf{1}$ and let $\lambda$ be its associated
eigenvalue. Since $\langle x,\mathbf{1}\rangle=0$ we have
$J_{\binom{e}{k}}x=0$ and thus (3.22) gives

$A_{e,k}^{2}x=-\binom{e-2}{k-1}A_{e,k}x,$

thus $\lambda^{2}=-\binom{e-2}{k-1}\lambda$ thus $\lambda=0$ or
$\lambda=-\binom{e-2}{k-1}$. Now if $m$ (respectively $m_{0}$) denotes the
multiplicity of the eigenvalue $-\binom{e-2}{k-1}$ (resp. $0$), what precedes
proves that $k\binom{e-1}{k}$ is of multiplicity $1$ and we have

$\displaystyle 1+m_{0}+m$ $\displaystyle=\binom{e}{k},$ $\displaystyle
k\binom{e-1}{k}-m\binom{e-2}{k-1}$ $\displaystyle=\mathrm{tr}(A_{e,k})=0,$

recalling that the diagonal entries of $A_{e,k}$ are $0$. We find that
$m=k\binom{e-1}{k}/\binom{e-2}{k-1}=e-1$ and $m_{0}=\binom{e}{k}-e$, which
concludes the proof. ∎

######  Proposition 3.23.

Let $k\in\\{1,\dots,e-1\\}$. For any $x\in\mathbb{R}^{\mathcal{P}_{e,k}}$ we
have

$q_{e,k}(x)\leq\frac{\lVert x\rVert_{1}^{2}}{2e}k(e-k).$

###### Proof.

For any $x\in\mathbb{R}^{\mathcal{P}_{e,k}}$ we have

$x=\frac{\langle
x,\mathbf{1}\rangle}{\lVert\mathbf{1}\rVert^{2}}\mathbf{1}+x_{\perp},$

where $x_{\perp}\in\mathbb{R}^{\mathcal{P}_{e,k}}$ is orthogonal to
$\mathbf{1}$. By Lemma 3.21 we have $q_{e,k}(x_{\perp})\leq 0$ thus

$q_{e,k}(x)=\frac{\langle
x,\mathbf{1}\rangle^{2}}{\lVert\mathbf{1}\rVert^{4}}q_{e,k}(\mathbf{1})+q_{e,k}(x_{\perp})\leq\frac{\langle
x,\mathbf{1}\rangle^{2}}{\lVert\mathbf{1}\rVert^{4}}q_{e,k}(\mathbf{1}).$

Now $\lvert\langle x,\mathbf{1}\rangle\rvert\leq\lVert x\rVert_{1}$ by the
triangle inequality and, again by Lemma 3.21,

$q_{e,k}(\mathbf{1})=\frac{1}{2}\langle\mathbf{1},A_{e,k}\mathbf{1}\rangle=\frac{k}{2}\binom{e-1}{k}\lVert\mathbf{1}\rVert^{2}.$

Hence, we have

$q_{e,k}(x)\leq\frac{\lVert
x\rVert_{1}^{2}}{\lVert\mathbf{1}\rVert^{2}}\frac{k}{2}\binom{e-1}{k}=\frac{\lVert
x\rVert_{1}^{2}}{\binom{e}{k}}\frac{k}{2}\binom{e-1}{k}.$

We obtain the announced result since
$\binom{e}{k}=\binom{e}{e-k}=\frac{e}{e-k}\binom{e-1}{e-k-1}=\frac{e}{e-k}\binom{e-1}{k}$.
∎

######  Corollary 3.24.

We have

$N_{r,e}\leq\frac{r^{2}}{2e}\left\lfloor\frac{e^{2}}{4}\right\rfloor.$

###### Proof.

For any $k\in\\{1,\dots,e-1\\}$, by Proposition 3.23 we know that
$q_{e,k}(x)\leq\frac{r^{2}}{2e}k(e-k)$ if $\lVert x\rVert_{1}=r$. We conclude
by (3.20) since $k(e-k)\leq\bigl{\lfloor}\frac{e^{2}}{4}\bigr{\rfloor}$ (see
the proof of Proposition 3.11). ∎

######  Definition 3.25.

For any $r\geq 0$ and $e\geq 1$ we define
$N^{\prime}_{r,e}\coloneqq\Bigl{\lfloor}\frac{r^{2}}{2e}\bigl{\lfloor}\frac{e^{2}}{4}\bigr{\rfloor}\Bigr{\rfloor}$.

Combining Corollaries 3.13 and 3.24 yields the following final bounds.

######  Corollary 3.26.

For any $e,r\geq 2$ we have

$\left\lfloor\frac{e}{2}\right\rfloor\left\lfloor\frac{r^{2}}{4}\right\rfloor\leq
N_{r,e}\leq N^{\prime}_{r,e}\leq\frac{er^{2}}{8}$

We have the following particular case.

######  Proposition 3.27.

Assume that $e,r\geq 2$ are both even. Then

$N_{r,e}=N^{\prime}_{r,e}=\frac{er^{2}}{8}.$

####  3.3.3 More values for small parameters

Let $r,e\geq 2$. We now aim to compute the value of $N_{r,e}$ for
$r\in\\{3,4\\}$ or $e\in\\{3,\dots,6\\}$. For these values, we will see that,
except for $r=3$, we have $N_{r,e}=N^{\prime}_{r,e}$ (see Proposition 3.43).

##### With small $r$

We begin by the following result, which will be in fact interesting only in
the case $r=3$.

######  Lemma 3.28.

If $r\geq 3$ then $N_{r,e}\leq\lfloor\frac{(r-1)re}{6}\rfloor$.

###### Proof.

Let $E_{1},\dots,E_{r}\subseteq\\{1,\dots,e\\}$ and define
$a_{jk}\coloneqq\mathsf{w}(E_{j},E_{k})=\min(\lvert E_{j}\rvert,\lvert
E_{k}\rvert)-\lvert E_{j}\cap E_{k}\rvert$ for all $1\leq j<k\leq r$. By (3.3)
and since $a_{jk}\in\mathbb{N}$, it suffices to show that

$\sum_{1\leq j<k\leq r}a_{jk}\leq\frac{(r-1)re}{6}.$

Let $1\leq j<k<l\leq r$; note that this is possible since $r\geq 3$. We have

$\displaystyle a_{jk}+a_{kl}+a_{jl}$
$\displaystyle=\begin{multlined}\min(\lvert E_{j}\rvert,\lvert
E_{k}\rvert)+\min(\lvert E_{k}\rvert,\lvert E_{l}\rvert)+\min(\lvert
E_{j}\rvert,\lvert E_{l}\rvert)\\\ -\lvert E_{j}\cap E_{k}\rvert-\lvert
E_{k}\cap E_{l}\rvert-\lvert E_{j}\cap E_{l}\rvert\end{multlined}\min(\lvert
E_{j}\rvert,\lvert E_{k}\rvert)+\min(\lvert E_{k}\rvert,\lvert
E_{l}\rvert)+\min(\lvert E_{j}\rvert,\lvert E_{l}\rvert)\\\ -\lvert E_{j}\cap
E_{k}\rvert-\lvert E_{k}\cap E_{l}\rvert-\lvert E_{j}\cap E_{l}\rvert$
$\displaystyle\leq\lvert E_{j}\rvert+\lvert E_{k}\lvert+\lvert
E_{l}\rvert-\lvert E_{j}\cap E_{k}\rvert-\lvert E_{k}\cap E_{l}\rvert-\lvert
E_{j}\cap E_{l}\rvert$ $\displaystyle\leq\lvert E_{j}\cup E_{k}\cup
E_{l}\rvert-\lvert E_{j}\cap E_{k}\cap E_{l}\rvert$ $\displaystyle\leq\lvert
E_{j}\cup E_{k}\cup E_{l}\rvert$ $\displaystyle\leq e.$

Now, in the sum $\sum_{1\leq j<k<l\leq r}(a_{jk}+a_{kl}+a_{jl})$ each couple
$(j,k)$ for $1\leq j<k\leq r$ appears exactly $r-2$ times, thus

$\sum_{1\leq j<k<l\leq r}(a_{jk}+a_{kl}+a_{jl})=(r-2)\sum_{1\leq j<k\leq
r}a_{jk}.$

Thus, we find that

$(r-2)\sum_{1\leq j<k\leq r}a_{jk}\leq e\binom{r}{3},$

and thus

$\sum_{1\leq j<k\leq r}a_{jk}\leq e\frac{r(r-1)}{6},$

whence the result. ∎

######  Proposition 3.29 (Case $r=3$).

We have $N_{3,e}=e$.

###### Proof.

By Corollary 3.13 and Lemma 3.28 we have

$2\left\lfloor\frac{e}{2}\right\rfloor\leq N_{3,e}\leq e,$ (3.30)

in particular $N_{3,e}=e$ if $e$ is even. (Note that if we use Corollary 3.13
instead of Lemma 3.28 we only obtain $N_{r,e}\leq\frac{9e}{8}$.) Thus, we can
now assume that $e$ is odd and let us write $e=2e^{\prime}+3$ with
$e^{\prime}\geq 0$. We first assume that $e^{\prime}\geq 1$. By Lemma 3.12 we
have

$N_{3,2e^{\prime}}+N_{3,3}\leq N_{3,e}.$

Since $2e^{\prime}$ is even we have $N_{3,2e^{\prime}}=2e^{\prime}$, thus we
obtain, using (3.30) for the right inequality,

$e-3+N_{3,3}\leq N_{3,e}\leq e.$

Thus, to conclude it suffices to prove that $N_{3,3}=3$. We already know by
(3.30) that $N_{3,3}\leq 3$, thus it suffices to find
$E_{1},E_{2},E_{3}\subseteq\\{1,2,3\\}$ such that $\mathsf{w}(\mathcal{E})=3$.
Using (3.3), this equality is satisfied with $E_{i}\coloneqq\\{i\\}$ for any
$i\in\\{1,2,3\\}$ and this concludes the proof. ∎

######  Proposition 3.31 (Case $r=4$).

We have:

$N_{4,e}=\begin{cases}2e,&\text{if }e\text{ is even},\\\ 2e-1,&\text{if
}e\text{ is odd}.\end{cases}$

###### Proof.

If $e$ is even then we know by Proposition 3.27 that
$N_{4,e}=\frac{e4^{2}}{8}=2e$. We thus now assume that $e$ is odd. By
Corollary 3.24 we obtain

$N_{4,e}\leq\frac{8}{e}\left\lfloor\frac{e^{2}}{4}\right\rfloor<\frac{8}{e}\frac{e^{2}}{4}=2e,$

thus $N_{4,e}\leq 2e-1$. If $e>3$, as in the proof of Proposition 3.29, using
Lemma 3.12 we have

$N_{4,e-3}+N_{4,3}\leq N_{4,e},$

thus, since $e-3$ is even,

$2e-6+N_{4,3}\leq N_{4,e}.$

Hence, to conclude it suffices to prove that $N_{4,3}=5$. We have seen that
$N_{4,3}\leq 2\cdot 3-1=5$, thus it suffices to find
$E_{1},\dots,E_{4}\subseteq\\{1,\dots,e\\}$ with $\mathsf{w}(\mathcal{E})=5$.
Setting $E_{i}\coloneqq\\{i\\}$ for all $i\in\\{1,2,3\\}$ and $E_{4}\coloneqq
E_{3}$, we have

$\mathsf{w}(E_{j},E_{k})=\begin{cases}1,&\text{if }j\in\\{1,2\\},\\\
0,&\text{otherwise},\end{cases}$

for any $1\leq j<k\leq 4$, thus $\mathsf{w}(\mathcal{E})=\sum_{1\leq j<k\leq
4}\mathsf{w}(E_{j},E_{k})=3+2=5$. This concludes the proof. ∎

##### With small $e$

The next results will mainly consist in using the fact that $\mathbf{1}$ is an
eigenvector of $A_{e,k}$.

######  Proposition 3.32 (Case $e=3$).

We have $N_{r,3}=\bigl{\lfloor}\frac{r^{2}}{3}\bigr{\rfloor}$.

###### Proof.

By Corollary 3.24 we have
$N_{r,3}\leq\frac{r^{2}}{6}\bigl{\lfloor}\frac{9}{4}\bigr{\rfloor}=\frac{r^{2}}{3}$,
hence $N_{r,3}\leq\bigl{\lfloor}\frac{r^{2}}{3}\bigr{\rfloor}$. Thus, it
suffices to find $x\in\mathbb{N}^{\mathcal{P}_{e,1}}$ with $\lVert
x\rVert_{1}=r$ such that
$q_{e,1}(x)=\bigl{\lfloor}\frac{r^{2}}{3}\bigr{\rfloor}$. To that extent,
write $r=3a+h$ with $a\geq 1$ and $h\in\\{-1,0,1\\}$ and consider $x\coloneqq
a\mathbf{1}+h\mathbf{1}_{E}\in\mathbb{Z}^{\mathcal{P}_{e,1}}$, where
$\mathbf{1}_{E}$ is any characteristic vector of an element of
$\mathcal{P}_{e,1}$. Note that $x\in\mathbb{N}^{\mathcal{P}_{e,1}}$ and
$\lVert x\rVert_{1}=a\binom{3}{1}+h=3a+h=r$ indeed. We have, recalling that
$a_{E,E}=0$ and from Lemma 3.21 that
$A_{e,1}\mathbf{1}=\binom{2}{1}\mathbf{1}=2\,\mathbf{1}$,

$\displaystyle q_{e,1}(x)$ $\displaystyle=\frac{1}{2}\langle
A_{e,1}x,x\rangle$ $\displaystyle=\frac{1}{2}\left(a^{2}\langle
A_{e,1}\mathbf{1},\mathbf{1}\rangle+h^{2}\langle
A_{e,1}\mathbf{1}_{E},\mathbf{1}_{E}\rangle+2ah\langle
A_{e,1}\mathbf{1},\mathbf{1}_{E}\rangle\right)$
$\displaystyle=\frac{1}{2}\left(2a^{2}\lVert\mathbf{1}\rVert^{2}+4ah\right)$
$\displaystyle=3a^{2}+2ah$ $\displaystyle=\frac{r^{2}-h^{2}}{3}$
$\displaystyle=\begin{dcases}\frac{r^{2}}{3},&\text{if }r=0\pmod{3},\\\
\frac{r^{2}-1}{3},&\text{if }r=\pm 1\pmod{3},\end{dcases}$
$\displaystyle=\left\lfloor\frac{r^{2}}{3}\right\rfloor.$

This concludes the proof. ∎

######  Proposition 3.33 (Case $e\in\\{4,6\\}$).

We have $N_{r,4}=\bigl{\lfloor}\frac{r^{2}}{2}\bigr{\rfloor}$ and
$N_{r,6}=\bigl{\lfloor}\frac{3r^{2}}{4}\bigr{\rfloor}=3\bigl{\lfloor}\frac{r^{2}}{4}\bigr{\rfloor}$.

###### Proof.

If $r$ is even then the result is true by Proposition 3.27. Now if $r$ is odd,
by Corollary 3.26 we have

$\displaystyle 2\left\lfloor\frac{r^{2}}{4}\right\rfloor\leq
N_{r,4}\leq\left\lfloor\frac{r^{2}}{2}\right\rfloor,$ $\displaystyle
3\left\lfloor\frac{r^{2}}{4}\right\rfloor\leq
N_{r,6}\leq\left\lfloor\frac{3r^{2}}{4}\right\rfloor.$

Now we have

$\displaystyle\left\lfloor\frac{r^{2}}{2}\right\rfloor$
$\displaystyle=\frac{r^{2}-1}{2},$
$\displaystyle\left\lfloor\frac{r^{2}}{4}\right\rfloor$
$\displaystyle=\frac{r^{2}-1}{4},$

which concludes the proof for $e=4$. For $e=6$ we note that

$3\left\lfloor\frac{r^{2}}{4}\right\rfloor=3\frac{r^{2}-1}{4}=\frac{3r^{2}}{4}-\frac{3}{4}=\left\lfloor\frac{3r^{2}}{4}\right\rfloor,$

which concludes the proof. ∎

We will now deal with the case $e=5$.

######  Definition 3.34.

For any $r\geq 0$ and $e\geq 2$, we define

$Q_{r,e}\coloneqq\max_{\begin{subarray}{c}x\in\mathbb{N}^{\mathcal{P}_{e,k}}\\\
\lVert x\rVert_{1}=r\end{subarray}}q_{e,k}(x),$

where $k\coloneqq\bigl{\lfloor}\frac{e}{2}\bigr{\rfloor}$.

Recalling (3.20) and Corollary 3.26, for any $r\geq 2$ and $e\geq 2$ we have

$Q_{r,e}\leq N_{r,e}\leq N^{\prime}_{r,e}.$ (3.35)

######  Remark 3.36.

We will see that when $e=5$ then $Q_{r,e}=N_{r,e}$ (Lemma 3.40). It is unclear
whether this equality holds in full generality, in particular we state the
next results with the quantity $Q_{r,e}$.

Recall from Definition 3.25 that
$N^{\prime}_{r,e}=\bigl{\lfloor}\frac{r^{2}}{2e}\bigl{\lfloor}\frac{e^{2}}{4}\bigr{\rfloor}\bigr{\rfloor}$.

######  Lemma 3.37.

Let $r\geq 0$ and $e\geq 2$. Assume that $Q_{r,e}=N^{\prime}_{r,e}$. Then
$Q_{r+\binom{e}{k},e}=N^{\prime}_{r+\binom{e}{k},e}$, where
$k\coloneqq\bigl{\lfloor}\frac{e}{2}\bigr{\rfloor}$.

###### Proof.

By assumption, we can find $x\in\mathbb{N}^{\mathcal{P}_{e,k}}$ with $\lVert
x\rVert_{1}=r$ such that $q_{e,k}(x)=N^{\prime}_{r,e}$ (with
$k=\bigl{\lfloor}\frac{e}{2}\bigr{\rfloor}$). We have, where $\lambda\coloneqq
k\binom{e-1}{k}$ is the highest eigenvalue of $A_{e,k}$ (see Lemma 3.21),

$\displaystyle q_{e,k}(x+\mathbf{1})$
$\displaystyle=q_{e,k}(x)+q_{e,k}(\mathbf{1})+\langle
x,A_{e,k}\mathbf{1}\rangle$
$\displaystyle=Q_{r,e}+\frac{\lambda}{2}\binom{e}{k}+\lambda r,$

thus $Q_{r+\binom{e}{k},e}\geq Q_{r,e}+\frac{\lambda}{2}\binom{e}{k}+\lambda
r$. Note that $\frac{\lambda}{2}\binom{e}{k}=q_{e,k}(\mathbf{1})$ is an
integer. Now, recalling the identity
$\binom{e}{k}=\frac{e}{e-k}\binom{e-1}{k}$ (see the proof of Proposition
3.23), we have
$N^{\prime}_{r,e}=\left\lfloor\frac{r^{2}}{2\binom{e}{k}}\lambda\right\rfloor$
thus

$\displaystyle N^{\prime}_{r+\binom{e}{k},e}$
$\displaystyle=\left\lfloor\frac{r^{2}+\binom{e}{k}^{2}+2\binom{e}{k}r}{2\binom{e}{k}}\lambda\right\rfloor$
$\displaystyle=\left\lfloor\frac{r^{2}}{2\binom{e}{k}}\lambda+\frac{\lambda}{2}\binom{e}{k}+\lambda
r\right\rfloor$
$\displaystyle=\left\lfloor\frac{r^{2}}{2\binom{e}{k}}\lambda\right\rfloor+\frac{\lambda}{2}\binom{e}{k}+\lambda
r.$

Finally, we have

$\displaystyle N^{\prime}_{r+\binom{e}{k},e}$ $\displaystyle\geq
Q_{r+\binom{e}{k},e}$
$\displaystyle=Q_{e,k}+\left(Q_{r+\binom{e}{k},e}-Q_{r,e}\right)$
$\displaystyle\geq
N^{\prime}_{r,e}+\left(N^{\prime}_{r+\binom{e}{k},e}-N^{\prime}_{r,e}\right)$
$\displaystyle=N^{\prime}_{r+\binom{e}{k},e},$

which concludes the proof. ∎

######  Lemma 3.38.

Let $r\geq 0$ and $e\geq 2$. Let
$k\coloneqq\bigl{\lfloor}\frac{e}{2}\bigr{\rfloor}$ and assume that there
exists $x\in\\{0,1\\}^{\mathcal{P}_{e,k}}$ with $\lVert x\rVert_{1}=r$ such
that $q_{e,k}(x)=N^{\prime}_{r,e}$ where
$k\coloneqq\bigl{\lfloor}\frac{e}{2}\bigr{\rfloor}$ (in particular, we have
$Q_{r,e}=N^{\prime}_{r,e}$). Then
$Q_{\binom{e}{k}-r,e}=N^{\prime}_{\binom{e}{k}-r,e}$.

###### Proof.

As in Lemma 3.37, we have, noting that
$\mathbf{1}-x\in\mathbb{N}^{\mathcal{P}_{e,k}}$ since the entries of $x$ are
at most $1$,

$\displaystyle Q_{\binom{e}{k}-r,e}$ $\displaystyle\geq q_{e,k}(\mathbf{1}-x)$
$\displaystyle=q_{e,k}(x)+q_{e,k}(\mathbf{1})-\langle
x,A_{e,k}\mathbf{1}\rangle$
$\displaystyle=Q_{r,e}+\frac{\lambda}{2}\binom{e}{k}-\lambda r$
$\displaystyle=N^{\prime}_{r,e}+\frac{\lambda}{2}\binom{e}{k}-\lambda r$
$\displaystyle=N^{\prime}_{\binom{e}{k}-r,e},$

thus we conclude by (3.35). ∎

######  Remark 3.39.

Assume that $r=1$. We have $N^{\prime}_{1,e}\leq\frac{e}{8}$ by Corollary
3.26, thus if $e\leq 7$ we have $N^{\prime}_{1,e}=0$. In particular, the
assumption $Q_{r,e}=N^{\prime}_{r,e}$ of Lemma 3.37 is satisfied. Moreover,
since any $x\in\mathbb{N}^{\mathcal{P}_{e,k}}$ with $\lVert x\rVert_{1}=1$ has
entries in $\\{0,1\\}$, the assumption of Lemma 3.38 is also satisfied (using
(3.35)).

We give a last preliminary result, which is a refinement of (3.20) (see Remark
3.36).

######  Lemma 3.40.

Assume that $e=5$. For any $r\geq 2$ we have $Q_{r,e}=N_{r,e}$.

###### Proof.

By (3.20), Proposition 3.23 and Corollary 3.13, it suffices to prove that

$\left\lfloor\frac{r^{2}}{2e}k^{\prime}(e-k^{\prime})\right\rfloor<\left\lfloor\frac{e}{2}\right\rfloor\left\lfloor\frac{r^{2}}{4}\right\rfloor,$

for all
$k^{\prime}\in\bigl{\\{}1,\dots,\bigl{\lfloor}\frac{e}{2}\bigr{\rfloor}-1\bigr{\\}}$.
In our setting, this reduces to proving

$\left\lfloor\frac{2r^{2}}{5}\right\rfloor<2\left\lfloor\frac{r^{2}}{4}\right\rfloor.$
(3.41)

If $r$ is even, we have
$\bigl{\lfloor}\frac{2r^{2}}{5}\bigr{\rfloor}\leq\frac{2r^{2}}{5}<\frac{r^{2}}{2}=2\bigl{\lfloor}\frac{r^{2}}{4}\bigr{\rfloor}$
thus (3.41) holds. If $r$ is odd we have
$\bigl{\lfloor}\frac{r^{2}}{4}\bigr{\rfloor}=\frac{r^{2}-1}{4}$ and

$\frac{2r^{2}}{5}<\frac{r^{2}-1}{2}\iff 4r^{2}<5r^{2}-5,$

thus (3.41) holds since $r\geq 2$ is odd. ∎

######  Proposition 3.42 (Case $e=5$).

For any $r\geq 2$ we have
$N_{r,5}=N^{\prime}_{r,5}=\bigl{\lfloor}\frac{3r^{2}}{5}\bigr{\rfloor}$.

###### Proof.

By Lemma 3.37 and Lemma 3.40, it suffices to prove the equality for $2\leq
r<\binom{5}{2}=10$, together with $Q_{r,5}=N^{\prime}_{r,5}$ for
$r\in\\{0,1\\}$. For $r=0$ we have $Q_{r,5}=N^{\prime}_{r,5}=0$, and the
equality also holds for $r=1$ by Remark 3.39. By Lemma 3.38 and again Remark
3.39, we thus have $N_{r,5}=N^{\prime}_{r,5}$ for $r=9$.

It remains to prove $N_{r,5}=N^{\prime}_{r,5}$ for $r\in\\{2,\dots,8\\}$. It
follows from Propositions 3.10 and 3.29 that the equality
$N_{r,5}=N^{\prime}_{r,5}$ holds for $r\in\\{2,3\\}$. Moreover, the proofs of
these propositions guarantee that Lemma 3.38 can be applied for
$r\in\\{2,3\\}$, so that the equality $N_{r,5}=N^{\prime}_{r,5}$ also holds
for $r\in\\{7,8\\}$ (using Lemma 3.40). The same can be said for $r=4$
(Proposition 3.31) and thus $r=6$, however we need to carefully check that the
conditions of Lemma 3.38 can be satisfied when $r=4$. Following the proofs of
Propositions 3.11 and 3.31, taking

$\displaystyle E_{1}\coloneqq\\{1,3\\},\quad E_{2}\coloneqq\\{2,4\\},$
$\displaystyle E_{3}\coloneqq\\{1,5\\},\quad E_{4}\coloneqq\\{2,5\\},$

by (3.3) we have $\mathsf{w}(\mathcal{E})=\sum_{1\leq j<k\leq
4}\bigl{(}2-\lvert E_{j}\cap E_{k}\rvert\bigr{)}=9=N_{4,5}$ indeed and the
corresponding $x\in\mathbb{N}^{\mathcal{P}_{5,2}}$ has entries in $\\{0,1\\}$.
We could have also noticed that $N^{\prime}_{6,5}=\lfloor\frac{3\cdot
36}{5}\rfloor=\lfloor\frac{3\cdot 35}{5}+\frac{3}{5}\rfloor=21$, and by Lemma
3.12 we have

$N_{6,5}\geq N_{6,3}+N_{6,2},$

thus by Propositions 3.11 and 3.32 we have

$N_{6,5}\geq\left\lfloor\frac{6^{2}}{3}\right\rfloor+\left\lfloor\frac{6^{2}}{4}\right\rfloor=2\cdot
6+3^{2}=21,$

thus $N_{6,5}=N^{\prime}_{6,5}$ by Corollary 3.26 (note that this alternative
proof does not work for $r\in\\{7,8\\}$).

It thus remains to treat the case $r=5$. We define:

$\displaystyle E_{1}\coloneqq\\{1,2\\},\quad E_{2}\coloneqq\\{1,3\\},$
$\displaystyle E_{5}\coloneqq\\{2,4\\},\quad\quad\quad\quad
E_{3}\coloneqq\\{3,5\\},$ $\displaystyle E_{4}\coloneqq\\{4,5\\},$

so that $\lvert E_{j}\cap E_{j+1}\rvert=1$ for all $j\in\\{1,\dots,5\\}$ (with
$E_{6}\coloneqq E_{1}$), every other pairwise intersection being empty. By
(3.3), we deduce that

$\mathsf{w}(\mathcal{E})=5\cdot 1+5\cdot 2=15,$

thus $N_{5,5}\geq 15$. Since $N^{\prime}_{5,5}=\bigl{\lfloor}\frac{3\cdot
5^{2}}{5}\bigr{\rfloor}=15$, by Corollary 3.26 we obtain that
$N_{5,5}=N^{\prime}_{5,5}$ and this concludes the proof. ∎

##### All in one

We now gather all the results we have proven in these small cases.

######  Proposition 3.43.

Let $e,r\geq 2$. We have

$N_{r,e}=N^{\prime}_{r,e},$

in (at least) the following cases:

$r\in\\{2,4\\}\quad\text{or}\quad e\in\\{2,\dots,6\\}.$

Note that if $r=3$ then $N_{2,e}=e$ by Proposition 3.29, however if for
instance $e$ is even we have
$N^{\prime}_{3,e}=\bigl{\lfloor}\frac{9e}{8}\bigr{\rfloor}=e+\bigl{\lfloor}\frac{e}{8}\bigr{\rfloor}>N_{3,e}$
as soon as $e\geq 8$.

###### Proof.

First, note that the result holds if both $r$ and $e$ are even, by Proposition
3.27. We have

$N^{\prime}_{r,e}=\begin{dcases}\left\lfloor\frac{r^{2}e}{8}\right\rfloor,&\text{if
}e\text{ is even},\\\
\left\lfloor\frac{r^{2}(e^{2}-1)}{8e}\right\rfloor,&\text{if }e\text{ is
odd}.\end{dcases}$

For $r=2$, we deduce that (with $e\geq 2$ odd)

$N^{\prime}_{2,e}=\left\lfloor\frac{e^{2}-1}{2e}\right\rfloor=\left\lfloor\frac{e}{2}-\frac{1}{2e}\right\rfloor=\left\lfloor\frac{e}{2}\right\rfloor,$

thus $N^{\prime}_{2,e}=N_{2,e}$ by Proposition 3.10. For $r=4$, we have (with
$e\geq 2$ odd)

$N^{\prime}_{4,e}=\left\lfloor\frac{2(e^{2}-1)}{e}\right\rfloor=2e-\left\lfloor\frac{2}{e}\right\rfloor=2e-1,$

thus $N^{\prime}_{4,e}=N_{4,e}$ by Proposition 3.31. The equalities
$N^{\prime}_{r,e}=N_{r,e}$ for $e\in\\{2,\dots,6\\}$ are straightforward from
Propositions 3.11, 3.32, 3.33 and 3.42. ∎

####  3.3.4 Asymptotics

In this section, we will describe the asymptotic behaviour of the two
sequences $(N_{r,e})_{e\geq 2}$ and $(N_{r,e})_{r\geq 2}$ for $r,e\geq 2$. We
first recall the following basic fact, which will be used without any further
reference.

######  Lemma 3.44.

Let $(u_{n})_{n\geq 1}$ be a sequence of real numbers that converges to
$+\infty$ and let $x\in\mathbb{R}$. Then the sequence $\bigl{(}\frac{\lfloor
xu_{n}\rfloor}{u_{n}}\bigr{)}_{n\geq 1}$ converges to $x$.

We also recall the following standard result.

######  Lemma 3.45 (Fekete’s superadditive lemma).

Let $(u_{n})_{n\geq 1}$ be a superadditive sequence. Then
$\bigl{(}\frac{u_{n}}{n}\bigr{)}_{n\geq 1}$ has a limit in $[-\infty,+\infty]$
as $n\to+\infty$.

By Lemmas 3.12 and 3.45, we know that the sequence
$\left(\frac{N_{r,e}}{e}\right)_{e\geq 1}$ has a limit
$N_{r,\infty}\in[0,+\infty]$. By Corollary 3.26 we have

$\frac{1}{2}\left\lfloor\frac{r^{2}}{4}\right\rfloor\leq
N_{r,\infty}\leq\frac{r^{2}}{8},$ (3.46)

in particular if $r$ is even then $N_{r,\infty}=\frac{r^{2}}{8}$.

######  Remark 3.47.

By Propositions 3.10, 3.29 and 3.31, we have $N_{2,\infty}=\frac{1}{2}$,
$N_{3,\infty}=1$ and $N_{4,\infty}=2$.

Our aim is now to give a similar result when $r$ grows to infinity.

######  Proposition 3.48.

The sequence $\left(\frac{N_{r,e}}{r^{2}}\right)_{r\geq 2}$ converges to
$N_{\infty,e}\coloneqq\max_{\begin{subarray}{c}y\in\mathbb{R}^{{\mathcal{P}_{e}}}\\\
\lVert y\rVert_{1}=1\end{subarray}}q_{e}(y)$.

###### Proof.

By Corollary 3.26, we know that the sequence
$\bigl{(}N_{r,e}/r^{2}\bigr{)}_{r\geq 2}$ is bounded. Hence, to prove that it
converges to $N_{\infty,e}$ it suffices to prove that any converging
subsequence converges to $N_{\infty,e}$. We thus consider a converging
subsequence of $\bigl{(}N_{r,e}/r^{2}\bigr{)}_{r\geq 2}$ and let
$\ell\in\mathbb{R}$ be its limit. To avoid the notation being overloaded, we
still denote this subsequence by $\bigl{(}N_{r,e}/r^{2}\bigr{)}_{r\geq 2}$. By
Proposition 3.14, we have $N_{r,e}/r^{2}\leq N_{\infty,e}$ for any $r\geq 2$
thus $\ell\leq N_{\infty,e}$. We will now prove the reverse inequality.

Let $x=(x_{E})\in\mathbb{R}^{{\mathcal{P}_{e}}}$. Defining $\lvert
x\rvert\coloneqq(\lvert x_{E}\rvert)_{E\in{\mathcal{P}_{e}}}$ we have
$\lVert\lvert x\rvert\rVert_{1}=\lVert x\rVert_{1}$. Since the entries of
$A_{e}$ are non-negative we have $q_{e}(x)\leq q_{e}(\lvert x\rvert)$. Hence,
if $x\in\mathbb{R}^{{\mathcal{P}_{e}}}$ is such that $q_{e}(x)=\max_{\lVert
y\rVert_{1}=1}q_{e}(y)$ then we can assume that $x$ has only non-negative
coordinates. In particular, we can take a sequence of rational vectors with
non-negative entries $(x_{n})_{n\geq 1}$ with limit $x$ and satisfying $\lVert
x_{n}\rVert_{1}=1$ for all $n\geq 1$, where the entries of $x_{n}$ have the
same denominator $b_{n}$. Noting that $b_{n}$ is not necessarily coprime with
its associated numerator, we can always assume that $b_{n}\to\infty$. Then by
Proposition 3.14 we have $N_{b_{n},e}\geq
q_{e}(b_{n}x_{n})=b_{n}^{2}q_{e}(x_{n})$, thus since $q_{e}$ is continuous we
obtain $\ell\geq q_{e}(x)=N_{\infty,e}$. This concludes the proof. ∎

By Corollary 3.26 we have

$\frac{1}{4}\left\lfloor\frac{e}{2}\right\rfloor\leq
N_{\infty,e}\leq\frac{1}{2e}\left\lfloor\frac{e^{2}}{4}\right\rfloor.$

We will be able here to determine the exact value of the limit $N_{\infty,e}$.

######  Corollary 3.49.

For any $e\geq 2$ we have

$N_{\infty,e}=\frac{1}{2e}\left\lfloor\frac{e^{2}}{4}\right\rfloor.$

###### Proof.

Since $Q_{0,e}=N^{\prime}_{0,e}=0$, we deduce from Lemma 3.37 and (3.35) that
$Q_{r,e}=N_{r,e}=N^{\prime}_{r,e}=\bigl{\lfloor}\frac{r^{2}}{2e}\lfloor\frac{e^{2}}{4}\rfloor\bigr{\rfloor}$
as soon as $r$ is of the form $r=\alpha\binom{e}{\lfloor\frac{e}{2}\rfloor}$
for some $\alpha\in\mathbb{Z}_{\geq 1}$. This proves that
$(N_{r,e}/r^{2})_{r}$ has a subsequence that converges to
$\frac{1}{2e}\bigl{\lfloor}\frac{e^{2}}{4}\bigr{\rfloor}$ and this concludes
the proof. ∎

##  4 Application in level one