primarymsc202005E05

# Symmetric functions in noncommuting variables in superspace

Diego Arcis
Camilo González
Sebastián Márquez  Departamento de Matemáticas, Universidad de La Serena,
Cisternas 1200 – 1700000 La Serena, Chile.white$\underbrace{1}$
Departamento de Matemáticas, Universidad de Concepción, Casilla 160–C –
4030000 Concepción, Chile. white$\underbrace{.}$
Departamento de Matemáticas, Universidad Autónoma de Chile, Sede Talca, 4
Norte 95 – 3460000 Talca, Chile<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In 2004, Rosas and Sagan developed the theory of symmetric functions in
noncommuting variables, achieving results analogous to classical symmetric
functions. On the other hand, in 2004, Desrosiers, Lapointe and Mathieu
introduced the theory of symmetric functions in superspace, which involve both
commuting and anticommuting variables, extending the classic theory. Here, we
introduce symmetric functions in noncommuting variables in superspace. We
define the classical symmetric functions in noncommuting variables to
superspace: monomials, power sums, elementaries and complete homogeneous,
which generalize both the ones studied by Rosas and Sagan and the ones studied
by Desrosiers, Lapointe and Mathieu. We also define Schur–type functions in
noncommuting variables in superspace.

##  1 Introduction

Symmetric functions play a significant role in mathematics, in areas such as
algebraic combinatorics, representation theory, Hopf algebras, among others
[23, 31]. They have been widely studied and are important not only in
mathematics, but also in connection with integrable models in physics [24],
particularly, symmetric functions turn out to be eigenfunctions of the
Hamiltonian of the Calogero–Moser–Sutherland model [29, 15]. Roughly speaking,
symmetric functions are formal power series in infinitely many commutative
variables, which remain invariant under any permutation of their variables.
Symmetric functions form an algebra whose bases are indexed by partitions of
integers. Some of these bases are: the monomial symmetric functions, the power
sum symmetric functions, the elementary symmetric functions and the complete
homogeneous symmetric functions.

An important family of symmetric functions that form a basis are the so–called
Schur functions, which appear as limits of characters of the general linear
group and are associated to the irreducible characters of the symmetric group
via the Schur–Weyl duality. Schur functions can be defined through various
methods: by using semistandard Young tableaux, via the Jacobi–Trudy identity,
or directly through a quotient involving Vandermonde–type determinants [31,
Chapter 7].

Several generalizations of the algebra of symmetric functions $\mathrm{\bf
Sym}$ have been studied, such as the algebra of quasisymmetric functions
$\mathrm{\bf QSym}$ [19] and its dual Hopf algebra of noncommutative symmetric
functions $\mathrm{\bf NSym}$ [18], among others [25, 16, 20]. Quasisymmetric
functions can be regarded as a refinement of symmetric functions in the sense
that remain invariant under a weaker form of symmetry.

Another generalization of symmetric functions are the ones involving
noncommuting variables, which were introduced by Wolf in 1936 [32]. The
algebra of symmetric functions in noncommuting variables $\mathrm{\bf NCSym}$
was studied by Rosas and Sagan [27], establishing results similar to the
commutative case. In particular, they defined bases analogous to the monomial,
power, elementary and complete homogeneous functions, which in this case are
indexed by set partitions. They showed relations between these bases and study
their projection to the algebra of symmetric functions in commutative
variables. Also, by using combinatorics of tableaux, they introduced a family
of Schur–type functions in noncommuting variables, satisfying some properties
similar to the classical Schur functions, however this family does not
constitute a basis of $\mathrm{\bf NCSym}$. By using a noncommutative version
of the Jacobi–Trudi determinant, it was introduced in [2] a new family of
Schur–type functions, which refines the one in [27] and constitutes a basis of
$\mathrm{\bf NCSym}$. Several results related to this algebra are known, such
as the isomorphism with the algebra of rook placements [9] and the connection
with supercharacter theory [1, 32]. Some results related to the Hopf structure
of $\mathrm{\bf NCSym}$ were obtained in [5, 4, 6].

Given the connection of symmetric functions with integrable models, and
motivated by a supersymmetric generalization of the Calogero–Moser–Sutherland
model [10, 13], it was developed in [11, 12] a new family of symmetric
functions called _symmetric functions in superspace_ involving commutative
variables $x=(x_{1},x_{2},\ldots)$ together with anticommuting variables
$\theta=(\theta_{1},\theta_{2},\ldots)$, in which the symmetric group acts
simultaneously. These functions have been widely studied in the last years,
extending results of classic symmetric functions. Classic bases as monomial,
power sum, elementary and complete homogeneous symmetric functions were
extended to superspace in [14], which can be are indexed by _superpartitions_.
Several definitions of Schur–type functions in superspace were given in [22].
On the other hand, it was proved in [17] that $\mathrm{\bf QSym}$ and
$\mathrm{\bf NSym}$ can be generalized to superspace as the Hopf algebras of
quasisymmetric functions in superspace $\mathrm{\bf sQSym}$ and of
noncommutative symmetric functions in superspace $\mathrm{\bf sNSym}$,
respectively. Some results related to the Hopf structure of $\mathrm{\bf
sNSym}$ were obtained in [3].

A natural problem is to consider symmetric functions involving noncommuting
and anticommuting variables, which generalize both symmetric functions in
noncommuting variables and symmetric functions in superspace. Here, we lead
with this problem. Specifically, we extend the classical bases and the
Schur–type functions in noncommuting variables to superspace, which project
naturally on their commutative counterpart in superspace. The paper is
organized as follows.

In Section 2, we introduce two new classes of combinatorial objects called
_partial set supercompositions_ and _set superpartitions_ , which generalize
the classic set partitions and are closely related to superpartitions. These
objects will be used to study the bases of symmetric functions in noncommuting
variables in superspace. We define _strongly coarsening_ and _coarsening_
relations on these combinatorial objects, which give them structures of
partially ordered sets. Also, we describe actions of the symmetric group on
these objects.

In Section 3, we introduce the algebra of symmetric functions in noncommuting
variables in superspace $\mathrm{\bf sNCSym}$. We define monomial symmetric
functions in superspace and show that these ones form a basis of $\mathrm{\bf
sNCSym}$. Also, we give combinatorial formulas for the product of these
functions. Other bases of $\mathrm{\bf sNCSym}$ are studied in Section 4. In
particular, we define power sum, elementary and complete homogeneous symmetric
functions in superspace. We also show the relations between these bases.

Section 5 is devoted to study how symmetric functions in noncommuting
variables in superspace project to the algebra functions in commuting
variables in superspace $\mathrm{\bf sSym}$. First, we recall the classic
symmetric functions in commuting variables in superspace. Then, we show
formulas that project our bases in their commuting counterpart, and we prove
that there is an isometry from $\mathrm{\bf sNCSym}$ to $\mathrm{\bf sSym}$.

In Section 6, we define Schur–type functions in noncommuting variables in
superspace in the sense of [27], by using super semistandard Young tableaux,
which were introduced in [22]. We express these functions in terms of the
monomial basis and show their projection on Schur–type functions in
commutative variables in superspace, which were given in [22].

Here, $\mathbb{N}$ denotes the set of positive integers, and
$\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$.

Given integers $m,n$, denote by $[m,n]$ the interval $\\{m,\ldots,n\\}$, and
$[m,n]_{0}:=[m,n]\cup\\{0\\}$. In particular, if $m=1$ we will denote them,
respectively, by $[n]$ and $[n]_{0}$ instead. Note that $[0]$ is empty and
$[0]_{0}=\\{0\\}$.

For every integer $n\geq 1$, we denote permutations $\sigma:[n]\to[n]$ as
words $\sigma(1)\cdots\sigma(n)$ in $[n]$, and we denote by
$\operatorname{inv}(\sigma)$ the _number of inversions_ of $\sigma$, that is,
the minimal number of commutations needed to get $12\cdots n$ from
$\sigma(1)\cdots\sigma(n)$. The _symmetric group_ of permutations of $[n]$,
will be denoted by $\mathfrak{S}_{n}$, and we set
${\displaystyle\mathfrak{S}=\bigsqcup_{n\geq 1}\mathfrak{S}_{n}}$.

##  2 Set superpartitions

Here we introduce a new class of combinatorial objects, which will index the
bases of the algebra of symmetric functions in superspace. These ones can be
regarded as a extension of the classic set partitions. It is worth to mention
that set superpartitions introduced here are different of the ones defined in
[26].

###  2.1 Preliminaries

Recall that a _set partition_ of a set $X$ is a collection $I$ of nonempty
subsets $I_{1},\ldots,I_{k}$ of $X$, called _blocks_ , such that
$I_{1}\cup\cdots\cup I_{k}=X$ and $I_{i}\cap I_{j}=\emptyset$ for all $i<j$.
If $X$ is a set of integers, it is usual to denote
$I=\\{I_{1},\ldots,I_{k}\\}$ by $I=(I_{a_{1}},\ldots,I_{a_{k}})$ instead,
where $\min(I_{a_{i}})<\min(I_{a_{j}})$ whenever $i<j$. The collection of set
partitions of $X$ is denoted by $\mathcal{P}(X)$. For $I,J\in\mathcal{P}(X)$,
write $I\preceq J$ if each block of $J$ is a union of blocks of $I$. It is
well known that $\preceq$ is a partial order that gives to $\mathcal{P}(X)$ a
structure of lattice with _infimum operation_ $\wedge$ and _supremum
operation_ $\vee$. For instance, for $X=[4]$ the lattice structure of
$\mathcal{P}(X)$ is given as in Figure 1.

Figure 1: Images of partial set supercompositions of bidegree $(2,2)$ in the
lattice of set partitions of $[4]$.

For every nonnegative integer $n$, the set $\mathcal{P}([n])$ is commonly
denoted by $\mathcal{P}_{n}$ instead. It is well known that the number of set
partitions of $[n]$ into $1\leq k\leq n$ blocks is the _Stirling number_ (of
the second kind) $S(n,k)$ [21, A008277], and that the number of all set
partitions of $[n]$ is the $n$th Bell number $b_{n}$ [21, A000110].

###  2.2 Partial set supercompositions

In what follows, for every subset $A$ of $\mathbb{N}_{0}$, we denote
$A\backslash\\{0\\}$ by $A^{+}$ instead. Also, for subsets $A,B$ of
$\mathbb{N}_{0}$, we set $A\oplus B$ as empty if $0\in A\cap B$ and $A\oplus
B=A\cup B$ otherwise. This operation is not associative in general, indeed
$(\\{1\\}\oplus\\{0,1\\})\oplus\\{0,1\\}=\\{\\}$ and
$\\{1\\}\oplus(\\{0,1\\}\oplus\\{0,1\\})=\\{1\\}$. However, if either none or
at most one of the sets $A,B,C$ contains zero, then $(A\oplus B)\oplus
C=A\oplus(B\oplus C)$.

A _partial set supercomposition_ $K$, of _degree_ $n=:\deg(K)$ and _fermionic
degree_ $m=:\operatorname{df}(K)\leq n+1$, is a tuple $(K_{1},\ldots,K_{q})$
of nonempty subsets of $[n]_{0}$, called _blocks_ , satisfying the following
conditions:

  1. 1.

$(K_{1}\cup\cdots\cup K_{q})^{+}=[n]$ and $K_{i}\cap K_{j}\subseteq\\{0\\}$
for all $i\neq j$.

  2. 2.

$0\in K_{i}$ for all $i\in[m]$ and $\min(K_{i})<\min(K_{j})$ for all
$m<i<j\leq q$.

Each block of $K$ containing $0$ is called a _fermionic block_. We say that
$(n,m)$ is the _bidegree_ of $K$. The number of blocks of $K$ is called the
_length_ of it and is denoted by $\ell(K)$. The collection $K^{+}$ formed by
the nonempty sets $K_{i}^{+}$ with $i\in[q]$ is a set partition of $[n]$ which
we call the _underlying set partition_ of $K$. For instance, below a partial
set supercomposition of bidegree $(7,3)$:

(2.1) $K=(\\{0,4\\},\\{0,2,7\\},\\{0\\},\\{1,5,6\\},\\{3\\}).$

Observe that, by definition, partial set supercompositions may contain many
blocks $\\{0\\}$. For instance, the following is a partial set
supercomposition of bidegree $(n,m)$:

$\hat{0}_{n,m}=(\underbrace{\\{0\\},\ldots,\\{0\\}}_{m\text{--times}},\\{1\\},\ldots,\\{n\\}).$

The collection of partial set supercompositions of degree $n$ will be denoted
by $s\mathcal{C}_{n}$, and the subcollection of it formed by the ones of
fermionic degree $m$ will be denoted by $s\mathcal{C}_{n,m}$.

For every $K=(K_{1},\ldots,K_{q})\in s\mathcal{C}_{n,m}$ and every pair of
permutations $(\sigma,\delta)\in\mathfrak{S}_{m}\times\mathfrak{S}_{n}$, we
define

$\sigma\triangleright
K=(K_{\sigma(1)},\ldots,K_{\sigma(m)},K_{m+1},\ldots,K_{q})\qquad\text{and}\qquad\delta\circ
K=(K_{1},\ldots,K_{m},\delta(K_{m+1}),\ldots,\delta(K_{q})).$

Observe that the operators $\triangleright$ and $\circ$ define group actions
of $\mathfrak{S}_{n}$ and $\mathfrak{S}_{m}$, respectively, on the collection
of partial set supercompositions of bidegree $(n,m)$. Thus
$\sigma\triangleright(\sigma^{\prime}\triangleright
K)=(\sigma\sigma^{\prime})\triangleright K$ and
$\delta\circ(\delta^{\prime}\circ K)=(\delta\delta^{\prime})\circ K$.

###  2.3 Coarsering and strongly coarsering

Here we stablish two partial orders on partial super set compositions.

Given $K,L\in s\mathcal{C}_{n,m}$, we say that $L$ is _strongly coarser_ than
$K$, or that $K$ is _strongly finer_ than $L$, denoted by $K\sqsubseteq L$, if
the following conditions hold:

  1. 1.

The $i$th fermionic block of $L$ is the $i$th fermionic block of $K$ joined
with some possible nonfermionic blocks of $K$.

  2. 2.

Each nonfermionic block of $L$ is a union of nonfermionic blocks of $K$.

In this case $K$ is called a _strongly refinement_ of $L$. The _strongly
coarsering_ relation $\sqsubseteq$ is a partial order on $s\mathcal{C}_{n,m}$
that gives to it a structure of complete meet–semilattice with operation
$\sqcap$ and infimum $\hat{0}_{n,m}$. See Figure 2. Observe that partial set
supercomposition whose blocks are all fermionic are maximal with respect to
$\sqsubseteq$.

Figure 2: Set superpartitions in the meet–semilattice of partial set
supercompositions of bidegree $(2,2)$.

For $r\in\Z$, the _$r$ -shift_ of $A\subseteq\mathbb{N}_{0}$ is the set $A[r]$
obtained by adding $r$ to each nonzero element of $A$. The _$r$ -shift_ of a
partial set supercomposition $K=(K_{1},\ldots,K_{q})$ is the collection
$K[r]=(K_{1}[r],\ldots,K_{q}[r])$. For instance, if
$K=(\\{0\\},\\{0,3,5\\},\\{1,2,4\\})$ then
$K[2]=(\\{0\\},\\{0,5,7\\},\\{3,4,6\\})$.

The _standardization_ of $K\in s\mathcal{C}_{n,m}$ is the set partition
$\operatorname{std}(K)\in\mathcal{P}_{n+m}$ obtained from $K[m]$ by
renumbering the zeros from $1$ to $m$ with respect to the sorting of the
blocks of $K$. This operator defines an injective order preserving map of
$(s\mathcal{C}_{n,m},\sqsubseteq)$ into $(\mathcal{P}_{n+m},\preceq)$. Observe
that $I\in\mathcal{P}_{n+m}$ belongs to
$\operatorname{std}(s\mathcal{C}_{n,m})$ if and only if the elements of $[m]$
belong to different blocks of $I$. For instance
$\operatorname{std}(\\{0,1\\},\\{0\\},\\{2\\})=(\\{1,3\\},\\{2\\},\\{4\\})$.

######  Proposition 2.1.

The image of $s\mathcal{C}_{n,m}$ under the standardization map is a convex
subposet of $\mathcal{P}_{n+m}$. In consequence, due to [28, Lemma, p. 359],
for every partial set supercomposition $K=(K_{1},\ldots,K_{q})$, we have

$[\hat{0}_{n,m},K]\simeq\prod_{i=1}^{q}\mathcal{P}_{|K_{i}|}.$

###### Proof.

Let $K,L\in s\mathcal{C}_{n,m}$, and let $I\in\mathcal{P}_{n+m}$ such that
$\operatorname{std}(K)\preceq I\preceq\operatorname{std}(L)$. Since $L$
belongs to $s\mathcal{C}_{n,m}$ and $I\preceq\operatorname{std}(L)$, then
every pair of elements of $[m]$ belong to different blocks of $I$. This
implies that $I=\operatorname{std}(N)$ for some $N\in s\mathcal{C}_{n,m}$.
Therefore $\operatorname{std}(s\mathcal{C}_{n,m})$ is a convex subposet of
$\mathcal{P}_{n+m}$. See Figure 1. ∎

For $K,L\in s\mathcal{C}_{n}$, we say that $L$ is _coarser_ than $K$, or that
$K$ is _finer_ than $L$, denoted by $K\preceq L$, if each block of $L$ is
obtained by $\oplus$–operating some blocks of $K$. In this case $K$ is called
a _refinement_ of $L$. Observe that as $L$ contains no empty blocks, then it
cannot be obtained by $\oplus$–operating more than one fermionic block of $K$.
With this _coarsering relation_ , the collection $s\mathcal{C}_{n}$ becomes a
partially ordered set. Note that every partial set supercomposition whose
blocks are all fermionic is maximal with respect to $\preceq$.

###  2.4 Set superpartitions

We say that $I=(I_{1},\ldots,I_{k})\in s\mathcal{C}_{n}$ is a _set
superpartition_ if $I_{i}\neq I_{j}$ and $\min(I_{i}\backslash
I_{j})<\min(I_{j}\backslash I_{i})$ for all $i,j\in[k]$ with $i<j$, where
$\min(\emptyset):=0$. The collection of set superpartitions of degree $n$ will
be denoted by $s\mathcal{P}_{n}$, and the subcollection of it formed by the
ones of fermionic degree $m$ will be denoted by $s\mathcal{P}_{n,m}$.

Observe that every set superpartition that contains no the block $\\{0\\}$ is
obtained by inserting, possibly, a zero in some blocks of a usual set
partition. This implies that, in $s\mathcal{P}_{n}$, there are $sb_{n}^{*}$
[21, A001861] of these set superpartitions. Note that $sb_{n}^{*}$ is obtained
by multiplying every addend of $b_{n}$ by $2^{k}$. Thus
$|s\mathcal{P}_{n}|=2sb_{n}^{*}=:sb_{n}$.

We say that $I\in s\mathcal{P}_{n}$ is _convex_ if each of its blocks is an
interval, that is, for each $B\in I$ there are $a,b\in[n]$ such that, either
$B=[a,b]$ or $B=[a,b]_{0}$. For instance, the following is convex of bidegree
$(7,3)$:

$I=(\\{0\\},\\{0,1,2\\},\\{0,3\\},\\{4,5,7\\},\\{6\\}).$

Observe that every $I\in s\mathcal{P}_{n,m}$ can be written by means
$\circ$–action of a convex set superpartition $J$, that is, $I=\delta\circ J$.
For instance, if $I=(\\{0\\},\\{0,4\\},\\{0,6\\},\\{1,3,5\\},\\{2\\})$, then

$I=\sigma\circ\underbrace{(\\{0\\},\\{0,1\\},\\{0,2\\},\\{3,4,5\\},\\{6\\})}_{J},\quad\text{where}\quad\sigma=461352.$

For every partial set supercomposition $K$, we will denote by $\bar{K}$ the
unique set superpartition obtained from $K$ by removing its possible repeated
blocks and so sorting the fermionic blocks. For instance:

$K=(\\{0,3\\},\\{0\\},\\{0,2\\},\\{0\\},\\{1,5\\},\\{4\\})\quad\to\quad\bar{K}=(\\{0\\},\\{0,2\\},\\{0,3\\},\\{1,5\\},\\{4\\}).$

We say that $K\in s\mathcal{C}_{n,m}$ is _nontrivial_ if it has at most one
block $\\{0\\}$, that is, there are $\sigma\in\mathfrak{S}_{m}$ and $I\in
s\mathcal{P}_{n,m}$ such that $K=\sigma\triangleright I$. See Figure 3.

Figure 3: Nontrivial elements in the meet–semilattice of partial set
supercompositions of bidegree $(2,2)$.

Observe that as nontrivial partial set supercompositions are obtained by
resorting the fermionic blocks of set superpartitions, then $K\sqsubseteq L$
implies that $\bar{K}\preceq\bar{L}$.

In what follows, for every $A\subseteq[n]_{0}$ and every collection
$K=(K_{1},\ldots,K_{q})$ of subsets of $[n]_{0}$, we set

$K\backslash A=(K_{1}\backslash A,\ldots,K_{q}\backslash
A)\,\backslash\,\emptyset.$

######  Proposition 2.2.

Given $K=(K_{1},\ldots,K_{h})$ and $L=(L_{1},\ldots,L_{k})$ in
$s\mathcal{C}_{n,m}$, we have $K\sqcap L=(N_{1},\ldots,N_{q})$, where
$N_{i}=K_{i}\cap L_{i}$ for all $i\in[m]$, and
$(N_{m+1},\ldots,N_{q})=(K^{+}\backslash A)\wedge(L^{+}\backslash A)$ with
$A=N_{1}\cup\cdots\cup N_{m}$.

###### Proof.

By definition, we have $N\sqsubseteq K$ and $N\sqsubseteq L$, where
$N=(N_{1},\ldots,N_{q})$ as on the statement. Thus $N\sqsubseteq K\sqcap L$.
Assume that $N\sqsubset K\sqcap L$, then $K\sqcap L$ contains a block
$B=N_{i}\cup N_{j}$ for some $i,j\in[q]$ with $i<j$. Now, if $N_{i}$ is
fermionic, then $B\subset K_{i}$ and $B\subset L_{i}$, which is a
contradiction because $N_{i}=K_{i}\cap L_{i}$. On the other hand, if $N_{i}$
is not fermionic, then $B\cap A=\emptyset$ and $B\subseteq K_{a}\cap H_{b}$
for some $a,b\geq m$. This implies that $B$ is contained in some block of
$(K^{+}\backslash A)\wedge(L^{+}\backslash A)$, which also is a contradiction.
Therefore $N=K\sqcap L$. ∎

##  3 Functions in superspace

In this section, we define the algebra of symmetric functions in noncommuting
variables in superspace. We introduce monomial symmetric functions in
superspace and show that these ones form a basis of it. Also, we give
combinatorial formulas for the product of these functions.

Let $\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle x\rangle\kern-2.5pt\rangle$
be the $\Q$–algebra of formal power series of bounded degree presented by
infinite generators $x_{1},x_{2},\ldots$ and $\theta_{1},\theta_{2},\ldots$,
called _variables_ , subject to the relations
$x_{i}\theta_{j}=\theta_{j}x_{i}$ and
$\theta_{i}\theta_{j}=-\theta_{j}\theta_{i}$ for all $i,j\in\mathbb{N}$. Every
element of $\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle
x\rangle\kern-2.5pt\rangle$ will be called a _function (in noncommuting
variables) in superspace_. Generators in $x:=(x_{1},x_{2},\ldots)$ will be
called _$x$ –variables_ and generators in
$\theta:=(\theta_{1},\theta_{2},\ldots)$ will be called _fermionic variables_
or simply _$\theta$ –variables_. Observe that $\theta_{i}^{2}=0$ for all
$i\in\mathbb{N}$.

Due to the relations that define
$\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle x\rangle\kern-2.5pt\rangle$,
every monomial occurring in a function in superspace can be uniquely written
as a product $q\theta_{a_{1}}\cdots\theta_{a_{r}}x_{b_{1}}\cdots x_{b_{s}}$ up
the relations that define $\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle
x\rangle\kern-2.5pt\rangle$, where $q\in\Q$ and $a_{i},b_{j}\in\mathbb{N}$.
For every nonnegative integer $n$ we will denote by
$\Q_{n}^{\theta}\langle\kern-2.5pt\langle x\rangle\kern-2.5pt\rangle$ the
subspace formed by the functions in superspace whose monomials have exactly
$n$ occurrences of $x$–variables. Similarly, for every nonnegative integer $m$
we denote by $\Q_{n,m}^{\theta}\langle\kern-2.5pt\langle
x\rangle\kern-2.5pt\rangle$ the subspace of functions in
$\Q_{n}^{\theta}\langle\kern-2.5pt\langle x\rangle\kern-2.5pt\rangle$ whose
monomials have exactly $m$ occurrences of $\theta$–variables. Hence, we obtain
the following decompositions:

$\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle
x\rangle\kern-2.5pt\rangle=\bigoplus_{n\geq
0}\Q_{n}^{\theta}\langle\kern-2.5pt\langle
x\rangle\kern-2.5pt\rangle=\bigoplus_{n,m\geq
0}\\!\Q_{n,m}^{\theta}\langle\kern-2.5pt\langle x\rangle\kern-2.5pt\rangle.$

We call $n$ the _degree_ and $m$ the _fermionic degree_ of the elements of
$\Q_{n,m}^{\theta}\langle\kern-2.5pt\langle x\rangle\kern-2.5pt\rangle$.

###  3.1 Symmetric functions in superspace

For every $n\geq 1$ the group $\mathfrak{S}_{n}$ acts linearly on
$\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle x\rangle\kern-2.5pt\rangle$ via
permutation of variables, that is, $s(x_{i})=x_{s(i)}$ and
$s(\theta_{i})=\theta_{s(i)}$ for all
$(s,i)\in\mathfrak{S}_{n}\times\mathbb{N}$, where $s(i):=i$ whenever $i>n$. An
element of $\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle
x\rangle\kern-2.5pt\rangle$ is called _symmetric_ if it is invariant under the
action of $\mathfrak{S}_{n}$ for all $n\in\mathbb{N}$. We will denote by
$\mathrm{\bf sNCSym}$ the subalgebra of
$\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle x\rangle\kern-2.5pt\rangle$
formed by the symmetric functions in superspace. Thus, we have

(3.1) $\mathrm{\bf sNCSym}=\bigoplus_{n\geq 0}\mathrm{\bf
sNCSym}_{n}=\bigoplus_{n,m\geq 0}\mathrm{\bf sNCSym}_{n,m},$

where $\mathrm{\bf sNCSym}_{n}=\mathrm{\bf
sNCSym}\cap\Q_{n}^{\theta}\langle\kern-2.5pt\langle
x\rangle\kern-2.5pt\rangle$ and $\mathrm{\bf sNCSym}_{n,m}=\mathrm{\bf
sNCSym}\cap\Q_{n,m}^{\theta}\langle\kern-2.5pt\langle
x\rangle\kern-2.5pt\rangle$ for all $n,m\in\mathbb{N}_{0}$.

Clearly, the action of the symmetric group on $\mathrm{\bf sNCSym}$ keep the
number of occurrences of variables in monomials, that is, $\sigma(\mathrm{\bf
sNCSym}_{n,m})\subseteq\mathrm{\bf sNCSym}_{n,m}$ for all permutation $\sigma$
and all $n,m\in\mathbb{N}_{0}$.

A monomial $q\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots x_{b_{n}}$ is
said to be _null–symmetric_ if
$\\{a_{1},\ldots,a_{m}\\}\backslash\\{b_{1},\ldots,b_{n}\\}$ has more than one
element. For instance, the monomial $\theta_{1}\theta_{2}x_{2}$ is non
null–symmetric, however $\theta_{1}\theta_{3}x_{2}$ is null–symmetric.

######  Proposition 3.1.

There are no symmetric functions in superspace involving null–symmetric
monomials. In consequence, $\mathrm{\bf sNCSym}_{n,m}=\\{0\\}$ for all
$n,m\in\mathbb{N}_{0}$ with $m>n+1$. Thus,

$\mathrm{\bf sNCSym}_{n}=\bigoplus_{m=0}^{n+1}\mathrm{\bf sNCSym}_{n,m}.$

###### Proof.

Let $f$ be a symmetric function in superspace, and let
$u=q\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots x_{b_{n}}$ be a monomial
occurring in it. If $u$ is null–symmetric, there are $i,j\in[m]$ such that
$a_{i},a_{j}\not\in\\{b_{1},\ldots,b_{n}\\}$. Consider $\sigma$ be the
transposition exchanging $a_{i}$ with $a_{j}$. As $f$ is symmetric,
$\sigma(u)=-u$ also occurs in $f$. This holds for all null–symmetric monomial
occurring in $f$. The equality $\mathrm{\bf sNCSym}_{n,m}=\\{0\\}$ when
$m>n+1$ is a consequence of the fact that every monomial with $n$
$x$–variables and $m>n+1$ $\theta$–variables is null–symmetric. ∎

In order to study the bases of $\mathrm{\bf sNCSym}$, we will consider other
actions of the symmetric group on
$\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle x\rangle\kern-2.5pt\rangle$ in
which $\mathfrak{S}_{n}$ acts independently on the $x$–variables and on the
$\theta$–variables. More specifically, given a monomial
$u=q\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots x_{b_{n}}$ and
permutations $(\sigma,\delta)\in\mathfrak{S}_{m}\times\mathfrak{S}_{n}$, we
define

(3.2) $\sigma\triangleright
u=q\theta_{a_{\sigma(1)}}\cdots\theta_{a_{\sigma(m)}}x_{b_{1}}\cdots
x_{b_{n}}\qquad\text{and}\qquad\delta\circ
u=q\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{\delta^{-1}(1)}}\cdots
x_{b_{\delta^{-1}(n)}}.$

Observe that, as $\theta$–variables anticommute, we have $\sigma\triangleright
u=(-1)^{\operatorname{inv}(\sigma)}u$.

###  3.2 Monomial functions in superspace

Here we introduce the basis of monomial functions in superspace.

######  Definition 3.2.

The _monomial function_ of a partial set supercomposition
$K=(K_{1},\ldots,K_{q})$ of bidegree $(n,m)$ is the function in superspace,
defined as follows:

$m_{K}\,\,\,\,=\sum_{(a_{1},\ldots,a_{m},b_{1},\ldots,b_{n})}\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots
x_{b_{n}},$

where

  1. 1.

For each $i\in[m]$, $a_{i}=b_{j}$ if and only if $j\in K_{i}^{+}$.

  2. 2.

For each $i,j\in[n]$, $b_{i}=b_{j}$ if and only if $i,j$ belong to the same
block of $K$.

We will denote by $M(K)$ the set of all monomials satisfying conditions (1)
and (2) above.

For instance, if $K=(\\{0,2\\},\\{0,1,3\\},\\{4\\})$ in $s\mathcal{C}_{4,2}$,
we have

$m_{K}=\theta_{2}\theta_{1}x_{1}x_{2}x_{1}x_{3}+\theta_{3}\theta_{1}x_{1}x_{3}x_{1}x_{2}+\theta_{1}\theta_{2}x_{2}x_{1}x_{2}x_{3}+\theta_{3}\theta_{2}x_{2}x_{3}x_{2}x_{1}+\theta_{1}\theta_{3}x_{3}x_{1}x_{3}x_{2}+\theta_{2}\theta_{3}x_{3}x_{2}x_{3}x_{1}+\cdots$

######  Proposition 3.3.

For every partial set supercomposition $K$, the function $m_{K}$ is symmetric.
Moreover, for every monomial $u\in M(K)$, we have

$m_{K}=\sum_{\sigma\in\mathfrak{S}}\sigma(u).$

###### Proof.

It is a direct consequence of the fact that $\sigma(M(K))=M(K)$ for all
$\sigma\in\mathfrak{S}$. ∎

Observe that if $K\in s\mathcal{C}_{n,m}$ contains at least two blocks
$\\{0\\}$, then $m_{K}$ involves only null–symmetric monomials, thus, by
Proposition 3.1, $m_{K}=0$. Now, recall that if $K$ is nontrivial, then
$K=\sigma\triangleright I$ for some $\sigma\in\mathfrak{S}_{m}$ and $I\in
s\mathcal{P}_{n,m}$. Thus, the monomial function $m_{K}$ can be characterized
by applying the $\triangleright$–action of $\sigma$, that is

$\sigma\triangleright
m_{I}\,\,\,\,=\sum_{(a_{1},\ldots,a_{m},b_{1},\ldots,b_{n})}\theta_{a_{\sigma(1)}}\cdots\theta_{a_{\sigma(m)}}x_{b_{1}}\cdots
x_{b_{n}}=m_{\sigma\,\triangleright\,I}=m_{K}.$

######  Proposition 3.4.

We have

  1. 1.

Every non null–symmetric monomial belongs to a unique $M(\sigma\triangleright
I)$.

  2. 2.

The set $\\{m_{\sigma\,\triangleright\,I}\mid\sigma\in\mathfrak{S}_{m},\,I\in
s\mathcal{P}_{n,m}\\}$ spans $\mathrm{\bf sNCSym}_{n,m}$.

  3. 3.

For every $I\in s\mathcal{P}_{n,m}$ and $\sigma\in\mathfrak{S}_{m}$,
$m_{\sigma\,\triangleright\,I}=(-1)^{\operatorname{inv}(\sigma)}m_{I}$.

  4. 4.

The set $\\{m_{I}\mid I\in s\mathcal{P}_{n,m}\\}$ is a basis of $\mathrm{\bf
sNCSym}_{n,m}$.

###### Proof.

Let $u=\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots x_{b_{n}}$ be a non
null–symmetric monomial, let $I^{+}$ be the unique set partition of $[n]$ in
which $i,j$ belong to the same block if and only if $b_{i}=b_{j}$, and let $I$
to be the unique set superpartition of $[n]$ obtained by adding $0$ to each
block of $I$ containing $i$ such that $b_{i}\in\\{a_{1},\ldots,a_{m}\\}$. As
$\theta$–variables are not necessarily sorted as the blocks of $I$, then there
is a unique permutation $\sigma\in\mathfrak{S}_{m}$ such that $u\in
M(\sigma\triangleright I)$.

For every $f\in\mathrm{\bf sNCSym}_{n,m}$, Proposition 3.1 and Proposition 3.3
imply that if $qu$ occurs in $f$ for some $q\in\Q$ and $u\in
M(\sigma\triangleright I)$ with $(\sigma,\,I)\in\mathfrak{S}_{m}\times
s\mathcal{P}_{n,m}$ as in (1), then $qm_{J}$ occurs in $f$ as well. As it
holds for every monomial occurring in $f$ and the set $\mathfrak{S}_{m}\times
s\mathcal{P}_{n,m}$ is finite, then $f$ is a linear combination of monomial
functions in
$\\{m_{\sigma\,\triangleright\,I}\mid\sigma\in\mathfrak{S}_{m},\,I\in
s\mathcal{P}_{n,m}\\}$. Thus (2) holds.

To show (3), it is enough to note that, by Definition 3.2, $m_{I}$ is obtained
by permuting by $\sigma^{-1}$ the $\theta$–variables of every monomial of
$M(\sigma\triangleright I)$, that is,
$m_{I}=(-1)^{\operatorname{inv}(\sigma)}m_{\sigma\,\triangleright\,I}$.

Finally, since $M(I)\cap M(J)$ is empty for all $I,J\in s\mathcal{P}_{n,m}$
with $I\neq J$, then $\\{m_{I}\mid I\in s\mathcal{P}_{n,m}\\}$ is linearly
independent. So, as a consequence of (2) and (3), we get that it is a basis of
$\mathrm{\bf sNCSym}_{n,m}$. ∎

###  3.3 Product

Let $K=(K_{1},\ldots,K_{h})$ and $L=(L_{1},\ldots,L_{k})$ be partial set
supercompositions. The _over product_ of $K$ with $L$ is the partial set
supercomposition $K/L=K\cup L[n]$, where . Observe that,

$\deg(K/L)=\deg(K)+\deg(L),\qquad\operatorname{df}(K/L)=\operatorname{df}(K)+\operatorname{df}(L),\qquad\ell(K/L)=\ell(K)+\ell(L).$

For instance, below the over product of two set superpartitions:

$(\\{0\\},\\{0,4\\},\\{1,2,3\\})/(\\{0\\},\\{0,3\\},\\{1,2\\})=(\\{0\\},\\{0,4\\},\\{0\\},\\{0,7\\},\\{1,2,3\\},\\{5,6\\}).$

We will denote by $I\shuffle J$ the collection of nontrivial partial set
supercompositions obtained from $I/J$, possibly by $\oplus$–operating some
pairs of blocks $I_{i}\oplus J_{j}[n]$ maintaining the order, of the fermionic
blocks, inherited from $I/J$. Note that $K\sqsupseteq I/J$ for all $K\in
I\shuffle J$, and $I/J\in I\shuffle J$ if and only if $\\{0\\}\not\in I\cap
J$. For instance, if $I=(\\{0\\},\\{0,3\\},\\{1,2\\})$ and
$J=(\\{{\color{blue}0},{\color{blue}2}\\},\\{{\color{blue}1}\\})$, the
collection $I\shuffle J$ is formed by the following elements.

$\begin{array}[]{c}(\\{0\\},\\{0,3\\},\\{{\color{blue}0},{\color{blue}5}\\},\\{1,2\\},\\{{\color{blue}4}\\})\\\\[5.69046pt]
\begin{array}[]{c}(\\{0,{\color{blue}4}\\},\\{0,3\\},\\{{\color{blue}0},{\color{blue}5}\\},\\{1,2\\})\\\
(\\{0\\},\\{0,3,{\color{blue}4}\\},\\{{\color{blue}0},{\color{blue}5}\\},\\{1,2\\})\\\
(\\{0\\},\\{0,3\\},\\{{\color{blue}0},1,2,{\color{blue}5}\\},\\{{\color{blue}4}\\})\\\
(\\{0\\},\\{0,3\\},\\{{\color{blue}0},{\color{blue}5}\\},\\{1,2,{\color{blue}4}\\})\\\
\end{array}\qquad\begin{array}[]{c}(\\{0,{\color{blue}4}\\},\\{0,3\\},\\{{\color{blue}0},1,2,{\color{blue}5}\\})\\\
(\\{0\\},\\{0,3,{\color{blue}4}\\},\\{{\color{blue}0},1,2,{\color{blue}5}\\})\\\
(\\{0,{\color{blue}4}\\},\\{0,3\\},\\{{\color{blue}0},1,2,{\color{blue}5}\\})\\\
(\\{0\\},\\{0,3,{\color{blue}4}\\},\\{{\color{blue}0},1,2,{\color{blue}5}\\})\end{array}\end{array}$

As it was shown in [5, Proposition 3.3] for $\mathrm{\bf NCSym}$. In
superspace, the collection of monomial functions, indexed by set
superpartitions, can be described as follows.

######  Proposition 3.5.

For every pair of set superpartitions $I,J$, we have

$m_{I}m_{J}=\sum_{K\in I\shuffle J}m_{K}.$

###### Proof.

Let $(n,m)$ and $(p,q)$ be the bidegrees of $I$ and $J$ respectively. As
$x$–variables commute with $\theta$–variables, we have

$\begin{array}[]{rcl}m_{I}m_{J}&=&{\displaystyle\left(\sum_{u\in
M(I)}\underbrace{\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots
x_{b_{n}}}_{u}\right)\\!\left(\sum_{v\in
M(J)}\underbrace{\theta_{c_{1}}\cdots\theta_{c_{q}}x_{d_{1}}\cdots
x_{d_{p}}}_{v}\right)},\\\\[22.76228pt] &=&{\displaystyle\sum_{uv\in
M(I)M(J)}\underbrace{\theta_{a_{1}}\cdots\theta_{a_{m}}\theta_{c_{1}}\cdots\theta_{c_{q}}x_{b_{1}}\cdots
x_{b_{n}}x_{d_{1}}\cdots x_{d_{p}}}_{uv}}.\end{array}$

Observe that, in the sum above, the coefficient of $uv\in M(I)M(J)$ is zero,
if and only if, either $a_{i}=c_{j}$ for some $i,j$, or there are $i,j$ such
that $a_{i}\neq b_{k}$ for all $k\in[n]$ and $c_{i}\neq d_{k}$ for all
$k\in[p]$.

Assume that the coefficient of $uv$ is nonzero, and let $K$ be the partial set
supercomposition such that $uv\in M(K)$. If
$\\{a_{1},\ldots,a_{m},b_{1},\ldots,b_{n}\\}\cap\\{c_{1},\ldots,c_{q},d_{1},\ldots,d_{p}\\}$
is empty, then $K=I/J$. Otherwise, we have $a_{i}=d_{j}$ or $b_{i}=c_{j}$ or
$b_{i}=d_{j}$ for some $i,j$. Then, the block of $K$ containing $i$ and $j+n$
is given by $A\cup B[n]$, where $A$ is the block of $I$ containing $i$, and
$B$ is the block of $J$ containing $j$. As the coefficient of $uv$ is nonzero,
then $A,B$ are not fermionic at once. Therefore $K\in I\shuffle J$.

Reciprocally, by definition each $K\in I\shuffle J$ maintains the order of the
fermionic blocks of $I/J$. This implies that every $w\in M(K)$ can be written
as a product $w=uv$, where $u\in M(I)$ and $v\in M(J)$. ∎

##  4 Other bases

In this section, we introduce power sum, elementary and complete homogeneous
symmetric functions in superspace, which constitute bases of $\mathrm{\bf
NCSym}$. Also, we give explicit relations between these bases.

###  4.1 Power sum functions in superspace

Here we introduce the basis of power sum functions in superspace.

######  Definition 4.1.

The _power sum function_ of a partial set supercomposition
$K=(K_{1},\ldots,K_{q})$ of bidegree $(n,m)$ is the function in superspace,
defined as follows:

$p_{K}\,\,\,\,=\sum_{(a_{1},\ldots,a_{m},b_{1},\ldots,b_{n})}\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots
x_{b_{n}},$

where

  1. 1.

For each $i\in[m]$, $a_{i}=b_{j}$ if $j\in K_{i}^{+}$.

  2. 2.

For each $i,j\in[n]$, $b_{i}=b_{j}$ if $i,j$ belong to the same block of $K$.

We will denote by $P(K)$ the set of all monomials satisfying conditions (1)
and (2) above.

For instance, for the set superpartition $K=(\\{0,2\\},\\{0,1,3\\},\\{4\\})$
in $s\mathcal{C}_{4,2}$, we have

$p_{I}=\theta_{2}\theta_{1}x_{1}x_{2}x_{1}^{2}+\theta_{1}\theta_{2}x_{2}x_{1}x_{2}x_{1}+\theta_{2}\theta_{1}x_{1}x_{2}x_{1}x_{2}+\theta_{1}\theta_{2}x_{2}x_{1}x_{2}^{2}+\theta_{3}\theta_{2}x_{2}x_{3}x_{2}^{2}+\theta_{2}\theta_{3}x_{3}x_{2}x_{3}x_{2}+\cdots$

######  Proposition 4.2.

For every nontrivial partial set supercomposition $K$ of bidegree $(n,m)$, we
have

$p_{K}=\sum_{L\,\sqsupseteq\,K}m_{L}.$

###### Proof.

Let $K\in s\mathcal{C}_{n,m}$. We must show that $P(K)$ is the union of $M(L)$
such that $L\sqsupseteq K$.

Let $u=\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots x_{b_{n}}\in P(K)$.
Since $K$ has at most one trivial fermionic block, then $u$ is non
null–symmetric. So, by Proposition 3.4(1), there is a unique nontrivial $L\in
s\mathcal{C}_{n,m}$ such that $u\in M(L)$. By Definition 4.1(1) and Definition
3.2(1), for each $i\in[m]$, we have $a_{i}=b_{j}$ for all $j\in K_{i}^{+}$.
Thus $K_{i}\subseteq L_{i}$. On the other hand, Definition 4.1(2) and
Definition 3.2(2) imply that each nonfermionic block of $K$ is contained in
one of $L$. Therefore $L\sqsupseteq K$.

Conversely, let $L\in s\mathcal{C}_{n,m}$ such that $L\sqsupseteq K$, and let
$u=\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots x_{b_{n}}\in M(L)$. By
definition, for each $i\in[m]$, we have $K_{i}\subseteq L_{i}$ and
$a_{i}=b_{j}$ for all $j\in L_{i}^{+}$. Thus $a_{i}=b_{j}$ for all $j\in
K_{i}^{+}$. Further, if $b_{i},b_{j}$ belong to the same block of $K$ for some
$i,j\in[n]$, then $b_{i},b_{j}$ belong to the same block of $L$ because
$L\sqsupseteq K$. These facts imply that $u$ satisfies conditions (1) and (2)
of Definition 4.1. Therefore $u\in P(K)$. ∎

######  Remark 4.3.

As $I\preceq J$ for some $I,J\in s\mathcal{P}_{n,m}$ implies that each
fermionic block of $J$ contains a unique fermionic block of $I$, we may
consider $\sigma\\!_{{}_{(I,\,J)}}$ to be the permutation given by these
contentions, that is, $I_{i}\subseteq J_{\sigma\\!_{{}_{(I,\,J)}}(i)}$ for all
$i\in[m]$. The number of inversions of $\sigma\\!_{{}_{(I,\,J)}}$ will be
denoted by $\operatorname{inv}(I,J)$. For instance,

$I=(\stackrel{{\scriptstyle 1}}{{\\{0\\}}},\stackrel{{\scriptstyle
2}}{{\\{0,3\\}}},\stackrel{{\scriptstyle
3}}{{\\{0,5\\}}},\\{1\\},\\{2,4\\}),\quad J=(\stackrel{{\scriptstyle
1}}{{\\{{\color{red}0},1,{\color{red}5}\\}}},\stackrel{{\scriptstyle
2}}{{\\{{\color{red}0},2,4\\}}},\stackrel{{\scriptstyle
3}}{{\\{{\color{red}0},{\color{red}3}\\}}}),\quad\sigma\\!_{{}_{(I,\,J)}}=231,\quad\operatorname{inv}(I,J)=2.$

Observe that $I\preceq J$ for some $I,J\in s\mathcal{P}_{n,m}$ do not imply
that $I\sqsubseteq J$, however $\sigma\triangleright I\sqsubseteq J$ and
$I\sqsubseteq\sigma^{-1}\triangleright J$, where $\sigma=\sigma_{{}_{(I,J)}}$.
Moreover if $I\sqsubseteq K\sqsubseteq\sigma^{-1}\triangleright J$ for some
$K\in s\mathcal{C}_{n,m}$, then $\sigma\triangleright
I\sqsubseteq\sigma\triangleright K\sqsubseteq J$. Reciprocally if
$\sigma\triangleright I\sqsubseteq K\subseteq J$, then
$I\sqsubseteq\sigma^{-1}\triangleright K\sqsubseteq\sigma^{-1}\triangleright
J$. For instance, $I=(\\{0\\},\\{0,1\\},\\{2\\})$ is not strongly finer than
$J=(\\{0,1\\},\\{0,2\\})$, see Figure 3, however, $21\triangleright
I\sqsubseteq J$ and $I\sqsubseteq 21\triangleright J$.

Due to Proposition 4.2 and Remark 4.3 we obtain the following consequence.

######  Corollary 4.4.

For every set superpartition $I$ of bidegree $(n,m)$, we have

$p_{I}=\sum_{J\,\succeq I}(-1)^{\operatorname{inv}(I,\,J)}m_{J}.$

###  4.2 Elementary functions in superspace

Here we introduce the basis of elementary functions in superspace.

######  Definition 4.5.

The _elementary function_ of a partial set supercomposition
$K=(K_{1},\ldots,K_{q})$ of bidegree $(n,m)$ is the function in superspace,
defined as follows:

$e_{K}\,\,\,\,=\sum_{(a_{1},\ldots,a_{m},b_{1},\ldots,b_{n})}\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots
x_{b_{n}},$

where

  1. 1.

For each $i\in[m]$, $a_{i}\neq b_{j}$ if $j\in K_{i}^{+}$.

  2. 2.

For each $i,j\in[n]$, $b_{i}\neq b_{j}$ if $b_{i},b_{j}$ belong to the same
block of $K$.

We will denote by $E(K)$ the set of all monomials satisfying conditions (1)
and (2) above.

######  Proposition 4.6.

For every $K=(K_{1},\ldots,K_{q})\in s\mathcal{C}_{n,m}$, we have

$e_{K}\,\,\,=\\!\sum_{K\,\sqcap\,L\,=\,\hat{0}_{n,m}}\\!\\!m_{L}.$

###### Proof.

Let $L=(L_{1},\ldots,L_{r})\in s\mathcal{C}_{n,m}$ satisfying $K\sqcap
L=\hat{0}_{n,m}$. Proposition 2.2 implies that $K_{i}\cap L_{i}=\\{0\\}$ if
$i\in[m]$ and $K^{+}\wedge L^{+}=\hat{0}$. Consider
$u=\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots x_{b_{n}}$ be a monomial
in $M(L)$. If $a_{i}=b_{j}$ for some $i\in[m]$ and $j\in K_{i}^{+}$,
Definition 3.2(1) implies that $j\in L_{i}$, which is impossible because
$K_{i}\cap L_{i}=\\{0\\}$. Thus $a_{i}\neq b_{j}$. Now, if $b_{i}=b_{j}$ for
some $i,j$ belonging to a block of $K$, Definition 3.2(2) implies that $i,j$
belong to the same block of $L$, which contradicts the fact that $K^{+}\wedge
L^{+}=\hat{0}$. Thus $b_{i}\neq b_{j}$. Therefore $u\in E(K)$.

Conversely, consider $u=\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots
x_{b_{n}}$ be a monomial in $E(K)$. Since $K$ has at most one trivial
fermionic block, then $u$ is non null–symmetric. So, by Proposition 3.4(1),
there is a unique $L\in s\mathcal{C}_{n,m}$ such that $u\in M(L)$. Let $j\in
K_{i}\cap L_{i}$ for some $i\in[m]$. If $j>0$, Definition 3.2(1) implies that
$a_{i}=b_{j}$ which contradicts Definition 4.5(1). Thus $j=0$. Now, if $i,j$
belong to the same block of $K^{+}\wedge L^{+}$, Definition 3.2(2) implies
that $b_{i}=b_{j}$ which contradicts Definition 4.5(2). Therefore $K^{+}\wedge
L^{+}=\hat{0}$. ∎

As a consequence, by using Proposition 3.4(3), we have the following
characterization in terms of set superpartitions.

######  Corollary 4.7.

For every set superpartition $I=(I_{1},\ldots,I_{k})$ with
$\operatorname{df}(I)=m$, we have

$e_{I}=\sum_{(\sigma,\,J)}(-1)^{\operatorname{inv}(\sigma)}m_{J},$

where $J=(J_{1},\ldots,J_{q})$ with $\operatorname{df}(J)=m$ satisfying
$I^{+}\wedge J^{+}=\hat{0}$ and $I_{i}\cap J_{\sigma(i)}=\\{0\\}$ for all
$i\in[m]$.

For instance, for the set superpartition $I=(\\{0\\},\\{0,2\\},\\{1,3\\})$ in
$s\mathcal{P}_{3,2}$, we have

$\begin{array}[]{rcl}e_{I}&=&m_{{}_{(\\{0,1,2\\},\\{0,3\\})}}-m_{{}_{(\\{0\\},\\{0,1,2\\},\\{3\\})}}-m_{{}_{(\\{0,1\\},\\{0,2,3\\})}}-m_{{}_{(\\{0\\},\\{0,2,3\\},\\{1\\})}}\\\
&&\textcolor{white}{.}-m_{{}_{(\\{0\\},\\{0,2\\},\\{1\\},\\{3\\})}}-m_{{}_{(\\{0,1\\},\\{0,2\\},\\{3\\})}}+m_{{}_{(\\{0,2\\},\\{0,3\\},\\{1\\})}}.\end{array}$

###  4.3 Complete homogeneous functions in superspace

Here we introduce the basis of complete homogeneous functions in superspace.

######  Definition 4.8.

The _complete homogeneous function_ of a partial set supercomposition
$K=(K_{1},\ldots,K_{q})$ of bidegree $(n,m)$ is the function in superspace,
defined as follows:

$h_{K}\,\,\,\,=\sum_{u}c_{u}u$

where every $u$ is a non null-symmetric monomial with $m$ $\theta$–variables
and $n$ $x$–variables, and $c_{u}$ is the number of ways to sort the elements
of the blocks of $K\sqcap L$ simultaneously, with $L$ the unique partial set
supercomposition of bidegree $(n,m)$ satisfying $u\in M(L)$.

We will denote by $H(I)$ the set of all monomials satisfying the conditions of
Definition 4.8.

For every $A=(A_{1},\ldots,A_{k})$, with $A_{1},\ldots,A_{k}$ subsets of
$[n]_{0}$, we define $A!=|A_{1}|!\cdots|A_{k}|!$.

######  Proposition 4.9.

For every partial set supercomposition $K$, we have

$h_{K}=\sum_{L}(K\sqcap L)!m_{L},$

where $L$ is a partial set supercomposition with the same bidegree of $K$.

###### Proof.

It is a direct consequence of fact that $c_{u}=(K\sqcap L)!$ and $c_{u}=c_{v}$
for all $u,v\in M(L)$. ∎

For $I,J\in s\mathcal{P}_{n,m}$ and $\sigma\in\mathfrak{S}_{m}$, we set
$I\wedge_{\sigma}J=I\sqcap(\sigma\triangleright J)$. By Proposition 2.2, we
can describe $I\wedge_{\sigma}J$ as the tuple $(N_{1},\ldots,N_{q})$, where
$N_{i}=I_{i}\cap J_{\sigma(i)}$ for all $i\in[m]$ and
$(N_{m+1},\ldots,N_{q})=(I\backslash A)\wedge(J\backslash A)$ with
$A=N_{1}\cup\cdots\cup N_{m}$. For instance, if
$I=(\\{0,1,3\\},\\{0,2\\},\\{4,5\\})$ and
$J=(\\{0,2,4\\},\\{0,3\\},\\{1\\},\\{5\\})$, then:

$A=\\{0,2,3\\},\quad I\backslash A=(\\{1\\},\\{4,5\\}),\quad J\backslash
A=(\\{1\\},\\{4\\},\\{5\\}),\quad
I\wedge_{21}J=(\\{0,3\\},\\{0,2\\},\\{1\\},\\{4\\},\\{5\\}).$

Thus, we obtain the following consequence.

######  Corollary 4.10.

For every set superpartition $I\in s\mathcal{P}_{n,m}$, we have

$h_{I}=\sum_{(\sigma,\,J)\in\mathfrak{S}_{m}\times
s\mathcal{P}_{n,m}}(I\wedge_{\sigma}J)!(-1)^{\operatorname{inv}(\sigma)}m_{J}.$

For instance, for the set superpartition $I=(\\{0\\},\\{0,2\\},\\{1,3\\})$ in
$s\mathcal{P}_{3,2}$, we have

$\begin{array}[]{rcl}h_{I}&=&2m_{{}_{(\\{0\\},\\{0,1,2,3\\})}}-m_{{}_{(\\{0,1,2\\},\\{0,3\\})}}+2m_{{}_{(\\{0,1,3\\},\\{0,2\\})}}+m_{{}_{(\\{0,1\\},\\{0,2,3\\})}}+m_{{}_{(\\{0\\},\\{0,1,2\\},\\{3\\})}}\\\
&&\textcolor{white}{.}+m_{{}_{(\\{0\\},\\{0,2,3\\},\\{1\\})}}+2m_{{}_{(\\{0\\},\\{0,2\\},\\{1,3\\})}}+m_{{}_{(\\{0\\},\\{0,2\\},\\{1\\},\\{3\\})}}+m_{{}_{(\\{0,1\\},\\{0,2\\},\\{3\\})}}-m_{{}_{(\\{0,2\\},\\{0,3\\},\\{1\\})}}.\end{array}$

###  4.4 Relationship between bases

Here we show that the sets of power sum functions, elementary functions and
homogeneous functions are indeed bases of the algebra $\mathrm{\bf sNCSym}$.

As it occurs with monomial functions, for each
$(\sigma,I)\in\mathfrak{S}_{m}\times s\mathcal{P}_{n,m}$, we have

(4.1) $p_{\sigma\triangleright
I}=(-1)^{\operatorname{inv}(\sigma)}p_{I},\qquad e_{\sigma\triangleright
I}=(-1)^{\operatorname{inv}(\sigma)}e_{I},\qquad h_{\sigma\triangleright
I}=(-1)^{\operatorname{inv}(\sigma)}h_{I}.$

We begin by writing monomial functions in terms of sums of power sum
functions, elementary functions and homogeneous functions. For this, we will
use the lattice structure of partial set supercompositions.

The _Möbius function_ [30, Section 3.7] of $(s\mathcal{C}_{n,m},\sqsubseteq)$
is the map $\mu:s\mathcal{C}_{n,m}\times s\mathcal{C}_{n,m}\to\Q$ such that
$\mu(K,K)=1$, and

$\mu(K,L)=-\sum_{K\,\sqsubseteq\,N\,\sqsubset\,L}\mu(K,N)\quad\text{if}\quad
K\sqsubset L.$

Observe that

(4.2) $\sum_{K\,\sqsubseteq\,N\,\sqsubseteq\,L}\mu(K,N)=\delta_{K,L}$

A well known formula in terms of _chains_ can be given for the Möbius function
$\mu$. Indeed, given two partial set supercompositions $K,L$, a _chain_
joining $K$ with $L$ is a finite sequence of partial set supercompositions
$F_{1}\sqsubset\cdots\sqsubset F_{q}$ such that $F_{1}=K$ and $F_{q}=L$. A
chain as above will be denoted by $[F_{1},\ldots,F_{q}]$. Thus,

(4.3) $\mu(K,L)=\sum_{[F_{1},\ldots,\,F_{q}]}(-1)^{q-1},$

where the sum is taken over all chains joining $K$ with $L$. For instance, if
$K=\hat{0}_{2,2}$ and $L=(\\{0\\},\\{0,1,2\\})$ as in Figure 3, there are
three chains joining $K$ with $L$, that is,

$[K,L],\qquad[K,(\\{0\\},\\{0,2\\},\\{1\\}),L],\qquad[K,(\\{0\\},\\{0\\},\\{1,2\\}),L],\qquad[K,(\\{0\\},\\{0,1\\},\\{2\\}),L].$

Thus $\mu(K,L)=(-1)+3(-1)^{2}=2$.

Proposition 2.1 and [28, Section 7, Proposition 3] imply that, for every
$K=(K_{1},\ldots,K_{l})\in s\mathcal{C}_{n,m}$, we have the following
formulas:

(4.4)
$\mu(\hat{0}_{n,m},K)=\prod_{i=1}^{l}(-1)^{|K_{i}|-1}(|K_{i}|-1)!\quad\text{and}\quad\sum_{K\sqsubseteq
L}|\mu(\hat{0}_{n,m},K)|=L!$

The _sign_ $(-1)^{K}$ of $K$ is defined as $\prod_{i=1}^{l}(-1)^{|K_{i}|-1}$.
Observe that $(-1)^{K/L}=(-1)^{K}(-1)^{L}$, and

(4.5) $\mu(\hat{0}_{n,m},K)=(-1)^{K}|\mu(\hat{0}_{n,m},K)|.$

The following proposition is a consequence of Equation (4.4) and the inversion
Möbius formula [28, Section 3, Proposition 2, Proposition 3]. The proof is
analogous to the one of [27, Theorem 3.3].

######  Proposition 4.11.

For every set superpartition $I$ of bidegree $(n,m)$, we have

$m_{I}=\sum_{K\,\sqsupseteq\,I}\mu(I,K)p_{K},\qquad
m_{I}=\sum_{L\,\sqsupseteq\,I}\frac{\mu(I,L)}{\mu(\hat{0}_{n,m},L)}\sum_{K\,\sqsubseteq\,L}\mu(K,L)e_{K},\qquad
m_{I}=\sum_{L\,\sqsupseteq\,I}\frac{\mu(I,L)}{|\mu(\hat{0}_{n,m},L)|}\sum_{K\,\sqsubseteq\,L}\mu(K,L)h_{K},$

where $K,L$ are partial set supercompositions of bidegree $(n,m)$.

######  Theorem 4.12.

The sets of power sum functions, elementary functions and complete homogeneous
functions indexed by set superpartitions are bases of the algebra $\mathrm{\bf
sNCSym}$.

###### Proof.

It is a consequence of Proposition 4.11, Equation 4.1 and the fact the
functions are null whenever they are indexed by trivial partial set
supercompositions. ∎

Observe that, specifically, the formula for power sum functions in Proposition
4.11 can be written as follows

$m_{I}=\sum_{I\preceq J}\mu(I,J)(-1)^{\operatorname{inv}(I,J)}p_{J},\qquad
I,J\in s\mathcal{P}_{n,m}.$

######  Example 4.13.

By Figure 3, Equation 4.1, Equation 4.3 and Equation 4.4, if
$I=(\\{0\\},\\{0,1\\},\\{2\\})$, Proposition 4.11 implies the following

$\begin{array}[]{c}m_{I}=p_{{}_{(\\{0\\},\\{0,1\\},\\{2\\})}}-p_{{}_{(\\{0\\},\\{0,1,2\\})}}+p_{{}_{(\\{0,1\\},\\{0,2\\})}},\\\\[4.2679pt]
m_{I}=\frac{1}{2}e_{{}_{(\\{0\\},\\{0,1\\},\\{2\\})}}-\frac{1}{2}e_{{}_{(\\{0\\},\\{0,2\\},\\{1\\})}}-\frac{1}{2}e_{{}_{(\\{0\\},\\{0,1,2\\})}}+e_{{}_{(\\{0,1\\},\\{0,2\\})}},\\\\[5.69054pt]
m_{I}=\frac{5}{2}h_{{}_{(\\{0\\},\\{0,1\\},\\{2\\})}}-\frac{1}{2}h_{{}_{(\\{0\\},\\{0,2\\},\\{1\\})}}-\frac{1}{2}h_{{}_{(\\{0\\},\\{0,1,2\\})}}+h_{{}_{(\\{0,1\\},\\{0,2\\})}}.\end{array}$

The following result is a consequence of Proposition 4.11 and the inversion
Möbius formula.

######  Proposition 4.14.

For every set superpartition $I$ of bidegree $(n,m)$, we have:

$\begin{array}[]{rclrcl}e_{I}&=&{\displaystyle\sum_{K\,\sqsubseteq\,I}\mu(0_{n,m},K)p_{K}},&\kern
40.00006ptp_{I}&=&{\displaystyle\frac{1}{\mu(0_{n,m},I)}\sum_{K\,\sqsubseteq\,I}\mu(K,I)e_{K}},\\\\[19.91684pt]
h_{I}&=&{\displaystyle\sum_{K\,\sqsubseteq\,I}|\mu(0_{n,m},K)|p_{K}},&\kern
40.00006ptp_{I}&=&{\displaystyle\frac{1}{|\mu(0_{n,m},I)|}\sum_{K\,\sqsubseteq\,I}\mu(K,I)h_{K}},\\\\[19.91684pt]
e_{I}&=&{\displaystyle\sum_{K\,\sqsubseteq\,I}(-1)^{K}\sum_{L\,\sqsubseteq\,K}\mu(L,K)h_{L}},&\kern
40.00006pth_{I}&=&{\displaystyle\sum_{K\,\sqsubseteq\,I}(-1)^{K}\sum_{L\,\sqsubseteq\,K}\mu(L,K)e_{L}}.\end{array}$

where $K$ is a partial supercomposition.

As in the classic case, by using Proposition 4.14, we may define a linear map
$\omega:\mathrm{\bf sNCSym}\to\mathrm{\bf sNCSym}$ satisfying
$\omega(e_{I})=h_{I}$ for all $I\in s\mathcal{P}_{n}$. Due to (4.1), for every
$K=\sigma\triangleright I$ with $(\sigma,I)\in\mathfrak{S}_{m}\times
s\mathcal{P}_{n,m}$, we have

(4.6) $\omega(e_{K})=\omega(e_{\sigma\triangleright
I})=\omega((-1)^{\operatorname{inv}(\sigma)}e_{I})=(-1)^{\operatorname{inv}(\sigma)}\omega(e_{I})=(-1)^{\operatorname{inv}(\sigma)}h_{I}=h_{\sigma\triangleright
I}=h_{K}.$

By using Proposition 4.14 and (4.6), we have $\omega^{2}=1$ and
$\omega(p_{I})=(-1)^{I}p_{I}$.

As it was shown in [4, Lemma 4.1] and [7, Proposition 2.39 and Corollary 2.41]
for $\mathrm{\bf NCSym}$. In superspace, the bases of power sum, elementary
and complete homogeneous functions are multiplicative.

######  Proposition 4.15.

For set superpartitions $I\in s\mathcal{P}_{n,m}$ and $J\in
s\mathcal{P}_{p,q}$, we have

$p_{I}p_{J}=p_{I/J},\qquad e_{I}e_{J}=e_{I/J},\qquad h_{I}h_{J}=h_{I/J}.$

###### Proof.

First, we show the multiplicativity of power sum functions. Let $A=\\{H\mid
H\in K\shuffle L,\,K\succeq I,\,L\succeq J\\}$. By definition, $H\succeq I/J$
for all $H\in A$. Conversely, let $H=(H_{1},\ldots,H_{k})\succeq I/J$. Note
that two different fermionic blocks of $I/J$ cannot be contained in the same
fermionic block of $H$. Then, for each $i\in[k]$, we define the following
sets:

$\displaystyle K_{i}=\left\\{\begin{array}[]{ll}H_{i}\cap[n]_{0}&\text{if
$H_{i}$ contains a fermionic block of $I$,}\\\
H_{i}\cap[n]&\text{otherwise}.\end{array}\right.$ $\displaystyle
L_{i}=\left\\{\begin{array}[]{ll}H_{i}\cap[n+1,n+p]_{0}&\text{if $H_{i}$
contains a fermionic block of $J[n]$,}\\\
H_{i}\cap[n+1,n+p]&\text{otherwise}.\end{array}\right.$

Observe that $H_{i}=K_{i}\sqcup L_{i}$ for all $i\in[k]$. This implies that
$H\in K\shuffle L$, where $K$ and $L$ are the set superpartitions obtained by
removing the empty components from $(K_{1},\ldots,K_{k})$ and
$(L_{1},\ldots,L_{k})$ respectively. Further $K\succeq I$ and $L\succeq J$.
Thus, by applying Proposition 3.5, we obtain the following

$p_{I}p_{J}=\sum_{K\succeq I}\sum_{L\succeq J}m_{K}m_{L}=\sum_{K\succeq
I}\sum_{L\succeq J}\left(\sum_{H\in K\shuffle L}m_{H}\right)=\sum_{H\in
A}m_{H}=\sum_{H\succeq I/J}m_{H}=p_{I/J}.$

To show the multiplicativity of elementary functions, we first observe that if
$K\sqsubseteq I$ and $L\sqsubseteq J$, then $K/L\sqsubseteq I/J$. Conversely,
if $H\sqsubseteq I/J$, we consider $K\in s\mathcal{C}_{n}$ obtained by
removing the blocks of $H$ contained in blocks of $J[n]$, and $L\in
s\mathcal{C}_{p}$ obtained by $(-n)$–shifting the collection produced by
removing the blocks of $H$ contained in blocks of $I$. Then $H=K/L$. Thus, by
applying Proposition 4.14, and since the Möbius function is multiplicative, we
obtain the following

$e_{I}e_{J}=\sum_{K\,\sqsubseteq\,I}\sum_{L\,\sqsubseteq\,J}\mu(0_{n,m},K)\mu(0_{p,q},L)p_{K/L}=\sum_{H\,\sqsubseteq\,I/J}\mu(0_{n,m}/0_{p,q},K/L)p_{H}=\sum_{H\,\sqsubseteq\,I/J}\mu(0_{n+p,m+q},H)p_{H}=e_{I/J}.$

Finally, to show the multiplicativity of complete homogeneous functions, we
first note that $\omega$ is an algebra morphism. By regarding power sun
functions, we have

$\omega(p_{K}p_{L})=\omega(p_{K/L})=(-1)^{K/L}p_{K/L}=(-1)^{K}p_{K}(-1)^{L}p_{L}=\omega(p_{K})\omega(p_{L}).$

Therefore
$h_{I}h_{J}=\omega(e_{I})\omega(e_{J})=\omega(e_{I}e_{J})=\omega(e_{I/J})=h_{I/J}$.
∎

##  5 Commutative variables and inner product

In this section, we show formulas that project our bases in their commuting
counterpart, and we prove that there is an isometry from $\mathrm{\bf sNCSym}$
to $\mathrm{\bf sSym}$.

Let $\Q^{\theta}[x]=\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle
x\rangle\kern-2.5pt\rangle/I$, where $I$ is the ideal of
$\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle x\rangle\kern-2.5pt\rangle$
generated by the elements $x_{i}x_{j}-x_{j}x_{i}$ for all $i,j\in\mathbb{N}$.
We denote by $\rho$ the associated natural projection
$\Q_{\\!\\!}^{\theta}\langle\kern-2.5pt\langle
x\rangle\kern-2.5pt\rangle\to\kern-8.00003pt\to\Q^{\theta}[x]$. The image of
$\mathrm{\bf sNCSym}$ under this projection is the algebra $\mathrm{\bf sSym}$
of symmetric functions in commutative variables in superspace, introduced in
[10]. The bases of this algebra are indexed by the so–called superpartitions.

###  5.1 Symmetric functions in commutative variables in superspace

Recall that bases of the classic symmetric functions in commutative variables
are indexed by partitions of numbers [23, 31]. A _partition_ $\lambda$ of
_degree_ $n=:\deg(\lambda)$ is a decreasing sequence
$(\lambda_{1},\ldots,\lambda_{k})$ of nonnegative integers, called
_components_ , such that $\lambda_{1}+\cdots+\lambda_{k}=n$. In general, every
finite sequence of nonnegative integers $u$ can be converted into a partition
$\operatorname{sort}(u)$ by sorting its components in decreasing order. The
partition $\lambda$ can also be denoted by $(1^{r_{1}},\ldots,n^{r_{n}})$,
where $r_{i}$ is the _multiplicity_ of $i$, that is, the number of times that
$i$ occurs in $\lambda$. We set $\lambda!=\lambda_{1}!\cdots\lambda_{k}!$ and
$\lambda^{!}=r_{1}!\cdots r_{n}!$. For instance, if
$\lambda=(5,5,5,3,3,1,1,1,1)$, then

$\lambda=(1^{4},3^{3},5^{2}),\qquad\lambda!=5!\,5!\,5!\,3!\,3!,\qquad\lambda^{!}=4!\,3!\,2!.$

As we mentioned above, the bases of $\mathrm{\bf sSym}$ are indexed by
superpartitions [14]. A _superpartition_ $\Lambda$ of degree $n$ [10,
Subsection 4.2.2] is an array
$(\Lambda_{1},\ldots,\Lambda_{m}\,;\Lambda_{m+1},\ldots,\Lambda_{k})$ of
nonnegative integers, called _components_ , satisfying
$\Lambda_{1}+\cdots+\Lambda_{k}=n$, where
$\Lambda^{a}:=(\Lambda_{1},\ldots,\Lambda_{m})$ is a partition of different
components and $\Lambda^{s}:=(\Lambda_{m+1},\ldots,\Lambda_{k})$ is a usual
partition. Components of $\Lambda^{a}$ are called _fermionic components_ of
$\Lambda$, and the numbers $m,k$ are called the _fermionic degree_ and the
_length_ of $\Lambda$, respectively. Set $\Lambda!=\Lambda^{a}!\Lambda^{s}!$.
The degree, fermionic degree and length of $\Lambda$ will be denoted as with
set superpartitions, that is, $\deg(\Lambda)$, $\operatorname{df}(\Lambda)$
and $\ell(\Lambda)$ respectively. The collection of superpartitions of degree
$n$ will be denoted by $\mathrm{sPar}_{n}$, and the subcollection of elements
with fermionic degree $m$ will be denoted by $\mathrm{sPar}_{n,m}$. As with
usual partitions, every array of nonnegative integers
$u=(u_{1},\ldots,u_{m};u_{m+1},\ldots,u_{k})$, with $u_{1},\ldots,u_{m}$ all
different, can be converted into a superpartition $\operatorname{sort}(u)$,
where $\operatorname{sort}(u)^{a}=\operatorname{sort}(u_{1},\ldots,u_{m})$ and
$\operatorname{sort}(u)^{s}=\operatorname{sort}(u_{m+1},\ldots,u_{k})$.

Some of the classical symmetric functions in superspace in commutating
variables were defined in [14]. Indeed, given a superpartition
$\Lambda=(\Lambda_{1},\ldots,\Lambda_{m};\Lambda_{m+1},\ldots,\Lambda_{n})$,
we have the following

  * •

_Monomial symmetric functions in superspace_ :

$m_{\Lambda}(x,\theta)=\sum_{\sigma\in\mathfrak{S}_{n}}\theta_{\sigma(1)}\cdots\theta_{\sigma(m)}x_{\theta(1)}^{\Lambda_{1}}\cdots
x_{\sigma(n)}^{\Lambda_{n}},$

where the sum consider no repeated terms.

  * •

_Power–sum symmetric functions in superspace_ :
$p_{\Lambda}=\tilde{p}_{\Lambda_{1}}\cdots\tilde{p}_{\Lambda_{m}}p_{\Lambda_{m+1}}\cdots
p_{\Lambda_{n}}$, where

$\tilde{p}_{k}=\sum_{i=1}^{n}\theta_{i}x_{i}^{k}\quad\text{and}\quad
p_{r}=\sum_{i=1}^{n}x_{i}^{r},\quad\text{for}\quad k\geq 0,r\geq 1.$

  * •

_Elementary symmetric functions in superspace_ :
$e_{\Lambda}=\tilde{e}_{\Lambda_{1}}\cdots\tilde{e}_{\Lambda_{m}}e_{\Lambda_{m+1}}\cdots
e_{\Lambda_{n}}$, where

$\tilde{e}_{k}=m_{(0;1^{k})}\quad\text{and}\quad
e_{r}=m_{(\emptyset;1^{r})},\quad\text{for}\quad k\geq 0,r\geq 1.$

  * •

_Complete homogeneous symmetric functions in superspace_ :
$h_{\Lambda}=\tilde{h}_{\Lambda_{1}}\ldots\tilde{h}_{\Lambda_{m}}h_{\Lambda_{m+1}}\cdots
h_{\Lambda_{n}}$, where

$\tilde{h}_{k}=\sum_{|\Lambda|=k,\,\operatorname{df}(\Lambda)=1}(\Lambda_{1}+1)m_{\Lambda}\quad\text{
and}\quad
h_{r}=\sum_{|\Lambda|=r,\,\operatorname{df}(\Lambda)=0}m_{\Lambda},\quad\text{for}\quad
k\geq 0,r\geq 1.$

Observe that the bases of $\mathrm{\bf Sym}$ are recovered whenever
$\Lambda^{a}=\emptyset$.

For every set superpartition $I=(I_{1},\ldots,I_{k})$ of bidegree $(n,m)$,
which has no fermionic blocks of the same size, we define the super partition:

$\Lambda(I)=\operatorname{sort}(|I_{1}|-1,\ldots,|I_{m}|-1;|I_{m+1}|,\ldots,|I_{k}|).$

Given a superpartition $\Lambda$, we say that $I$ is _of type_ $\Lambda$ if
$\Lambda(I)=\Lambda$. We denote by $\varepsilon(I)$ the minimal number of
transpositions needed to obtain $\Lambda(I)^{a}$ from
$(|I_{1}|-1,\ldots,|I_{m}|-1)$. For instance, the set superpartitions
$(\\{0,1,7\\},\\{0,5\\},\\{2\\},\\{3,4\\},\\{6\\})$ and
$(\\{0,2\\},\\{0,3,5\\},\\{1\\},\\{4\\},\\{6,7\\})$ are of type $(2,1;2,1,1)$.

To determine the number of set superpartitions of type
$\Lambda=(\Lambda_{1},\ldots,\Lambda_{m};\Lambda_{m+1},\ldots,\Lambda_{k})$,
we set $\Lambda^{\\!+}=\operatorname{sort}(\Lambda_{1},\ldots,\Lambda_{k})$.
Note that $\Lambda!=\Lambda^{\\!+}!$.

######  Lemma 5.1.

Let
$\Lambda=(\Lambda_{1},\ldots,\Lambda_{m};\Lambda_{m+1},\ldots,\Lambda_{k})$ be
a superpartition of degree $n$, and for each $i\in[k]$ denote by $k_{i}$ the
multiplicity of $\Lambda_{i}$ in $\Lambda^{\\!+}$. Then
$(\Lambda^{\\!+})^{!}=k_{1}\cdots k_{m}(\Lambda^{s})^{!}$.

###### Proof.

The result is a direct consequence of the fact that, for $i>m$, the
multiplicity of $\Lambda_{i}$ in $\Lambda^{s}$ is either $k_{i}-1$ whenever
$\Lambda_{i}\in\Lambda^{a}$, or it is $k_{i}$ otherwise. ∎

Recall that, for a partition $\lambda$ of degree $n$, the number of set
partitions of type $\lambda$ is
$\binom{n}{\lambda}=\frac{n!}{\lambda!\lambda^{!}}$. Below we generalize this
result for superpartitions.

######  Lemma 5.2.

Let
$\Lambda=(\Lambda_{1},\ldots,\Lambda_{m};\Lambda_{m+1},\ldots,\Lambda_{k})$ be
a superpartition of degree $n$, and for each $i\in[m]$ denote by $k_{i}$ the
multiplicity of $\Lambda_{i}$ in $\Lambda^{\\!+}$. Then, the number of set
superpartitions of type $\Lambda$ is

$\binom{n}{\Lambda}:=k_{1}\cdots
k_{m}\binom{n}{\,\,\Lambda^{\\!+}}=\frac{n!}{\Lambda!(\Lambda^{s})^{!}}.$

###### Proof.

Observe that every set superpartition of type $\Lambda$ can be obtained from a
usual set partition of type $\Lambda^{\\!+}$ by appending zeros to some of its
blocks. Indeed, if $I$ is a set superpartition of type $\Lambda$, then
$J=I^{+}$ is a set partition of type $\Lambda^{\\!+}$. Reciprocally, if $J$ is
a set partition of type $\Lambda^{+}$, then there are $B_{1},\ldots,B_{m}$ in
$J\cup\\{\emptyset\\}$ such that $|B_{i}|=\Lambda_{i}$ for all $i\in[m]$. We
get a set superpartition $I=K\backslash\\{\emptyset\\}$ of type $\Lambda$,
where $K$ is obtained from $J\cup\\{\emptyset\\}$ by replacing $B_{i}$ by
$B_{i}\cup\\{0\\}$. The proof follows from the fact that for each
$\Lambda_{i}$ with $i\in[m]$, there are $k_{i}$ blocks $B_{i}\in
J\cup\\{\emptyset\\}$ such that $|B_{i}|=\Lambda_{i}$ together with Lemma 5.1.
∎

Now, we will study the image of the bases of $\mathrm{\bf sNCSym}$ under the
projection $\rho$ onto the algebra of commutative variables $\mathrm{\bf
sSym}$. First, we prove that set superpartitions needed to describe $\rho$ on
the basis of monomial functions are the ones whose fermionic blocks have all
different sizes.

######  Proposition 5.3.

Let $I$ be a set superpartition having at least two fermionic blocks
$B,B^{\prime}$ such that $|B|=|B^{\prime}|$. Then $\rho(m_{I})=0$.

###### Proof.

Let $k=\ell(I)$, and let $i,j\in[k]$ such that $B,B^{\prime}$ are the $i$th
and $j$th blocks of $I$ respectively. Consider $g\in\\{m_{I},p_{I}\\}$, and
let
$r=\theta_{a_{1}}^{\epsilon_{1}}\cdots\theta_{a_{k}}^{\epsilon_{k}}x_{b_{1}}\cdots
x_{b_{n}}$ be a monomial occurring in $g$. Then
$r=\epsilon\theta_{a_{i}}\theta_{a_{j}}\theta_{d_{1}}\cdots\theta_{d_{r}}x_{b_{1}}\cdots
x_{b_{n}}$ for some $\epsilon\in\\{\pm 1\\}$, where
$r=\operatorname{df}(I)-2$. Since $g$ is symmetric, the monomial
$\sigma(r)=-\epsilon\theta_{a_{i}}\theta_{a_{j}}\theta_{d_{1}}\cdots\theta_{d_{r}}x_{\sigma(b_{1})}\cdots
x_{\sigma(b_{n})}$ also occurs in $g$, where $\sigma$ is the transposition
exchanging $a_{i}$ with $a_{j}$. Since $|B|=|B^{\prime}|$, then
$x_{\sigma(b_{1})}\cdots x_{\sigma(b_{n})}=x_{b_{1}}\cdots x_{b_{n}}$ in
$\Q^{\theta}[x]$. Thus $\rho(\sigma(r))=-\rho(r)$. As it holds for every
monomial occurring in $g$, then $\rho(g)=0$. ∎

######  Lemma 5.4.

For every $K\in s\mathcal{C}_{n,m}$ and $\delta\in\mathfrak{S}_{m}$, we have
$\rho(b_{\delta K})=\rho(b_{K})$, where $b$ denotes any of the basis elements
studied above.

###### Proof.

Since $\delta\circ b_{K}=b_{\delta\circ K}$, then, every monomial occurring in
$b_{\delta\circ K}$ can be written as

$\delta\circ(\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots
x_{b_{n}})=\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{\delta^{-1}(1)}}\cdots
x_{b_{\delta^{-1}(n)}},$

where $\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots x_{b_{n}}$ is a
monomial occurring in $b_{K}$. Now, since $x$–variables become commutative
under the action of $\rho$, then
$\rho(\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{\delta^{-1}(1)}}\cdots
x_{b_{\delta^{-1}(n)}})=\rho(\theta_{a_{1}}\cdots\theta_{a_{m}}x_{b_{1}}\cdots
x_{b_{n}})$. Thus, $\rho(b_{\delta\circ K})=\rho(b_{K})$. ∎

######  Proposition 5.5.

For every set superpartition $I$ with fermionic blocks of different sizes, we
have

  1. 1.

$\rho(m_{I})=(-1)^{\varepsilon(I)}(\Lambda(I)^{s})^{!}m_{\Lambda(I)}$.

  2. 2.

$\rho(p_{I})=(-1)^{\varepsilon(I)}p_{\Lambda(I)}$.

  3. 3.

$\rho(e_{I})=(-1)^{\varepsilon(I)}\Lambda(I)!e_{\Lambda(I)}$.

  4. 4.

$\rho(h_{I})=(-1)^{\varepsilon(I)}\Lambda(I)!h_{\Lambda(I)}$.

Furthermore, if $J$ is a set superpartition with blocks of the same size, we
have $\rho(b_{J})=0$, where $b$ is any of the basis elements above.

###### Proof.

Due to Lemma 5.4, it is enough to take $I=(I_{1},\ldots,I_{k})\in
s\mathcal{P}_{n,m}$ as a convex set superpartition.

To prove (1), consider $\Lambda(I)^{s}=(1^{k_{1}},\ldots,n^{k_{n}})$. Since
$I$ is convex, every monomial $u\in M(I)$ can be written as

$u=\theta_{a_{1}}\cdots\theta_{a_{m}}x_{a_{1}}^{|I_{1}|-1}\cdots
x_{a_{m}}^{|I_{m}|-1}x_{b_{m+1}}^{|I_{m+1}|}\cdots x_{b_{k}}^{|I_{k}|}.$

Observe that for every permutation $\sigma$ satisfying
$\sigma(\\{b_{m+1},\ldots,b_{k}\\})=\\{b_{m+1},\ldots,b_{k}\\}$ with
$|I_{i}|=|I_{j}|$ whenever $\sigma(b_{i})=b_{j}$, and $\sigma(b)=b$ for all
$b\not\in\\{b_{m+1},\ldots,b_{k}\\}$, we have

$\rho(\theta_{a_{1}}\cdots\theta_{a_{m}}x_{a_{1}}^{|I_{1}|-1}\cdots
x_{a_{m}}^{|I_{m}|-1}x_{\sigma(b_{m+1})}^{|I_{m+1}|}\cdots
x_{\sigma(b_{k})}^{|I_{k}|})=\rho(u).$

Since, for each $i\in[n]$, there are $k_{i}$ blocks of size $i$ in
$(I_{m+1},\ldots,I_{k})$, so there are $k_{1}!\cdots
k_{n}!=(\Lambda(I)^{s})^{!}$ such permutations. Finally, by sorting the
$\theta$–variables, we obtain
$\rho(m_{I})=(-1)^{\varepsilon(I)}(\Lambda(I)^{s})^{!}m_{\Lambda(I)}$.

To prove (2) to (4) we will use the fact that the bases are multiplicative
(Proposition 4.15).

Observe that $\rho(p_{[n]})=p_{n}$ and $\rho(p_{[n]_{0}})=\tilde{p}_{n}$,
where $p_{[n]}=p_{\\{[n]\\}}$ and $p_{[n]_{0}}=p_{\\{[n]_{0}\\}}$. For
$i\in[k]$, let $n_{i}=|I_{i}|-1$ if $i\leq m$, and $n_{i}=|I_{i}|$ otherwise.
Since $I$ is convex and the basis is multiplicative, we have

$\rho(p_{I})=\rho(p_{[n_{1}]_{0}}\cdots p_{[n_{m}]_{0}}p_{[n_{m+1}]}\cdots
p_{[n_{k}]})=\tilde{p}_{n_{1}}\cdots\tilde{p}_{n_{m}}p_{n_{m+1}}\cdots
p_{n_{k}}=(-1)^{\varepsilon(I)}p_{\Lambda(I)}.$

The argument is similar for the remaining bases by observing that

$\rho(e_{[n]})=n!e_{n},\qquad\rho(e_{[n]_{0}})=n!\tilde{e}_{n},\qquad\rho(h_{[n]})=n!h_{n},\qquad\rho(h_{[n]_{0}})=n!\tilde{h}_{n}.$

∎

We may define a _lifting map_ $\tilde{\rho}:\mathrm{\bf sSym}\to\mathrm{\bf
sNCSym}$, which result to be a right inverse of $\rho$, as the linear map
satisfying

(5.1)
$\tilde{\rho}(m_{\Lambda})=\frac{\Lambda!}{n!}\sum_{\Lambda(I)=\Lambda}(-1)^{\varepsilon(I)}m_{I}.$

######  Proposition 5.6.

The composition $\rho\circ\tilde{\rho}$ is the identity map on $\mathrm{\bf
sSym}$.

###### Proof.

By (1) of Proposition 5.5 together with Lemma 5.1 and Lemma 5.2, we have

$\rho(\tilde{\rho}(m_{\Lambda}))=\frac{\Lambda!}{n!}\sum_{\Lambda(I)=\Lambda}(-1)^{\varepsilon(I)}\rho(m_{I})=\frac{\Lambda!}{n!}\sum_{\Lambda(I)=\Lambda}(\Lambda(I)^{s})^{!}m_{\Lambda(I)}=\frac{\Lambda!}{n!}\binom{n}{\Lambda}(\Lambda(I)^{s})^{!}m_{\Lambda}=m_{\Lambda}.$

Therefore $\rho\circ\tilde{\rho}$ is the identity map on $\mathrm{\bf sSym}$.
∎

###  5.2 Inner product

It was shown in [14] that $\mathrm{\bf sSym}$ can be given with an inner
product $\left\langle\,,\right\rangle$ satisfying $\langle
m_{\Lambda},h_{\Omega}\rangle=(-1)^{\binom{m}{2}}\delta_{\Lambda,\Omega}$,
where $\Lambda,\Omega$ are superpartitions, $m=\operatorname{df}(\Lambda)$ and
$\delta$ is the Dirac delta function.

Here we extend the inner product of $\mathrm{\bf NCSym}$ [27] to superspace
and show that, with respect to this inner product, $\tilde{\rho}$ is an
isometry from $\mathrm{\bf sNCSym}$ to $\mathrm{\bf sSym}$.

Let $\left\langle\,,\right\rangle$ be the bilinear form on $\mathrm{\bf
sNCSym}$ satisfying $\left\langle
m_{I},h_{J}\right\rangle=(-1)^{\binom{m}{2}}n!\delta_{I,J}$, where $I,J$ are
set superpartitions and $(n,m)$ is the bidegree of $I$.

######  Lemma 5.7.

For every $I,J\in s\mathcal{P}_{n,m}$ and $\sigma\in\mathfrak{S}_{m}$, we have
$(I\wedge_{\sigma}J)!=(J\wedge_{\sigma^{-1}}I)!$.

###### Proof.

Let $(K_{1},\ldots,K_{p})=I\wedge_{\sigma}J$, and let
$(K_{1}^{\prime},\ldots,K_{q}^{\prime})=J\wedge_{\sigma^{-1}}I$. By
definition, we have $K^{\prime}_{\sigma(i)}=J_{\sigma(i)}\cap
I_{\sigma^{-1}(\sigma(i))}=J_{\sigma(i)}\cap I_{i}=I_{i}\cap
J_{\sigma(i)}=K_{i}$. This implies that $K_{1}\cup\cdots\cup
K_{m}=K_{1}^{\prime}\cup\cdots\cup K_{m}^{\prime}=:A$. Hence,
$(K_{m+1},\ldots,K_{p})=(K_{m+1}^{\prime},\ldots,K_{q}^{\prime})=(I\backslash
A)\wedge(J\backslash A)$ with $p=q$. Therefore
$(I\wedge_{\sigma}J)!=(J\wedge_{\sigma^{-1}}I)!$. ∎

######  Proposition 5.8.

Let $I,J$ be two set superpartitions of bidegree $(n,m)$. Then

$\left\langle
h_{I},h_{J}\right\rangle=n!\sum_{\sigma\in\mathfrak{S}_{m}}(I\wedge_{\sigma}J)!(-1)^{\operatorname{inv}(\sigma)}$

Furthermore, $\left\langle h_{I},h_{J}\right\rangle=\left\langle
h_{J},h_{I}\right\rangle$.

###### Proof.

Due to Corollary 4.10, we have

$\left\langle
h_{I},h_{J}\right\rangle=\left\langle\sum_{(\sigma,\,K)}(I\wedge_{\sigma}K)!(-1)^{\operatorname{inv}(\sigma)}m_{K},h_{J}\right\rangle=\sum_{(\sigma,K)}(I\wedge_{\sigma}K)!(-1)^{\operatorname{inv}(\sigma)}\left\langle
m_{K},h_{J}\right\rangle,$

where $\sigma\in\mathfrak{S}_{m}$ and $J\in s\mathcal{P}_{n}$. Then, because
of $\left\langle
m_{K},h_{J}\right\rangle=(-1)^{\binom{m}{2}}n!\delta_{K,\,J}$, we obtain the
result

$\left\langle
h_{I},h_{J}\right\rangle=\sum_{(\sigma,K)}(I\wedge_{\sigma}K)!(-1)^{\operatorname{inv}(\sigma)}(-1)^{\binom{m}{2}}n!\delta_{K,\,J}=(-1)^{\binom{m}{2}}n!\sum_{\sigma\in\mathfrak{S}_{m}}(I\wedge_{\sigma}K)!(-1)^{\operatorname{inv}(\sigma)}.$

Lemma 5.7 and the fact that $\operatorname{inv}(\sigma)$ and
$\operatorname{inv}(\sigma^{-1})$ have the same parity imply that
$\left\langle h_{I},h_{J}\right\rangle=\left\langle h_{J},h_{I}\right\rangle$.
∎

######  Proposition 5.9.

Let $I,J$ be two set superpartitions of bidegree $(n,m)$. Then

$\left\langle\,p_{I},p_{J}\right\rangle=\frac{(-1)^{\binom{m}{2}}n!}{|\mu(\hat{0}_{n,m},I)|}\delta_{I,J}.$

###### Proof.

Due to Corollary 4.4 and Proposition 4.14, we have

$p_{I}=\sum_{K\succeq I}m_{K},\qquad
p_{J}=\frac{1}{|\mu(\hat{0}_{n,m},J)|}\sum_{L\preceq J}\mu(L,J)h_{L},$

where $K,L$ are nontrivial partial set supercompositions of bidegree $(n,m)$.
Then,

$\left\langle\,p_{I},p_{J}\right\rangle=\frac{1}{|\mu(\hat{0}_{n,m},J)|}\sum_{K\succeq
I}\sum_{L\preceq J}\mu(L,J)\left\langle m_{K},h_{L}\right\rangle.$

Since $K,L$ are nontrivial, then $K=(\sigma,\bar{K})$ and $L=(\tau,\bar{L})$
for some $\sigma,\tau\in\mathfrak{S}_{m}$. Thus

$\left\langle
m_{K},h_{L}\right\rangle=(-1)^{\operatorname{inv}(\sigma)+\operatorname{inv}(\tau)}\left\langle
m_{\bar{K}},h_{\bar{L}}\right\rangle.$

We will distinguish four cases. If $I\succ J$ as set superpartitions, then
$\bar{L}\preceq J\prec I\preceq\bar{K}$. Thus $\left\langle
m_{K},h_{L}\right\rangle=0$. Therefore
$\left\langle\,p_{I},p_{J}\right\rangle=0$.

If $I,J$ are not comparable as set superpartitions, either $\bar{K}$ is not
comparable with $J$, or $\bar{K}\succ J$, otherwise $I\preceq\bar{K}\preceq
J$, which is a contradiction. Thus, $\mu(K,J)h_{K}$ is not an addend of
$p_{J}$, which implies $\left\langle\,p_{I},p_{J}\right\rangle=0$.

Now, if $I\prec J$ as set superpartitions, Remark 4.3 implies that
$I\prec(\alpha^{-1},J)$, as partial set supercomposition, for some
$\alpha\in\mathfrak{S}_{m}$.

$\begin{array}[]{rcl}\left\langle\,p_{I},p_{J}\right\rangle&=&{\displaystyle\frac{1}{|\mu(\hat{0}_{n,m},J)|}\sum_{(\alpha,\,I)\preceq
L\preceq J}\mu(L,J)\left\langle
m_{(\alpha^{-1},L)},h_{L}\right\rangle},\\\\[14.22636pt]
&=&{\displaystyle\frac{1}{|\mu(\hat{0}_{n,m},J)|}\sum_{(\alpha,\,I)\preceq
L\preceq J}(-1)^{\operatorname{inv}(\alpha)}\mu(L,J)\left\langle
m_{L},h_{L}\right\rangle},\\\\[14.22636pt]
&=&{\displaystyle\frac{(-1)^{\operatorname{inv}(\alpha)}(-1)^{\binom{m}{2}}n!}{|\mu(\hat{0}_{n,m},J)|}\sum_{(\alpha,I)\preceq
L\preceq J}\mu(L,J)=0.}\end{array}$

Therefore
${\displaystyle\left\langle\,p_{I},p_{J}\right\rangle=\frac{(-1)^{\binom{m}{2}}n!}{|\mu(\hat{0}_{n,m},I)|}}\delta_{I,J}$.
∎

######  Lemma 5.10.

For every $K,L\in s\mathcal{C}_{n,m}$ and
$(\sigma,\tau)\in\mathfrak{S}_{n}\times\mathfrak{S}_{m}$, we have $(K\sqcap
L)!=\left(\tau\triangleright\sigma(K)\,\sqcap\,\tau\triangleright\sigma(L)\right)!$.

###### Proof.

Write $K=(K_{1},\ldots,K_{h})$ and $L=(L_{1},\ldots,L_{k})$. Let $A=(K_{1}\cap
L_{1})\cup\cdots\cup(K_{m}\cap L_{m})$, and let $(K^{+}\backslash
A)\wedge(L^{+}\backslash A))=(N_{m+1},\ldots,N_{q})$, then
$(\sigma(K)^{+}\backslash\sigma(A))\wedge(\sigma(L)^{+}\backslash\sigma(A)))=(\sigma(N_{m+1}),\ldots,\sigma(N_{q}))$.
Since $\sigma,\tau$ are bijections, then $|K_{i}\cap
L_{i}|=|\sigma(K_{\tau(i)})\cap\sigma(L_{\tau(i)})|$ for all $i\in[m]$, and
$|\sigma(N_{i})|=|N_{i}|$ for all $i\in[m+1,q]$. So, Proposition 2.2 implies
that $(K\sqcap
L)!=\left(\tau\triangleright\sigma(K)\,\sqcap\,\tau\triangleright\sigma(L)\right)!$.
∎

######  Proposition 5.11.

The bilinear form $\left\langle\,,\right\rangle$ is an inner product on
$\mathrm{\bf sNCSym}$. Furthermore, with this inner product, the lifting map
$\tilde{\rho}$ is an isometry from $\mathrm{\bf sNCSym}$ to $\mathrm{\bf
sSym}$.

###### Proof.

Proposition 5.8 and Proposition 5.9 imply that $\left\langle\,,\right\rangle$
is an inner product. To show that $\tilde{\rho}$ is an isometry, for every
$\Lambda\in\mathrm{sPar}_{n,m}$, we define

$H_{\Lambda}=\sum_{\Lambda(I)=\Lambda}(-1)^{\varepsilon(I)}h_{I},\quad\text{where}\quad
I\in s\mathcal{P}_{n,m}.$

By Proposition 5.5(4) and Lemma 5.2, we have

(5.2)
$\rho(H_{\Lambda})=\sum_{\Lambda(I)=\Lambda}(-1)^{\varepsilon(I)}\rho(h_{I})=\sum_{\Lambda(I)=\Lambda}\Lambda(I)!h_{\Lambda(I)}=\sum_{\Lambda(I)=\Lambda}\Lambda!h_{\Lambda}=\binom{n}{\Lambda}\Lambda!h_{\Lambda}=\frac{n!}{(\Lambda^{s})^{!}}h_{\Lambda}.$

On the other hand, Proposition 4.9 implies that

$H_{\Lambda}=\sum_{\Lambda(I)=\Lambda}(-1)^{\varepsilon(I)}\sum_{K\in
s\mathcal{C}_{n,m}}(I\sqcap K)!m_{K}=\sum_{K\in
s\mathcal{C}_{n,m}}\underbrace{\sum_{\Lambda(I)=\Lambda}(-1)^{\varepsilon(I)}(I\sqcap
K)!}_{C_{K}}m_{K}=\sum_{K\in s\mathcal{C}_{n,m}}C_{K}m_{K}.$

By Lemma 5.10, we have $C_{K}=C_{L}$ for all $K,L\in s\mathcal{C}_{n,m}$ with
$\bar{K}$ and $\bar{L}$ of the same type. Hence,

$H_{\Lambda}=\sum_{\Omega\in\mathrm{sPar}_{n,m}}C_{\Omega}\tilde{\rho}(m_{\Omega})=\tilde{\rho}\left(\underbrace{\sum_{\Omega\in\mathrm{sPar}_{n,m}}C_{\Omega}m_{\Omega}}_{F}\right)=\tilde{\rho}(F),\quad\text{where}\quad
C_{\Omega}=\sum_{\Lambda(\bar{K})=\Omega}C_{K}.$

By using (5.2) together with Proposition 5.6, we get ${\displaystyle
F=\rho(\tilde{\rho}(F))=\rho(H_{\Lambda})=\frac{n!}{(\Lambda^{s})^{!}}h_{\Lambda}}$.
Thus

$\begin{array}[]{rcl}\left\langle\tilde{\rho}(m_{\Omega}),\tilde{\rho}(h_{\Lambda})\right\rangle&=&{\displaystyle\left\langle\frac{\Omega!}{n!}\sum_{\Lambda(I)=\Omega}(-1)^{\varepsilon(I)}m_{I},\frac{(\Lambda^{s})^{!}}{n!}H_{\Lambda}\right\rangle},\\\\[14.22636pt]
&=&{\displaystyle\left\langle\frac{\Omega!}{n!}\sum_{\Lambda(I)=\Omega}(-1)^{\varepsilon(I)}m_{I},\frac{(\Lambda^{s})^{!}}{n!}\sum_{\Lambda(J)=\Lambda}(-1)^{\varepsilon(J)}h_{J}\right\rangle},\\\\[14.22636pt]
&=&{\displaystyle\frac{\Lambda!(\Lambda^{s})^{!}}{(n!)^{2}}\binom{n}{\Lambda}(-1)^{\binom{m}{2}}n!}\delta_{\Omega,\Lambda},\\\\[11.38092pt]
&=&\left\langle m_{\Omega},h_{\Lambda}\right\rangle.\end{array}$

Therefore $\tilde{\rho}$ is an isometry. ∎

##  6 Schur symmetric functions in superspace

In this section, we define Schur–type functions in noncommuting variables in
superspace by means super semistandard Young tableaux. Also, we express these
functions in terms of the monomial basis and show their projection on
Schur–type functions in commutative variables in superspace.

Recall that partitions of integers can be represented by _Young tableaux_.
Indeed, the _Young diagram_ or simply the _diagram_ of a partition
$\lambda=(\lambda_{1},\ldots,\lambda_{k})$ is the array with $k$ rows of
left–justified boxes in which the $i$th row contains exactly $\lambda_{i}$
boxes. See Figure 4.

Figure 4: Diagram of the partition $\lambda=(7,4,4,2,1)$.

A partition $\mu$ is said to be _contained_ in a partition $\lambda$, denoted
by $\mu\subseteq\lambda$, if $\ell(\mu)\leq\ell(\lambda)$ and
$\mu_{i}\leq\lambda_{i}$ for all $i\in[\ell(\mu)]$. If $\mu\subseteq\lambda$,
the _skew diagram_ $\lambda/\mu$ is the diagram obtained by removing the boxes
from the diagram of $\lambda$ that belong to the diagram of $\mu$. The skew
diagram $\lambda/\mu$ is called an _horizontal $r$–strip_ (resp. _vertical
$r$–strip_) if $\deg(\lambda)-\deg(\mu)=r$ and each column (resp. row) has at
most one box. See Figure 5.

Figure 5: A horizontal $7$–strip $\lambda/\mu$, where $\lambda=(8,7,5,4,2,1)$
and $\mu=(7,5,5,3,1)$.

In this section, we identify
$\Lambda=(\Lambda_{1},\ldots,\Lambda_{m};\Lambda_{m+1},\ldots,\Lambda_{k})\in\mathrm{sPar}_{n,m}$
with the pair $(\Lambda^{\oplus},\Lambda^{+})$, where

$\Lambda^{\oplus}=\operatorname{sort}(\Lambda_{1}+1,\ldots,\Lambda_{m}+1,\Lambda_{m+1},\ldots,\Lambda_{k}).$

Observe that the skew diagram $\Lambda^{\oplus}/\Lambda^{+}$ is simultaneously
an horizontal $m$–strip and an vertical $m$–strip. The superpartition
$\Lambda$ can be represented either by the pair of diagrams of
$(\Lambda^{\oplus},\Lambda^{+})$, or by the diagram of $\Lambda^{\oplus}$ in
which the boxes of $\Lambda^{\oplus}/\Lambda^{+}$ has been replaced by
circles. These diagrams, accepting at most one round box at the end of each
row, will be called _super Young diagram_. See Figure 6.

$\left(\vbox{\hbox{\includegraphics{pics/002.pdf}}},\vbox{\hbox{\includegraphics{pics/003.pdf}}}\right)\kern
42.67912pt\vbox{\hbox{\includegraphics{pics/004.pdf}}}$ Figure 6: Diagrams
representing $\Lambda=(3,1;2,1)$ with $\Lambda^{+}=(3,2,1,1)$ and
$\Lambda^{\oplus}=(4,2,2,1)$.

Consider $\bar{\mathbb{N}}_{0}=\\{\bar{0},\bar{1},\bar{2},\ldots\\}$. For
every $\alpha\in\mathbb{N}_{0}\cup\bar{\mathbb{N}}_{0}$, we set
$|\alpha|=a\in\mathbb{N}_{0}$, if either $\alpha=a$ or $\alpha=\bar{a}$.

Let $\Lambda$ be a superpartition, and let
$\alpha_{1},\ldots,\alpha_{r}\in\mathbb{N}_{0}\cup\bar{\mathbb{N}}_{0}$. A
_super semistandard Young tableau_ of _shape_ $\Lambda$ and _weight_
$(\alpha_{1},\ldots,\alpha_{r})$, is a sequence of superpartitions
$(\Lambda_{(0)},\ldots,\Lambda_{(r)})$ with $\Lambda_{(0)}=\emptyset$ and
$\Lambda_{(r)}=\Lambda$, satisfying:

  1. 1.

For each $i\in[n]$, $\Lambda_{(i)}^{+}/\Lambda_{(i-1)}^{+}$ is a horizontal
$|\alpha_{i}|$–strip.

  2. 2.

If $\alpha_{i}\in\mathbb{N}_{0}$ and $\Lambda_{(i-1)}$ contains the $j$th
circle in the $r$th row, then

    1. (a)

$\operatorname{df}(\Lambda_{(i)})=\operatorname{df}(\Lambda_{(i-1)})$.

    2. (b)

$\Lambda_{(i)}$ contains the $j$th circle in the $r$th row whenever the $r$th
row of $\Lambda_{(i)}^{+}/\Lambda_{(i-1)}^{+}$ is empty.

    3. (c)

$\Lambda_{(i)}$ contains the $j$th circle in the $(r+1)th$ row whenever the
$r$th row of $\Lambda_{(i)}^{+}/\Lambda_{(i-1)}^{+}$ is nonempty.

  3. 3.

If $\alpha_{i}\in\bar{\mathbb{N}}_{0}$, then

    1. (a)

$\operatorname{df}(\Lambda_{(i)})=\operatorname{df}(\Lambda_{(i-1)})+1$.

    2. (b)

$\Lambda_{(i)}$ contains a unique circle, say in the $c$th column, such that
the $c$th column of $\Lambda^{+}_{(i)}/\Lambda^{+}_{(i-1)}$ is empty and every
other column to its left is nonemtpy.

    3. (c)

The diagram obtained from $\Lambda_{(i)}$ by removing the new circle above
satisfies conditions in (2).

See Figure 7 for an instance.

$\left(\,\,\emptyset,\quad\vbox{\hbox{\includegraphics{pics/007.pdf}}},\quad\vbox{\hbox{\includegraphics{pics/008.pdf}}},\quad\vbox{\hbox{\includegraphics{pics/009.pdf}}},\quad\vbox{\hbox{\includegraphics{pics/010.pdf}}},\quad\vbox{\hbox{\includegraphics{pics/011.pdf}}},\quad\vbox{\hbox{\includegraphics{pics/012.pdf}}}\,\,\right)$
Figure 7: A super semistandard Young tableau of shape
$(1,\bar{3},1,\bar{3},1,3)$.

Every super semistandard Young tableau $(\Lambda_{(0)},\ldots,\Lambda_{(r)})$
of weight $(\alpha_{1},\ldots,\alpha_{r})$ and shape $\Lambda$ can be
represented by filling with numbers the boxes and circles of $\Lambda$, such
that for each $i\in[r]$, the boxes of $\Lambda^{+}_{(i)}/\Lambda^{+}_{(i-1)}$
in $\Lambda$ are filled with the integer $i$, and if
$\alpha_{i}\in\bar{\mathbb{N}}_{0}$, the new circle of $\Lambda_{(i)}$ in
$\Lambda$ is filled with $i$ as well. See Figure 8. Observe that if $\Lambda$
is a usual partition, this representation coincides with the classic
definition of semistandard Young tableaux [31].

Figure 8: Super semistandard Young tableau in Figure 7 represented by filling
its boxes with numbers.

If $T$ is a super Young diagram, we will denote its boxes by
$T_{1}^{s},\ldots,T_{n}^{s}$, where $i<j$ if and only if $T_{j}^{s}$ is
located either to the right or below $T_{i}^{s}$. The circles of $T$ are
denoted by $T_{1}^{a},\ldots,T_{m}^{a}$, indexed from top to bottom. Given $t$
either a box or a circle of $T$, we denote by $c(t)$ its label.

######  Definition 6.1.

The _Schur function_ of a superpartition $\Lambda\in\mathrm{sPar}_{n,m}$ is
the function in superspace, defined as follows:

$S_{\Lambda}=\sum_{T\in
sSSYT(\Lambda)}\underbrace{\theta_{{}_{c(T^{a}_{1})}}\cdots\theta_{{}_{c(T^{a}_{m})}}\sum_{\sigma\in\mathfrak{S}_{n}}x_{{}_{c(T^{s}_{\sigma(1)})}}\cdots
x_{{}_{c(T^{s}_{\sigma(n)})}}}_{[\theta x]^{T}},$

where $sSSYT(\Lambda)$ is the collection of super semistandard Young tableux
of shape $\Lambda$.

For instance, for $\Lambda=(2,1;\,\,)$ and
$T=\vbox{\hbox{\includegraphics{pics/015.pdf}}}$ in $sSSYT(\Lambda)$ with
weight $(\bar{1},\bar{2})$, we have

$[\theta
x]^{T}=\theta_{2}\theta_{1}(x_{1}x_{2}x_{2}+x_{2}x_{1}x_{2}+x_{2}x_{2}x_{1}+x_{1}x_{2}x_{2}+x_{2}x_{1}x_{2}+x_{2}x_{2}x_{1}).$

Observe that if $\Lambda\in\mathrm{sPar}_{n,m}$ and $T\in sSSYT(\Lambda)$ with
weight $(\alpha_{1},\ldots,\alpha_{r})$, then there is
$\sigma\in\mathfrak{S}_{n}$ such that

$x_{{}_{c(T^{s}_{\sigma(1)})}}\cdots
x_{{}_{c(T^{s}_{\sigma(n)})}}=x_{1}^{|\alpha_{1}|}\cdots
x_{r}^{|\alpha_{r}|}=:x^{T}.$

Thus,

(6.1) $[\theta
x]^{T}=\theta^{T}\sum_{\sigma\in\mathfrak{S}_{n}}\sigma^{-1}\circ
x^{T},\quad\text{where}\quad\theta^{T}=\theta_{{}_{c(T^{a}_{1})}}\cdots\theta_{{}_{c(T^{a}_{m})}}.$

######  Lemma 6.2.

For every superpartition $\Lambda$, there is a bijection between the
subcollection of $sSSYT(\Lambda)$ with weight $(\alpha_{1},\ldots,\alpha_{r})$
and the subcollection of $sSSYT(\Lambda)$ with weight
$(\alpha_{1},\ldots,\alpha_{i-1},\alpha_{i+1},\alpha_{i},\alpha_{i+1},\ldots,\alpha_{r})$.

###### Proof.

Let $g$ be the classic bijection between semistandard Young tableaux of shape
$\lambda$ and weight $(a_{1},\ldots,a_{r})$ and semistandard Young tableaux of
shape $\lambda$ and weight
$(a_{1},\ldots,a_{i-1},a_{i+1},a_{i},a_{i+1},\ldots,a_{r})$ [31, Theorem
7.10.2].

Now, let $G$ be the map sending a super semistandard Young tableau
$(\Lambda_{(0)},\ldots,\Lambda_{(r)})$ of shape $\Lambda$ and weight
$(\alpha_{1},\ldots,\alpha_{r})$ to the super semistandard Young tableau
$(\Lambda_{(0)},\ldots,\Lambda_{(i-1)},\Omega,\Lambda_{(i+1)},\ldots,\Lambda_{(r)})$
of shape
$(\alpha_{1},\ldots,\alpha_{i-1},\alpha_{i+1},\alpha_{i},\alpha_{i+1},\ldots,\alpha_{r})$,
where
$g(\Lambda_{(0)}^{+},\ldots,\Lambda_{(r)}^{+})=(\Lambda_{(0)}^{+},\ldots,\Lambda_{(i-1)}^{+},\Omega^{+},\Lambda_{(i+1)}^{+},\ldots,\Lambda_{(r)}^{+})$,
such that the number of new circles from $\Lambda_{(i-1)}$ to $\Omega$ is the
same as the one from $\Lambda_{(i)}$ to $\Lambda_{(i+1)}$, and similarly the
number of new circles from $\Omega$ to $\Lambda_{(i+1)}$ is the same as the
one from $\Lambda_{(i-1)}$ to $\Lambda_{(i)}$. As $g$ is a bijection and new
circles are uniquely determined, then $G$ is a bijection as well. ∎

######  Lemma 6.3.

For every monomial $u$ in $x$–variables, we have $s_{i}(\sigma^{-1}\circ
u)=\sigma^{-1}\circ(s_{i}(u))$, where $s_{i}=(i\,\,i+1)$,
$\sigma\in\mathfrak{S}_{n}$ and $\circ$ is the action defined in (3.2).

###### Proof.

Let $u=x_{b_{1}}\cdots x_{b_{n}}$. It is enough to take $\sigma=(j\,\,j+1)$.
Then:

$\begin{array}[]{rcl}s_{i}(\sigma^{-1}\circ u)&=&s_{i}(x_{b_{1}}\cdots
x_{b_{j-1}}x_{b_{j+1}}x_{b_{j}}x_{b_{j+2}}\cdots x_{b_{n}}),\\\
&=&x_{s_{i}(b_{1})}\cdots
x_{s_{i}(b_{j-1})}x_{s_{i}(b_{j+1})}x_{s_{i}(b_{j})}x_{s_{i}(b_{j+2})}\cdots
x_{s_{i}(b_{n})},\\\ &=&\sigma^{-1}\circ(x_{s_{i}(b_{1})}\cdots
x_{s_{i}(b_{n})}),\\\ &=&\sigma^{-1}\circ s_{i}(u).\end{array}$

∎

######  Theorem 6.4.

For every $\Lambda\in\mathrm{sPar}_{n,m}$, the Schur function $S_{\Lambda}$ is
symmetric.

###### Proof.

We need to show that for every $T\in sSSYT(\Lambda)$ there is $T^{\prime}\in
sSSYT(\Lambda)$ such that $s_{i}([\theta x]^{T})=[\theta x]^{T^{\prime}}$ with
$s_{i}=(i\,\,i+1)$ for all $i$. Lemma 6.2 implies that there are
$\tau\in\mathfrak{S}_{n}$ and $T^{\prime}\in sSSYT(\Lambda)$, such that
$s_{i}(\theta^{T})=\theta^{T^{\prime}}$ and $s_{i}(x^{T})=\tau^{-1}\circ
x^{T^{\prime}}$. Thus, by applying Lemma 6.3, we have

$s_{i}([\theta
x]^{T})=s_{i}(\theta^{T})\sum_{\sigma\in\mathfrak{S}_{n}}s_{i}(\sigma^{-1}\circ
x^{T})=\theta^{T^{\prime}}\sum_{\sigma\in\mathfrak{S}_{n}}\sigma^{-1}\circ
s_{i}(x^{T})=\theta^{T^{\prime}}\sum_{\sigma\in\mathfrak{S}_{n}}\sigma^{-1}\circ(\tau^{-1}\circ
x^{T^{\prime}})=[\theta x]^{T^{\prime}}.$

∎

Given a super semistandard Young $T$ with $m$ circles, we denote by
$\operatorname{inv}(T)$, the number of inversions of the word
$c(T_{1}^{a})\cdots c(T_{m}^{a})$. As defined in [22], for
$\Lambda,\Omega\in\mathrm{sPar}_{n,m}$ with
$\Omega=(\Omega_{1},\ldots,\Omega_{m};\Omega_{m+1},\ldots,\Omega_{k})$, we set

(6.2) $K_{\Lambda,\Omega}=\sum_{T}(-1)^{\operatorname{inv}(T)},$

where $T$ is a super semistandard Young tableau of shape $\Lambda$ and weight
$(\bar{\Omega}_{1},\ldots,\bar{\Omega}_{m},\Omega_{m+1},\ldots,\Omega_{k})$.

######  Proposition 6.5.

For every superpartition $\Lambda\in\mathrm{sPar}_{n,m}$, we have

$S_{\Lambda}=\sum_{\Omega}K_{\Lambda,\Omega}\Omega!\sum_{\Lambda(I)=\Omega}(-1)^{\varepsilon(I)}m_{I}.$

###### Proof.

By Proposition 3.4(4) and Theorem 6.4, there are scalars $c_{I}$ such that

$S_{\Lambda}=\sum_{I\in s\mathcal{P}_{n,m}}c_{I}m_{I}.$

Let $u\in M(I)$ for some $I\in s\mathcal{P}_{n,m}$. Since $u$ occurs in
$[\theta x]^{T}$ for some $T\in sSSYT(\Lambda)$, then

$u=\theta_{{}_{c(T^{a}_{1})}}\cdots\theta_{{}_{c(T^{a}_{m})}}x_{{}_{c(T^{s}_{\sigma(1)})}}\cdots
x_{{}_{c(T^{s}_{\sigma(n)})}},\qquad\sigma\in\mathfrak{S}_{n}.$

Observe that $c_{I}$ is the coefficient of $u$ in $S_{\Lambda}$, that is,
$c_{I}=d_{I}\Lambda(I)!$, where $d_{I}$ is the number of super semistandard
Young tableaux $T$ such that, for each $i$, the $i$th part of $\Lambda(I)$ is
the number of $i$’s in the boxes of $T$, and $\Lambda(I)!$ is the number of
permutations fixing the set $\\{j\mid c(T_{j}^{s})=i\\}$. Thus,

$S_{\Lambda}=\sum_{I}\Lambda(I)!K_{\Lambda,\Lambda(I)}(-1)^{\varepsilon(I)}m_{I}=\sum_{\Omega}K_{\Lambda,\Omega}\Omega!\sum_{\Lambda(I)=\Omega}(-1)^{\varepsilon(I)}m_{I}.$

∎

For instance,  is the unique super semistandard Young tableau of shape
$(2,1;\,)$ and type $(\bar{2},\bar{1})$, thus

$S_{(2,1;\,)}=2\left(m_{\\{0,1,2\\},\\{0,3\\}}+m_{\\{0,1,3\\},\\{0,2\\}}-m_{\\{0,1\\},\\{0,2,3\\}}\right).$

A remarkable generalization of classic Schur functions are the so–called
_Macdonald polynomials_ [23, Chapter VI], which are symmetric functions with
coefficients given by rational functions in parameters $q$ and $t$. As shown
in [23, Chapter VI], Schur functions can be obtained as a special case of
Macdonald polynomials when setting $q=t$. In superspace, Macdonald polynomials
involving parameters $q$ and $t$ were defined in [8], and so _Schur–type
functions in commutative variables in superspace_ can be gotten as special
limits on these parameters. More precisely, we are interested in two
Schur–type functions $s_{\Lambda}$ and $\bar{s}_{\Lambda}$ obtained by setting
$q=t=0$ and $q=t=\infty$, respectively. Combinatorial formulas for
$s_{\Lambda}$ and $\bar{s}_{\Lambda}$ were given in [22]. Indeed, for a
superpartition $\Lambda$, we have:

$s_{\Lambda}=\sum_{\Omega}K_{\Lambda,\Omega}m_{\Omega},\qquad\bar{s}_{\Lambda}=\sum_{\Omega}\bar{K}_{\Lambda,\Omega}m_{\Omega},$

where $K_{\Lambda,\Omega}$ is defined in (6.2), and $\bar{K}_{\Lambda,\Omega}$
is an integer number related to another kind of super semistandard Young
tableaux which will be defined below.

Let $\Lambda$ be a superpartition, and let
$\alpha_{1},\ldots,\alpha_{r}\in\mathbb{N}_{0}\cup\bar{\mathbb{N}}_{0}$. A
_super semistandard Young tableau of the second kind_ of _shape_ $\Lambda$ and
_weight_ $(\alpha_{1},\ldots,\alpha_{r})$, is a sequence of superpartitions
$(\Lambda_{(0)},\ldots,\Lambda_{(r)})$ with $\Lambda_{(0)}=\emptyset$ and
$\Lambda_{(r)}=\Lambda$, satisfying:

  1. 1.

If $\alpha_{i}\in\mathbb{N}_{0}$ and $\Lambda_{(i-1)}$ contains the $j$th
circle in the $r$th row, then

    1. (a)

$\operatorname{df}(\Lambda_{(i)})=\operatorname{df}(\Lambda_{(i-1)})$.

    2. (b)

$\Lambda_{(i)}^{+}/\Lambda_{(i-1)}^{+}$ is a horizontal $\alpha_{i}$–strip.

    3. (c)

$\Lambda_{(i)}$ contains the $j$th circle in the $r$th row whenever the $r$th
row of $\Lambda_{(i)}^{+}/\Lambda_{(i-1)}^{+}$ is empty.

    4. (d)

$\Lambda_{(i)}$ contains the $j$th circle either in the $r$th row or in the
$(r+1)th$ row whenever the $r$th row of
$\Lambda_{(i)}^{+}/\Lambda_{(i-1)}^{+}$ is nonempty.

  2. 2.

If $\alpha_{i}\in\bar{\mathbb{N}}_{0}$, then

    1. (a)

$\operatorname{df}(\Lambda_{(i)})=\operatorname{df}(\Lambda_{(i-1)})+1$.

    2. (b)

$\Lambda_{(i)}^{\oplus}/\Lambda_{(i-1)}^{\oplus}$ is a horizontal
$(|\alpha_{i}|+1)$–strip.

    3. (c)

The rightmost box of $\Lambda_{(i)}^{\oplus}/\Lambda_{(i-1)}^{\oplus}$, say
the new circle, also belongs to $\Lambda_{(i)}^{\oplus}/\Lambda_{(i)}^{+}$.

    4. (d)

The diagram obtained from $\Lambda_{(i)}$ by removing the new circle above
satisfies conditions in (1).

See Figure 9 for an instance.

$\left(\,\,\emptyset,\quad\vbox{\hbox{\includegraphics{pics/017.pdf}}},\quad\vbox{\hbox{\includegraphics{pics/018.pdf}}},\quad\vbox{\hbox{\includegraphics{pics/019.pdf}}},\quad\vbox{\hbox{\includegraphics{pics/020.pdf}}}\,\,\right)$
Figure 9: A super semistandard Young tableau of the second kind of shape
$(2,0;3)$ and type $(\bar{2},\bar{0},2,1)$.

As with super semistandard Young tableaux, every super semistandard Young
tableau of the second kind can be represented, in the same manner, by filling
with numbers the boxes and circles of its Young diagram.

######  Definition 6.6.

The _Schur function of the second kind_ of a superpartition
$\Lambda\in\mathrm{sPar}_{n,m}$ is the function in superspace, defined as
follows:

$\bar{S}_{\Lambda}=\sum_{T\in\bar{s}SSYT(\Lambda)}\theta_{{}_{c(T^{a}_{1})}}\cdots\theta_{{}_{c(T^{a}_{m})}}\sum_{\sigma\in\mathfrak{S}_{n}}x_{{}_{c(T^{s}_{\sigma(1)})}}\cdots
x_{{}_{c(T^{s}_{\sigma(n)})}},$

where $\bar{s}SSYT(\Lambda)$ is the collection of super semistandard Young
tableux of the second kind of shape $\Lambda$.

The following proposition is analogous to Theorem 6.4 and Proposition 6.5.

######  Proposition 6.7.

For every $\Lambda\in\mathrm{sPar}_{n,m}$, the Schur function of the second
kind $\bar{S}_{\Lambda}$ is symmetric, and

$\bar{S}_{\Lambda}=\sum_{\Omega}\bar{K}_{\Lambda,\Omega}\Omega!\sum_{\Lambda(I)=\Omega}(-1)^{\varepsilon(I)}m_{I}.$

######  Proposition 6.8.

For every superpartitions $\Lambda,\Omega\in\mathrm{sPar}_{n,m}$, we have

$\begin{array}[]{rclrclrcl}\rho(S_{\Lambda})&=&n!s_{\Lambda},&\tilde{\rho}(n!s_{\Lambda})&=&S_{\Lambda},&\langle
S_{\Lambda},S_{\Omega}\rangle&=&(-1)^{\binom{m}{2}}n!^{2}\delta_{\Lambda,\Omega},\\\\[8.5359pt]
\rho(\bar{S}_{\Lambda})&=&n!\bar{s}_{\Lambda},&\tilde{\rho}(n!\bar{s}_{\Lambda})&=&\bar{S}_{\Lambda},&\langle\bar{S}_{\Lambda},\bar{S}_{\Omega}\rangle&=&(-1)^{\binom{m}{2}}n!^{2}\delta_{\Lambda,\Omega}.\end{array}$

###### Proof.

By Proposition 6.5 and the definition of $\tilde{\rho}$ in (5.1), we have

$S_{\Lambda}=\sum_{\Omega\preceq\Lambda}K_{\Lambda,\Omega}\Omega!\sum_{\Lambda(I)=\Omega}(-1)^{\sigma(I)}m_{I}=\sum_{\Omega\preceq\Lambda}K_{\Lambda,\Omega}\Omega!\tilde{\rho}(m_{\Omega})\frac{n!}{\Omega!}=n!\sum_{\Omega\preceq\Lambda}K_{\Lambda,\Omega}\,\tilde{\rho}(m_{\Omega}).$

Now, by applying Proposition 5.6, we obtain

$\rho(S_{\Lambda})=n!\sum_{\Omega\preceq\Lambda}K_{\Lambda,\Omega}m_{\Omega}=n!s_{\Lambda},\qquad\tilde{\rho}(n!s_{\Lambda})=\sum_{\Omega\preceq\Lambda}n!K_{\Lambda,\Omega}\,\tilde{\rho}(m_{\Omega})=\sum_{\Omega\preceq\Lambda}K_{\Lambda,\Omega}\Omega!\sum_{\Lambda(I)=\Omega}(-1)^{\varepsilon(I)}m_{I}=S_{\Lambda}.$

On the other hand, by applying Proposition 5.11, we have

$\left\langle
S_{\Lambda},S_{\Omega}\right\rangle=\left\langle\tilde{\rho}(n!s_{\Lambda}),\tilde{\rho}(n!s_{\Omega})\right\rangle=n!^{2}\left\langle
s_{\Lambda},s_{\Omega}\right\rangle=n!^{2}(-1)^{\binom{m}{2}}\delta_{\Lambda,\Omega}.$

The rest of the equalities are proved in the same manner. ∎

Observe that $\\{S_{\Lambda}\\}_{\Lambda}$ and
$\\{\bar{S}_{\Lambda}\\}_{\Lambda}$ are not bases for $\mathrm{\bf sNCSym}$.
Indeed, bases of $\mathrm{\bf sNCSym}$ are indexed by set superpartitions,
however there is no a bijection between superpartitions and set
superpartitions. In the non-superspace case, a first definition of Schur
functions was given in [27] by using classic semistandard Young tableaux,
nevertheless these ones do not constitute a basis of $\mathrm{\bf NCSym}$
either. A basis of Schur–type functions for $\mathrm{\bf NCSym}$ was given in
[2] by using a noncommuting version of the Jacobi–Trudi determinant. They also
proved that Schur functions in [27] are sums of them. In superspace, even in
the commutative case, there is no a version of the Jacobi–Trudi determinant.
Thus, the Jacobi–Trudi identity for Schur functions is still an open problem,
as does the case for our Schur–type functions.

#### Acknowledgements

The first author was supported by the grant DIDULS PR232146.

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