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# Rational Schur Superalgebras

Andrew Riesen

###### Abstract

We develop and study the generalization of rational Schur algebras to the
super setting. Similar to the classical case, this provides a new method for
studying rational supermodules of the general linear supergroup $GL(m|n)$.
Furthermore, we establish a Schur-Weyl duality result for rational Schur
superalgebras and conclude that under certain conditions these objects will be
semisimple.

##  1 Introduction

In the 1980s, Green demonstrated that understanding the polynomial
representations of $GL(n)$ over $\mathbb{C}$ is intimately linked to
understanding the representation theory of Schur algebras [5]. More precisely,
he showed that any finite-dimensional polynomial representation of $GL(n)$
over $\mathbb{C}$ can be split into homogeneous representations and the
category of such degree-$r$ homogeneous representations is equivalent to the
category of modules of the Schur algebra $S(n,r)$. In [4], Dotty and Dipper
presented a generalization of Schur algebras called rational Schur algebras.
These were defined over a field $k$ of arbitrary characteristic and shown to
have a similar connection with rational representations of $GL(n)$. A key
component to understanding the structure of (rational) Schur algebras is an
equivalence of three definitions. For example, the Schur algebra $S(n,r)$ is
defined as the dual of degree-$r$ homogeneous polynomials on $GL(n)$. It is
isomorphic to the centralizer of the action of the symmetric group $S_{r}$ on
$(k^{n})^{\otimes r}$, and to the image $\rho_{r}(U(\mathfrak{gl}(n)))$, where
$\rho_{r}$ denotes the canonical representation of $\mathfrak{gl}(n)$ on
$(k^{n})^{\otimes r}$. Analogous definitions are equivalent for the rational
Schur algebra. In this paper, we generalize the connection rational Schur
algebras have with the general linear group and demonstrate a similar
equivalence of definitions in the context of supermathematics.

Supermathematics was first studied in the 1980s and arose from the
mathematical framework required for supersymmetry in physics. It has been a
theme of modern mathematics to develop ideas from representation theory to the
super case and study how the properties change or stay the same. We continue
this theme by studying rational Schur algebras in the super case, which we
call rational Schur superalgebras. Similar notions were also considered in
[11].

Let $m,n\geq 0$. The super analogue of the general linear group is called the
general linear supergroup, denoted by $GL(m|n)$. Contrary to the classical
case, $GL(m|n)$ can only be described as a particular representable functor.
In addition, properties that hold in the classical case fail in the super
case, such as complete reducibility. Due to this, the representation theory of
the general linear supergroup is more difficult to understand. Though one can
still consider the super analogue of polynomial and rational representations
of $GL(m|n)$, unlike in the classical case, not all rational representations
can be obtained from polynomial representations alone. This is because the
super analogue of the determinant, the Berezinian, is not a polynomial
function of $GL(m|n)$. Thus, the classical argument of tensoring a finite-
dimensional rational representation by a large power of the determinant
representation will no longer produce a polynomial representation in the super
setting. For this reason, we believe that understanding the super analogue of
the rational Schur algebra will provide an important tool for studying
rational representations of $GL(m|n)$.

This paper is organized as follows. First, we review notions and definitions
from supermathematics. Then we introduce a superbialgebra
$\tilde{\mathcal{A}}(m|n)$, which will play the role of the polynomial ring
$k[x_{i,j}:1\leq i,j\leq n]$ in the classical setting, and study its
structure. From this superbialgebra we obtain rational Schur superalgebras,
introduce their category of supermodules, and clarify the connection they have
to representations of the general linear supergroup. The rest of this paper is
devoted to proving a semisimplicity result for rational Schur superalgebras
and an equivalence of three definitions. We approach this problem by
establishing a Hopf superalgebra isomorphism between
$\tilde{\mathcal{A}}(m|n)$ and the Hopf superalgebra of regular functions on
$GL(m|n)$, which was introduced by Scheunnert and Zhang in [14]. This
isomorphism then allows us to prove an equivalence of definitions and the
semisimplicity of rational Schur superalgebras under certain conditions.

In future work we hope to generalize the results of this paper to all rational
Schur superalgebras of a fixed $(m,n)$, similar to what has been achieved for
Schur superalgebras. By combining Theorem $3.1$ with the remarks in Section
$3.3$, this should provide a new way to understand the representation theory
of $GL(m|n)$.

##  2 Preliminaries

First, recall that a superspace $V$ over a field $k$ is a $k$-vector space
with a $\mathbb{Z}/2$-grading. That is, $V=V_{0}\oplus V_{1}$, where elements
of $V_{0}$ are referred to as even elements and elements of $V_{1}$ as odd. If
$x\in V_{0}$ or $x\in V_{1}$ it is called homogeneous and its parity is
defined by

$|x|=\begin{cases}0&\text{ if }x\in V_{0}\\\ 1&\text{ if }x\in
V_{1}\end{cases}$

If $V$ is a finite-dimensional superspace such that $\dim(V_{0})=m$ and
$\dim(V_{1})=n$, then its super-dimension is referred to as $(m|n)$. For
example, the standard $(m|n)$-dimensional superspace, $k^{m|n}$, is defined as
$k^{m}\oplus k^{n}$.

A morphism of superspaces $V$ and $W$ is a $k$-linear map $\phi:V\rightarrow
W$ such that $\phi(V_{i})\subseteq W_{i}$ for $i=0,1$. Such morphisms will be
referred to as even linear maps.

Given superspaces $V$ and $W$, we may endow their tensor product
$V\otimes_{k}W$ with a superspace structure by defining the even and odd
components as

$(V\otimes_{k}W)_{0}=(V_{0}\otimes_{k}W_{0})\oplus(V_{1}\otimes_{k}W_{1})$
$(V\otimes_{k}W)_{1}=(V_{0}\otimes_{k}W_{1})\oplus(V_{1}\otimes_{k}W_{0})$

An associative $k$-superalgebra is then a superspace $A$ which is an
associative $k$-algebra such that its structure maps:

$\Delta:A\otimes_{k}A\rightarrow A,\quad\ \eta:k\rightarrow A$

are even linear maps. Since we will be working exclusively with associative
$k$-superalgebras we will simply refer to them as superalgebras. A
homomorphism of superalgebras $\phi:A\rightarrow B$ is then a homomorphism of
associative algebras such that $\phi$ is an even linear map. We say that a
superalgebra $A$ is supercommutative if for all homogeneous $x,y\in A$ the
following holds:

$xy=(-1)^{|x|\cdot|y|}yx$

Also, if $A,B$ are superalgebras, then so is $A\otimes_{k}B$ with
multiplication defined on homogeneous elements $x,w\in A$ and $y,z\in B$ by

$(x\otimes y)\cdot(w\otimes z)=(-1)^{|y||w|}((xw)\otimes(yz))$

and then extended linearly to all of $A\otimes_{k}B$.

Supercoalgebras, supermodules, supercomodules and their morphisms are defined
in the exact same way as their classical counterparts, except the structure
maps must respect the $\mathbb{Z}/2$-gradings as demonstrated above by the
superalgebra definition. See [3] for details.

###  2.1 The General Linear Supergroup

In this section we recall for the sake of completeness some definitions and
facts pertaining to the general linear supergroup. All superspaces and
superalgebras are assumed to be over a fixed field $k$. See the introductions
of [16, 9] for a more complete treatment.

Contrary to the classical case, the general linear supergroup of a superspace
is most naturally defined as a functor, in the following way. Let $M$ be a
finite-dimensional superspace and fix a homogeneous basis
$\\{m_{1},\ldots,m_{t}\\}$. Then $M\otimes_{k}A$ is a right $A$-supermodule
and has a $A$-linear basis given by $\mathcal{C}=\\{m_{i}\otimes 1:1\leq i\leq
t\\}$. The general linear supergroup $GL(M)$ is a functor from the category of
finite dimensional supercommutative superalgebras to the category of groups.
It associates to every supercommutative superalgebra $A$ all of the invertible
even $A$-linear maps on $M\otimes_{k}A$, which we denote by $GL(M)(A)$.
Furthermore, if $f:A\rightarrow B$ is a morphism of supercommutative
superalgebras, then $GL(M)$ sends it to $GL(M)(f):GL(M)(A)\rightarrow
GL(M)(B)$, which is defined by writing an element of $T\in GL(M)(A)$ as a
$t\times t$ matrix (with respect to the basis $\mathcal{C}$) with entries in
$A$ and applying $f$ to each entry. In symbols, the general linear supergroup
$GL(M)$ is the functor which sends objects

$A\mapsto(\operatorname{End}_{A}(M\otimes_{k}A))^{\times}$

and morphisms

$f\in\operatorname{Hom}_{\mathbb{superalg}}(A,B)\ \text{ to }\ \
GL(M)(f)((a_{i,j})_{1\leq i,j\leq t})=(f(a_{i,j}))_{1\leq i,j\leq t}$

This definition is independent of the chosen basis of $M$.

In general $GL(M)$ will be a representable functor, but of particular interest
to us is the case where $M=k^{m|n}$. The representing object of $GL(k^{m|n})$
is referred to as its coordinate superalgebra and is defined in the following
way.

######  Definition 2.1.

[16] Let $\mathcal{A}(m|n)=k[x_{ij}:1\leq i,j\leq m+n]$ be the free
supercommutative superalgebra on $(m+n)\times(m+n)$ variables, with the parity
on the variables given by

$|x_{ij}|:=\overline{i}+\overline{j}\mod 2,\text{ where
}\overline{i}=\begin{cases}\overline{0}&\text{ if }1\leq i\leq m\\\
\overline{1}&\text{ if }m+1\leq i\leq m+n\end{cases}$

Let $d_{1}=\det(x_{ij})_{1\leq i,j\leq m}$ and $d_{2}=\det(x_{ij})_{m+1\leq
i,j\leq m+n}$, which are even elements of $\mathcal{A}(m|n)$. The _coordinate
superalgebra of $GL(k^{m|n})$_ is then defined as the localization of
$\mathcal{A}(m|n)$ at $d_{1}d_{2}$.

From now on we will denote $GL(k^{m|n})$ by $GL(m|n)$ and its coordinate
superalgebra by $k[GL(m|n)]$. Since the general linear supergroup is defined
as a functor, it is natural for its supermodules to be morphisms of functors.

######  Definition 2.2.

Let $M$ be a finite-dimensional superspace. Then it is a _left
$GL(m|n)$-supermodule_ if there is a natural transformation
$p:GL(m|n)\rightarrow GL(M)$.

The coordinate superalgebra of $GL(m|n)$ is also a Hopf superalgebra, with
comultiplication and counit structure maps given respectively by:

$\Delta(x_{i,j})=\sum_{k=1}^{n+m}x_{i,k}\otimes x_{k,j},\ \ \ \
\epsilon(x_{i,j})=\delta_{i,j}$

As described in [1] there is an equivalence between the category of left
$GL(m|n)$-supermodules and right $k[GL(m|n)]$-supercomodules. We use this
identification implicitly from now on when talking about
$GL(m|n)$-supermodules.

The following definitions and facts will be needed to define rational Schur
superalgebras.

######  Definition 2.3.

Suppose that $M$ is a finite-dimensional right supercomodule of $k[GL(m|n)]$,
with structure map $\tau:M\rightarrow M\otimes_{k}k[GL(m|n)]$. Let
$\\{v_{1},..,v_{r}\\}$ be a homogeneous basis of $M$. Then
$\tau(v_{j})=\sum_{i=1}^{r}v_{i}\otimes a_{ij}$, with $a_{ij}\in k[GL(m|n)]$.
The matrix formed from these $k[GL(m|n)]$ coefficients, $A:=(a_{ij})_{1\leq
i,j\leq r}$, is called the _defining matrix_ of the representation.

######  Definition 2.4.

Let $M$ be a right supercomodule of $k[GL(m|n)]$. Then the _coefficient space
of $M$_, denoted $cf(M)$, is the superspace spanned by the entries of its
defining matrix.

These constructions are independent of the choice of basis. Furthermore, if
$M,N$ are representations of $GL(m|n)$, then $M\otimes_{k}N$ is as well and
the following equality holds in $k[GL(m|n)]$:

$cf(M\otimes_{k}N)=cf(M)\cdot cf(N)$

###  2.2 Coefficient Spaces of Superalgebras

The following results will be needed in Section $4$.

Let $M$ be a finite-dimensional superspace and $A$ a superalgebra. Using the
identification $M\otimes_{k}M^{*}\cong\operatorname{End}_{k}(M)$ we may endow
$\operatorname{End}_{k}(M)$ with the structure of a superspace. It can be
easily checked that composition turns $\operatorname{End}_{k}(M)$ into a
superalgebra. So, by a representation $M$ of $A$ we mean a superalgebra
homomorphism $\rho:A\rightarrow\operatorname{End}_{k}(M)$.

######  Definition 2.5.

Let $M$ be a finite-dimensional superspace with homogeneous basis
$B_{M}=\\{v_{1},...,v_{\ell}\\}$. If
$\rho:A\rightarrow\operatorname{End}_{k}(M)$ is a representation of $A$, then
with respect to the basis $B_{M}$, $\rho(a)$ is an $\ell\times\ell$ matrix
with entries in $k$. Let $1\leq i,j\leq\ell$. Then the $(i,j)$th entry of
$\rho(a)$ depends linearly on $a$ and so defines an element of $A^{*}$, which
we denote as $a_{i,j}$. The _defining matrix of $\rho$_ is the matrix
$(a_{i,j})_{1\leq i,j\leq\ell}\in\operatorname{Mat}_{\ell\times\ell}(A^{*})$.

######  Definition 2.6.

Let $A,M,\rho$ be as above. Then the _coefficient space of $M$_ is

$cf_{A}(M)=\operatorname{span}_{k}\\{a_{i,j}:1\leq i,j\leq l\\}$

It is well known that $cf_{A}(M)$ is independent of the choice of basis of $M$
and so $cf_{A}(M)$ is well-defined. We will also need the following fact
observed in [14, Pg. 53].

######  Lemma 2.1.

Let $\rho:A\rightarrow\operatorname{End}_{k}(M)$ be a finite-dimensional
representation of $A$. Then $(cf_{A}(M))^{*}$ is a superalgebra that is
isomorphic to $\rho(A)$.

##  3 Category of Supermodules

Throughout this section, $k$ will denote an algebraically closed field of
characteristic not equal to $2$, and all $GL(m|n)$-supermodules are left
$GL(m|n)$-supermodules.

###  3.1 Polynomial Representations of $GL(m|n)$

To motivate the results for rational representations of $GL(m|n)$, we recall
some basic facts about $\mathcal{A}(m|n)$ and define what polynomial
representations of $GL(m|n)$ are.

Let $\mathcal{A}(m|n)$ be as in Definition 2.1. Like the coordinate
superalgebra $k[GL(m|n)]$, it is a superbialgebra [12].

######  Definition 3.1.

For a positive integer $r$, define the _degree- $r$ homogeneous polynomials_
of $\mathcal{A}(m|n)$ to be:

$\mathcal{A}(m|n;r):=\operatorname{span}_{k}\\{x_{i_{i},j_{1}}...x_{i_{r},j_{r}}:1\leq
i_{r},j_{r}\leq m+n\\}$

If $r=0$, then $\mathcal{A}(m|n;0):=k$.

It follows easily from the definition of comultiplication in $k[GL(m|n)]$ that
$\mathcal{A}(m|n;r)$ is a sub-supercoalgebra of $k[GL(m|n)]$. As in the
classical case we have a supercoalgebra decomposition:

$\mathcal{A}(m|n)=\bigoplus_{r=0}^{\infty}\mathcal{A}(m|n;r)$

######  Definition 3.2.

Let $M$ be a $GL(m|n)$-supermodule. Then $M$ is a _polynomial representation_
if $cf(M)\subseteq\mathcal{A}(m|n)$.

A _degree- $r$ homogeneous representation_ of $GL(m|n)$ is a
$GL(m|n)$-supermodule $M$ such that $cf(M)\subseteq\mathcal{A}(m|n;r)$.

###  3.2 Rational Representations of Bidegree $(r,s)$

Suppose that $M$ is a left $GL(m|n)$-supermodule and let

$i:GL(m|n)\rightarrow GL(m|n)$

be the inversion morphism. This induces a dual map
$i^{*}:k[GL(m|n)]\rightarrow k[GL(m|n)]$ which is the anti-pode of
$k[GL(m|n)]$. Since defining rational Schur superalgebras requires the map
$i^{*}$ and coefficient spaces from a dual representation, we will discuss
these objects in more detail. Consider the $(m+n)\times(m+n)$ matrix
$(x_{k,\ell})_{1\leq k,\ell\leq m+n}$ with entires in $k[GL(m|n)]$. Then
$i^{*}(x_{p,q})$ is the $(p,q)th$ entry of $((x_{k,\ell})_{1\leq k,\ell\leq
m+n})^{-1}$, and these entries will be in $k[GL(m|n)]$. If $C$ is the defining
matrix of a supercomodule $M$ of $k[GL(m|n)]$, then $M^{*}$ is a supercomodule
of $k[GL(m|n)]$ and $(C^{-1})^{\mathbb{st}}$ is its defining matrix [16] .
Here $\mathbb{st}$ denotes the supertranspose:

######  Definition 3.3.

Suppose that $V$ is a finite-dimensional $A$-supermodule with ordered
homogeneous basis
$B_{V}=(v_{1},...,v_{\ell}|\overline{v_{1}},...,\overline{v_{\ell^{\prime}}})$,
and $T\in\operatorname{End}_{A}(V)$ is even. Write
$[T]_{B_{V}}=\begin{bmatrix}T_{1}&T_{2}\\\ T_{3}&T_{4}\end{bmatrix}$. Then the
_supertranspose_ of $[T]_{B_{V}}$ is

$[T]_{B_{V}}^{st}=\begin{bmatrix}(T_{1})^{t}&(T_{3})^{t}\\\
-(T_{2})^{t}&(T_{4})^{t}\end{bmatrix}$

For the sake of brevity, denote $V=k^{m|n}$ and $W=V^{*}$. Then $V$ is a left
$GL(m|n)$-supermodule with structure map given by

$\tau(e_{i})=\sum_{k=1}^{m+n}e_{k}\otimes x_{ki}$

Evidently, $cf(V^{\otimes r})=\mathcal{A}(m|n;r)$.

######  Definition 3.4.

Let $V$ and $W$ be as above. Then the $(r,s)$-homogeneous component of
$k[GL(m|n)]$ is

$\tilde{\mathcal{A}}(m|n;r,s):=cf(V^{\otimes r}\otimes_{k}W^{\otimes
s})=cf(V)^{r}cf(W)^{s}$

The space generated by all such components is

$\tilde{\mathcal{A}}(m|n):=\sum_{(r,s)\in\mathbb{Z}_{\geq
0}^{2}}\tilde{\mathcal{A}}(m|n;r,s)$

We would like to describe what elements of $\tilde{\mathcal{A}}(m|n;r,s)$ look
like for fixed $(r,s)$. To do this we need the notion of a parity functor and
the super analogue of the determinant, the Berezinian. The parity functor
$\Pi$ is a functor on superspaces which takes a superspace $V=V_{0}\oplus
V_{1}$ to the superspace $\Pi(V)$ with even component $V_{1}$ and odd
component $V_{0}$, while fixing all even linear maps. In other words, it
switches the parity of superspaces.

######  Definition 3.5.

Let $T,V$ be as in Definition 3.3, and set
$B_{\Pi(V)}=(\overline{v_{1}},...,\overline{v_{\ell^{\prime}}}|v_{1},...,v_{\ell})$.
Then the _Berezinian_ is

$Ber(T):=\det(T_{1}-T_{2}T_{4}^{-1}T_{3})\det(T_{4})^{-1}$

If $T$ is an even $A$-linear map, then
$[\Pi(T)]_{B_{\Pi(V)}}=\begin{bmatrix}T_{4}&T_{3}\\\
T_{2}&T_{1}\end{bmatrix}$, and

$Ber^{*}(T):=Ber(\Pi(T))$

This definition is independent of the homogeneous basis chosen for $V$.

######  Proposition 3.1.

[8]

  1. 1.

$Ber(ST)=Ber(S)Ber(T)$

  2. 2.

$Ber^{*}(ST)=Ber^{*}(S)Ber^{*}(T)$

Recall that $\mathcal{A}(m|n)=k[x_{ij}:1\leq i,j\leq m+n]$, and consider the
matrix $A=(x_{i,j})_{1\leq i,j\leq m+n}$. Then $Ber,Ber^{*}$ are elements of
$k[GL(m|n)]$ represented by $Ber(A),Ber^{*}(A)$. Let $D(i,j)$ be the matrix
formed by replacing the $i$th row of $A$ with zeros except the $(i,j)$th entry
which is $1$. Evidently, $Ber(D(i,j)),Ber^{*}(D(i,j))\in k[GL(m|n)]$, and we
denote these elements by $Ber_{i,j},Ber^{*}_{i,j}$ respectively. Lastly,
denote by $\tilde{x}_{i,j}$ the $(i,j)$th entry of $A^{-1}$.

We may now give an explicit description of $\tilde{\mathcal{A}}(m|n)$.

######  Lemma 3.1.

The following properties hold:

  1. 3.1, Part 1.

$\tilde{\mathcal{A}}(m|n)$ is generated as a $k$-superalgebra by the elements
$x_{i,j},\tilde{x}_{i,j}$

  2. 3.1, Part 2.

By the super Cramer rule [8], $cf(W)$ is the $k$-space spanned by
$\tilde{x}_{i,j}$ where

$\tilde{x}_{i,j}=\begin{cases}Ber_{i,j}(Ber)^{-1}&\text{ when
}i\in\\{1,...,m\\}\\\ Ber^{*}_{i,j}(Ber^{*})^{-1}&\text{ when
}i\in\\{m+1,...,m+n\\}\end{cases}$

  3. 3.1, Part 3.

$\tilde{\mathcal{A}}(m|n;r,s)$ is the span of the following set

$\\{x_{i_{1},j_{1}}^{m_{1}}\ldots
x_{i_{t},j_{t}}^{m_{t}}\tilde{x}_{i_{t+1},j_{t+1}}^{m_{t+1}}\ldots\tilde{x}_{i_{t+k},j_{t+k}}^{m_{t+k}}:1\leq
i_{l},j_{l}\leq m+n,\ m_{1}+...+m_{t}=r,m_{t+1}+...+m_{t+k}=s\\}$

  4. 3.1, Part 4.

$\sum_{k=1}^{m+n}x_{i,k}\tilde{x}_{k,j}=\delta_{i,j}\ \ \ \ \ \text{ and }\ \
\ \ \ \sum_{k=1}^{m+n}\tilde{x}_{i,k}x_{k,j}=\delta_{i,j}$

Note that $Ber^{*}$ is the multiplicative inverse of $Ber$, but in general
$Ber^{*}_{i,j}$ and $Ber_{i,j}$ are not invertible (see the example at the end
of section eight in [8]).

We would like to establish a supercoalgebra decomposition of
$\tilde{\mathcal{A}}(m|n)$ in terms of $\tilde{\mathcal{A}}(m|n;r,s)$, which
is similar to the one given for $\mathcal{A}(m|n)$. To do this we need to
establish that $\tilde{\mathcal{A}}(m|n;r,s)$ is a supercoalgebra and to that
end we prove the following:

######  Lemma 3.2.

(Compare with [4, Equation 2.2.2])

$\Delta(\tilde{x}_{i,j})=\sum_{k=1}^{m+n}(-1)^{(|i|+|k|)(|k|+|j|)}(\tilde{x}_{k,j}\otimes\tilde{x}_{i,k})\
\ \ \epsilon(\tilde{x}_{i,j})=\delta_{i,j}$

###### Proof.

Suppose that $1\leq i,j,s,t\leq m+n$, and set
$C_{i,j}:=\sum_{k=1}^{m+n}(-1)^{(|i|+|k|)(|k|+|j|)}\tilde{x}_{k,j}\otimes\tilde{x}_{i,k}$.
Then

$\Delta(x_{s,t})C_{i,j}=(\sum_{p=1}^{m+n}x_{s,p}\otimes
x_{p,t})(\sum_{k=1}^{m+n}(-1)^{(|i|+|k|)(|k|+|j|)}\tilde{x}_{k,j}\otimes\tilde{x}_{i,k})=$
$\sum_{p,k=1}^{m+n}(-1)^{(|i|+|k|)(|k|+|j|)}(-1)^{(|p|+|t|)(|k|+|j|)}(x_{s,p}\tilde{x}_{k,j})\otimes(x_{p,t}\tilde{x}_{i,k})$

Using this calculation and the relation
$\sum_{p=1}^{m+n}x_{i,p}\tilde{x}_{p,j}=\delta_{i,j}$ we find

$\displaystyle\sum_{w=1}^{m+n}\Delta(x_{s,w})C_{w,j}$
$\displaystyle=\sum_{w=1}^{m+n}\sum_{p,k=1}^{m+n}(-1)^{(|p|+|w|)(|k|+|j|)+|w|(|k|+|j|)+|k|(|k|+|j|)}(x_{s,p}\tilde{x}_{k,j})\otimes(x_{p,w}\tilde{x}_{w,k})$
$\displaystyle=\sum_{p,k=1}^{m+n}\sum_{w=1}^{m+n}(-1)^{|p|(|k|+|j|)+|k|(|k|+|j|)}(x_{s,p}\tilde{x}_{k,j})\otimes(x_{p,w}\tilde{x}_{w,k})$
$\displaystyle=\sum_{p,k=1}^{m+n}(-1)^{(|p|+|k|)(|k|+|j|)}(x_{s,p}\tilde{x}_{k,j})\otimes(\sum_{w=1}^{m+n}x_{p,w}\tilde{x}_{w,k})$
$\displaystyle=\sum_{p,k=1}^{m+n}(-1)^{(|p|+|k|)(|k|+|j|)}(x_{s,p}\tilde{x}_{k,j})\otimes(\delta_{p,k})$
$\displaystyle=\sum_{p=1}^{m+n}(-1)^{(|p|+|p|)(|p|+|j|)}(x_{s,p}\tilde{x}_{p,j})\otimes
1=(\sum_{p=1}^{m+n}x_{s,p}\tilde{x}_{p,j})\otimes 1=\delta_{s,j}$

Consider now the $(m+n)\times(m+n)$ matrix $A=(\Delta(x_{i,j}))_{1\leq i,j\leq
m+n}$ with entries in $k[GL(m|n)]\otimes_{k}k[GL(m|n)]$. This has an inverse
given by $(\Delta(\tilde{x}_{i,j}))_{1\leq i,j\leq m+n}$ as comultiplication
is a superalgebra morphism. If $C$ denotes the matrix $(C_{i,j})_{1\leq
i,j\leq m+n}$ then the calculation above shows that $AC=Id$. This implies that
$C=(\Delta(\tilde{x}_{i,j}))$ and proves the first part of the lemma. The
second part follows in a similar way. ∎

######  Proposition 3.2.

$\tilde{\mathcal{A}}(m|n;r,s)$ is a supercoalgebra and
$\tilde{\mathcal{A}}(m|n)$ is a superbialgebra

###### Proof.

This follows immediately from Lemma 3.2 and Lemma 3.1, Part 3. ∎

Lemma 3.1, Part 4 implies that

$\tilde{\mathcal{A}}(m|n;r,s)\subseteq\tilde{\mathcal{A}}(m|n;r+1,s+1)$

Following [4] we define for $z\in\mathbb{Z}_{\geq 0}$:

$\tilde{\mathcal{A}}(m|n)_{z}:=\bigcup_{t\geq
0}\tilde{\mathcal{A}}(m|n;z+t,t)\text{ and
}\tilde{\mathcal{A}}(m|n)_{-z}:=\bigcup_{t\geq
0}\tilde{\mathcal{A}}(m|n;t,z+t)$

Evidently, these are supercoalgebras. In the aforementioned paper, the authors
provide a coalgebra decomposition for the classical case
$\tilde{\mathcal{A}}(m|0)$ [4, 2.6.1]. The super analogue of this result holds
as well

$\tilde{\mathcal{A}}(m|n)=\bigoplus_{z\in\mathbb{Z}}\tilde{\mathcal{A}}(m|n)_{z}$
(1)

The proof of this is similar to the classical one, which we now briefly
outline. The only way to realize $w\in\tilde{\mathcal{A}}(m|n;r,s)$ as an
element of $\tilde{\mathcal{A}}(m|n;e,f)$ is to repeatedly use the relations
$w(\sum_{k=1}^{m+n}x_{i,k}\tilde{x_{k,i}})=w\cdot 1$ or
$(\sum_{k=1}^{m+n}\tilde{x}_{i,k}x_{k,i})w=1\cdot w$. This means there must
exist some $t\geq 0$, such that $r+t=e,s+t=f$. Thus, if $r\geq s$ this implies
that $w\in\tilde{\mathcal{A}}(m|n)_{r-s}$, and similarly if $s>r$ then
$w\in\tilde{\mathcal{A}}(m|n)_{-(s-r)}$. So in either case,
$\tilde{\mathcal{A}}(m|n;r,s),\tilde{\mathcal{A}}(m|n;e,f)\subseteq\tilde{\mathcal{A}}(m|n)_{r-s}$.
This shows the sum in Equation 1 is direct.

######  Definition 3.6.

If $M$ is a $GL(m|n)$-supermodule and
$cf(M)\subseteq\tilde{\mathcal{A}}(m|n)_{z}$, then it is a supermodule of
_rational degree $z$_. Furthermore, if
$cf(M)\subseteq\tilde{\mathcal{A}}(m|n;r,s)$ and
$cf(M)\not\subset\tilde{\mathcal{A}}(m|n;r-1,s-1)$, then it is a supermodule
of _bidegree $(r,s)$_.

######  Theorem 3.1.

(Compare with [4, Theorem 2.7]) If $V$ is a finite-dimensional
$GL(m|n)$-supermodule, then the following holds.

  1. 1.

$V=\bigoplus_{z\in\mathbb{Z}}V_{z}$, where $V_{z}$ is a supermodule of
rational degree $z$.

  2. 2.

Each $V_{z}$ is the union of an ascending chain of $GL(m|n)$-supermodules as
follows:

    1. (a)

If $z\geq 0$, then $V_{z,0}\subseteq V_{z+1,1}\subseteq....\subseteq
V_{z+t,t}\subseteq...$, where $V_{z+t,t}$ is a supermodule of bidegree
$(z+t,t)$ and $V_{z}=\bigcup_{t=0}^{\infty}V_{z,t}$.

    2. (b)

If $z<0$, then $V_{0,-z}\subseteq V_{1,-z+1}\subseteq...\subseteq
V_{t,-z+t}\subseteq....$, where $V_{t,-z+t}$ is a supermodule of bidegree
$(t,-z+t)$ and $V_{z}=\bigcup_{t=0}^{\infty}V_{t,-z+t}$.

###### Proof.

Lemma 8.3 from [11], states that $x_{i,j},\tilde{x}_{i,j}$ for $1\leq i,j\leq
m+n$ generate $k[GL(m|n)]$. Thus, $\tilde{\mathcal{A}}(m|n)=k[GL(m|n)]$ (See
also Corollary 4.1 for a self-contained proof of this fact). Similar to Dipper
and Doty’s proof of Theorem 2.7 in [4], by combining Equation 1 with Theorem
1.6c from [6] we obtain the $k[GL(m|n)]$-comodule decomposition

$V=\bigoplus_{z\in\mathbb{Z}}V_{z}$

This comodule decomposition is actually a supercomodule decomposition. For if
$\tau:V\rightarrow V\otimes_{k}k[GL(m|n)]$ denotes the supercomodule structure
map, then in Green’s proof of Theorem 1.6c,
$V_{z}=\tau^{-1}(V\otimes_{k}\tilde{\mathcal{A}}(m|n)_{z})$. Since the pre-
image of a superspace under an even linear map is a superspace, we see that
$V_{z}$ is a supercomodule. Thus completing the proof of the first statement.

The proof of the second statement is essentially identically to Dipper and
Doty’s proof in Theorem 2.7 of [4], and so is omitted.

∎

######  Definition 3.7.

Fix non-negative integers $m,n,r,s$. The _rational Schur superalgebra_ , which
we denote as $\mathcal{S}(m|n;r,s)$, is $(\tilde{\mathcal{A}}(m|n;r,s))^{*}$.

By Proposition 3.2 and Lemma 3.1, Part 3 this will indeed be a finite-
dimensional superalgebra.

###  3.3 Equivalence of Categories

It is well know that if $A$ is a finite-dimensional supercoalgebra, then its
category of supercomodules is equivalent to the category of supermodules over
$A^{*}$. As in the classical case we define,

$\mathcal{S}(m|n)_{z}:=(\tilde{\mathcal{A}}(m|n)_{z})^{*}$

This equals the following inverse limit when $z\geq 0$ or $z<0$ respectively:

$\varprojlim_{t\in\mathbb{Z}_{\geq 0}}\mathcal{S}(m|n;z+t,t),\ \ \ \ \ \
\varprojlim_{t\in\mathbb{Z}_{\geq 0}}\mathcal{S}(m|n;t,-z+t)$

Since every $GL(m|n)$-supermodule can be decomposed into rational supermodules
of varying degrees, this suggests that understanding the category of
supercomodules of $\tilde{\mathcal{A}}(m|n)_{z}$ will give some insight into
the representation theory of $GL(m|n)$. As we have just mentioned, this is
equivalent to understanding the representation theory of
$\mathcal{S}(m|n)_{z}$. Since this is built up from the finite-dimensional
superalgebras, $\mathcal{S}(m|n;r,s)$, the first step is to understand their
representation theory.

##  4 Equivalence of Definitions and Semisimplicity

In this section, $k$ will denote an algebraically closed field of
characteristic zero, $V=k^{m|n},W=V^{*}$, and $T(r,s)=V^{\otimes r}\otimes
W^{\otimes s}$. In the classical case, the symmetric group $S_{r}$ acts on the
space $(k^{n})^{\otimes r}$, and the Schur algebra $S(n,r)$ may be defined as
$\operatorname{End}_{S_{r}}((k^{n})^{\otimes r})$. This definition is
equivalent to taking the dual of the coalgebra given by degree-$r$ homogeneous
polynomials in $k[GL(n)]$. The Schur algebra $S(n,r)$ is also isomorphic to
the image of the canonical $\mathfrak{gl}(n)$ representation
$\rho_{r}:U(\mathfrak{gl}(n))\rightarrow\operatorname{End}((k^{n})^{\otimes
r})$. Thus, a Schur algebra may be defined through any of the three equivalent
definitions. This section aims to demonstrate a similar equivalence of
definitions for $\mathcal{S}(m|n;r,s)$. First, we prove that $k[GL(m|n)]$ is
isomorphic to the Hopf superalgebra of regular functions on $GL(m|n)$, which
was introduced in [14]. This allows us to demonstrate that
$\mathcal{S}(m|n;r,s)$ is isomorphic to the image of the universal enveloping
superalgebra $U(\mathfrak{gl}(m|n))$ under its natural action on $T(r,s)$. If
$m-n\geq r+s$, then the semisimplicity of this image is shown by proving
$T(r,s)$ is an irreducible $\mathcal{S}(m|n;r,s)$-supermodule on which the
superalgebra acts faithfully. Lastly, it is known that the walled Brauer
algebra $B_{r,s}(m-n)$ acts naturally on $T(r,s)$, and when $m-n\geq r+s$ it
is isomorphic to $\operatorname{End}_{\mathfrak{gl}(m|n)}(T(r,s))$ [10]. The
semisimplicity result in combination with this Schur-Weyl duality then implies
through the double centralizer theorem that
$\mathcal{S}(m|n;r,s)\cong\operatorname{End}_{B_{r,s}(m-n)}(T(r,s))$.

###  4.1 The Hopf Superalgebra of Regular Functions on $GL(m|n)$

The Lie superalgebra $\mathfrak{gl}(m|n)$ acts naturally on $T(r,s)$ through
the following action defined on homogeneous elements:

$\begin{split}x\cdot(v_{1}\otimes...\otimes v_{r}\otimes
w_{1}\otimes...\otimes
w_{s})&=\sum_{k=1}^{r}(-1)^{|x|\sum_{p=1}^{k}|v_{p}|}v_{1}\otimes...\otimes(x\cdot
v_{k})\otimes...\otimes v_{r}\otimes...\otimes w_{s}\\\
&+\sum_{k=1}^{s}(-1)^{|x|(\sum_{p=1}^{r}|v_{p}|+\sum_{t=1}^{k}|w_{t}|)}v_{1}\otimes...\otimes
v_{r}\otimes...\otimes(x\cdot w_{k})\otimes...\otimes w_{s}\end{split}$

We denote this representation by
$\rho_{r,s}:U(\mathfrak{gl}(m|n))\rightarrow\operatorname{End}_{k}(T(r,s))$.

######  Definition 4.1.

[14]

The _Hopf superalgebra of regular functions on $GL(m|n)$_ is the sub-
superalgebra of $(U(\mathfrak{gl}(m|n)))^{*}$ generated by:

$cf_{U(\mathfrak{gl}(m|n))}(V)=\operatorname{span}_{k}\\{t_{i,j}:1\leq i,j\leq
m+n\\}\quad\text{and}\quad
cf_{U(\mathfrak{gl}(m|n))}(W)=\operatorname{span}_{k}\\{\overline{t}_{i,j}:1\leq
i,j\leq m+n\\}$

We denote this sub-superalgebra by $\mathcal{B}$. In [14] it was demonstrated
that $\mathcal{B}$ is a Hopf superalgebra. We briefly recall its
superbialgebra structure maps. Let $\Delta_{0}$ and $\epsilon_{0}$ be the
canonical comultiplication and counit structure maps on
$U(\mathfrak{gl}(m|n))$, respectively. Let $f,g\in U(\mathfrak{gl}(m|n))^{*}$
and $a,b\in U(\mathfrak{gl}(m|n))$. The unit, multiplication,
comultiplication, and counit of $\mathcal{B}$ are then all given respectively
by:

$1_{U(\mathfrak{gl}(m|n))^{*}}:=\epsilon_{0}^{*},\quad\langle m(f\otimes
g),a\rangle:=\langle f\otimes
g,\Delta_{0}(a)\rangle,\quad\langle\Delta(f),a\otimes b\rangle:=\langle
f,ab\rangle,\quad\epsilon(-):=\langle-,1_{U(\mathfrak{gl}(m|n))}\rangle$

Furthermore, let $B$ denote the superalgebra generated by $\\{t_{i,j}:1\leq
i,j\leq m+n\\}$. It was shown in [12, Theorem 2.1f] that $B$ is isomorphic to
$\mathcal{A}(m|n)$ as a superbialgebra by sending $t_{i,j}\mapsto x_{i,j}$.
Although, comultiplication on $\mathcal{A}(m|n)$ in this case is defined by:

$\Delta^{\prime}(x_{i,j}):=\sum_{h=1}^{m+n}(-1)^{(|i|+|h|)\cdot(|h|+|j|)}x_{i,h}\otimes
x_{h,j}$

Denote $\mathcal{A}(m|n)$ with the superbialgebra structure given in our
context by $(\mathcal{A}(m|n),\Delta)$, and the superbialgebra structure in
[12] by $(\mathcal{A}(m|n),\Delta^{\prime})$. It may be easily checked that
the superalgebra isomorphism

$f:(\mathcal{A}(m|n),\Delta)\rightarrow(\mathcal{A}(m|n),\Delta^{\prime}),\quad
x_{i,j}\mapsto(-1)^{|j|(|i|+|j|)}x_{i,j}$

is also a supercoalgebra isomorphism. Thus, $(\mathcal{A}(m|n),\Delta)$ and
$B$ are isomorphic as superbialgebras. If $\alpha:=\det((t_{i,j})_{1\leq
i,j\leq m})$ and $\beta:=\det((t_{i,j})_{m+1\leq i,j\leq m+n})$, then by
localizing $B$ at $\alpha\cdot\beta$ it follows that $k[GL(m|n)]$ is
isomorphic to $B_{\alpha\cdot\beta}$ as superbialgebras. Since a
superbialgebra isomorphism of Hopf superalgebras is a Hopf superalgebra
isomorphism, this implies that $k[GL(m|n)]$ is isomorphic to
$B_{\alpha\cdot\beta}$. From [14, Theorem 3.3] we see that
$B_{\alpha\cdot\beta}$ is isomorphic to $\mathcal{B}$ as a Hopf superalgebra.
So, we have the following lemma:

######  Lemma 4.1.

The coordinate superalgebra $k[GL(m|n)]$ is isomorphic to $\mathcal{B}$ as a
Hopf superalgebra.

This lemma provides an alternative proof to Lemma $8.3$ from [11].

######  Corollary 4.1.

The coordinate superalgebra $k[GL(m|n)]$ is generated by
$x_{i,j},\tilde{x}_{i,j}$ for $1\leq i,j\leq m+n$. So,
$k[GL(m|n)]=\tilde{\mathcal{A}}(m|n)$.

###### Proof.

Note, the Hopf superalgebra isomorphism given in [14, Theorem 3.3]

$h:B_{\alpha\cdot\beta}\rightarrow\mathcal{B}$

sends $(-1)^{|j|(|i|+|j|)}x_{i,j}$ to
$\tilde{t}_{i,j}:=(-1)^{|j|(|i|+|j|)}t_{i,j}$. Let $s^{\prime}$ denote the
antipode of $\mathcal{B}$. Then recalling [14, pg.55], we see

$(s^{\prime}((h\circ f)(x_{i,j})))_{1\leq i,j\leq
m+n}=(\tilde{t}_{i,j})_{1\leq i,j\leq m+n}^{-1}$

Since $h,f$ are Hopf superalgebra morphisms this implies:

$(\tilde{t}_{i,j})_{1\leq i,j\leq m+n}^{-1}=((h\circ f)(s(x_{i,j})))_{1\leq
i,j\leq m+n}=((h\circ f)(\tilde{x}_{i,j}))_{1\leq i,j\leq m+n}$

But,

$(\tilde{t}_{i,j})_{1\leq i,j\leq m+n}^{-1}=(\overline{t}_{i,j})_{1\leq
i,j\leq m+n}$

Therefore,

$\overline{t}_{i,j}=(h\circ f)(\tilde{x}_{i,j})\quad\text{for}\quad 1\leq
i,j\leq m+n$

Since $\\{t_{i,j},\overline{t}_{i,j}:1\leq i,j\leq m+n\\}$ generates
$\mathcal{B}$, this implies $\\{x_{i,j},\tilde{x}_{i,j}:1\leq i,j\leq m+n\\}$
generates $k[GL(m|n)]$. Recalling, Lemma 3.1, Part 1 we see
$k[GL(m|n)]=\tilde{\mathcal{A}}(m|n)$. ∎

###  4.2 First equivalence of Definitions

To show that a rational Schur superalgebra $\mathcal{S}(m|n;r,s)$ is
isomorphic to $\rho_{r,s}(U(\mathfrak{gl}(m|n)))$ we need the following
result:

######  Proposition 4.1.

As supercoalgebras $\tilde{\mathcal{A}}(m|n;r,s)$ is isomorphic to
$\mathbb{cf}_{U(\mathfrak{gl}(m|n))}(T(r,s))$.

###### Proof.

By the isomorphism given in Corollary 4.1 and Lemma 3.1, Part 3, it suffices
to show that

$\mathbb{cf}_{U(\mathfrak{gl}(m|n))}(T(r,s))=\operatorname{span}\\{t_{i_{1},j_{1}}^{m_{1}}...t_{i_{p},j_{p}}^{m_{p}}\overline{t}_{i_{p+1},j_{p+1}}^{m_{p+1}}...\overline{t}_{i_{p+\ell},j_{p+\ell}}^{m_{p+\ell}}:\sum_{k=1}^{p}m_{k}=r,\sum_{k=1}^{\ell}m_{p+k}=s\\}$

Recalling the definition of $\mathcal{B}$ this follows immediately from the
following fact: If $A,B$ are finite-dimensional representations of
$\mathfrak{gl}(m|n)$, then

$\mathbb{cf}_{U(\mathfrak{gl}(m|n))}(A\otimes
B)=\mathbb{cf}_{U(\mathfrak{gl}(m|n))}(A)\cdot\mathbb{cf}_{U(\mathfrak{gl}(m|n))}(B)$

∎

######  Theorem 4.1.

(Compare with [14, Theorem 3.2]) The rational Schur superalgebra
$\mathcal{S}(m|n;r,s)$ is isomorphic to $\rho_{r,s}(U(\mathfrak{gl}(m|n)))$.

###### Proof.

This follows immediately from combining Proposition 4.1 and Lemma 2.1. ∎

###  4.3 Rational Schur Superalgebras as Centralizer Algebras and a
Semisimplicity Result

Recall that if $m\geq n$, then the walled Brauer algebra $B_{r,s}(m-n)$ acts
on $T(r,s)$, and when $m-n\geq r+s$ it is isomorphic to
$\operatorname{End}_{U(\mathfrak{gl}(m|n))}(T(r,s))$ [10]. Proceeding
similarly to the classical case we will demonstrate that a rational Schur
superalgebra is isomorphic to the centralizer of the action of a walled Brauer
algebra, or in symbols

$\mathcal{S}(m|n;r,s)\cong\operatorname{End}_{B_{r,s}(m-n)}(T(r,s))$

By the previous section, it suffices to show that $\mathcal{S}(m|n;r,s)$ is
semisimple, as once this has been proven we may apply the double centralizer
theorem [13, Theorem 11.1.1] to obtain the desired isomorphism. To do this it
suffices to find a semisimple faithful representation of
$\mathcal{S}(m|n;r,s)$. A natural candidate for this is $T(r,s)$.

It is known that the $T(r,s)$ decomposes as a $\mathfrak{gl}(m|n)$-supermodule
into indecomposable sub-supermodules and hence indecomposable
$\mathcal{S}(m|n;r,s)$ sub-supermodules. Brundan and Stroppel classified the
isomorphism types of these indecomposables in [2] through $(m|n)$-cross bi-
partitions.

######  Definition 4.2.

If $\lambda_{L}$ is a partition of $r$ and $\lambda_{R}$ a partition of $s$,
then $(\lambda_{L},\lambda_{R})$ is a _$(m|n)$ -cross bi-partition of $(r,s)$_
if there exists a $1\leq i\leq m+1$ such that
$\lambda^{L}_{i}+\lambda^{R}_{m+2-i}<n+1$.

The set of $(m|n)$-cross bi-partitions of $(r,s)$ is denoted by
$\Lambda_{r,s}$.

######  Theorem 4.2.

[2, Theorem 8.19] The indecomposable $\mathcal{S}(m|n;r,s)$ sub-supermodules
of $T(r,s)$ are parameterized up to isomorphism by the $(m|n)$-cross bi-
partitions of $(r,s)$.

Suppose $\lambda\in\Lambda_{r,s}$, and let $R(\lambda)$ denote the unique (up
to isomorphism) $\mathfrak{gl}(m|n)$ sub-supermodule of $T(r,s)$ associated to
$\lambda$. In [7], Heidersdorf associates to each such bi-partition an
invariant $d(\lambda)$ and shows that $R(\lambda)$ is an irreducible
$\mathfrak{gl}(m|n)$-supermodule if and only if $d(\lambda)=0$. Furthermore,
Proposition 10.3 of the same paper states that if $m-n\geq r+s$, then
$d(\lambda)=0$. Since we are assuming $m-n\geq r+s$ so that $B_{r,s}(m-n)$ is
isomorphic to $\operatorname{End}_{U(\mathfrak{gl}(m|n))}(T(r,s))$, we may
conclude that all indecomposable $\mathfrak{gl}(m|n)$ sub-supermodules of
$T(r,s)$ are also irreducible. Summarizing this discussion we have:

######  Lemma 4.2.

If $m-n\geq r+s$, then $T(r,s)$ is a semisimple
$\mathfrak{gl}(m|n)$-supermodule.

Since $\mathcal{S}(m|n;r,s)$ acts as $\rho_{r,s}(U(\mathfrak{gl}(m|n)))$ on
$T(r,s)$, this is evidently a faithful semisimple representation. Therefore,
$\mathcal{S}(m|n;r,s)$ is semisimple as an algebra.

######  Remark 1.

Assuming [15, Proposition 1.7 ], this implies that $\mathcal{S}(m|n;r,s)$ is a
semisimple superalgebra.

######  Theorem 4.3.

If $m-n\geq r+s$, then $\mathcal{S}(m|n;r,s)$ is the centralizer algebra of
the action of $B_{r,s}(m-n)$, or in symbols

$\mathcal{S}(m|n;r,s)\cong\operatorname{End}_{B_{r,s}(m-n)}(T(r,s))$

###### Proof.

By mixed Schur-Weyl duality it is known that
$\operatorname{End}_{\mathfrak{gl}(m|n)}(T(r,s))\cong\rho(B_{r,s}(m-n))$ where
$\rho$ denotes the right action of $B_{r,s}(m-n)$ on $T(r,s)$. By the double
centralizer theorem [13, Theorem 11.1.1] this implies that
$\operatorname{End}_{B_{r,s}(m-n)}(T(r,s))=\rho_{r,s}(U(\mathfrak{gl}(m|n)))$.

∎

## Acknowledgements

I would like to thank Dr. Nicolas Guay for his invaluable guidance and support
in this project.

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