The Symplectic $q$-Schur algebra

Remember that $A^{{\rm s}}_{R,q}(n,r)$ is a coalgebra for each $r$. Therefore, its dual $R$-module inherits the structure of an $R$-algebra. We define

$${S^{\rm s}_{R,q}(n,r)}$$$$\mbox{\index{${S^{\rm s}_{R,q}(n,r)}$}}$$$$:={\rm Hom}_{R}{(A^{{\rm s}}_{R,q}(n,r),R)}$$

and call it the symplectic $q$-Schur algebra. Two linear forms $\mu, \nu \in S^{\rm s}_{R,q}(n,r)$ are multiplied by convolution, that is

$$\mu \nu (a):= (\mu \otimes \nu) \circ \Delta (a)$$

for all $a \in A^{{\rm s}}_{R,q}(n,r)$. The reader may verify that one obtains the symplectic Schur algebra in the classical situation as defined in [O2]. This also is identical to the symplectic Schur algebra in the sense of S. Donkin, respectively S. Doty ([Do2] respectively [Dt]). One aim is to show that the construction is stable under base changes and that it is a free $R$-module. Both facts follow when we have shown that $A^{{\rm s}}_{R,q}(n,r)$ is free as an $R$-module. Further we want to initiate the study of the representation theory of this algebra. An easy way to do this is to check that the axioms of a cellular algebra given by J. Graham and G. Lehrer in [GL] hold. These axioms are as follows:

Let $A$ be an associative unital algebra over a commutative unital ring $R$ together with a partially ordered finite set $\Lambda$ and finite sets $M(\lambda)$ to each $\lambda \in \Lambda$ (the set of ``$\lambda$-tableaux''). $A$ is called a cellular algebra if the following properties hold:

(C1)

$A$ possesses an $R$-basis $\{{C^{\lambda}_{S,T}}$$\vert\; \lambda \in \Lambda , \; S,T \in M(\lambda) \}$.

(C2)

$A$ posesses an $R$-linear involution $^*$ which is an algebra anti-automorphism such that ${C^{\lambda}_{S,T}}^*= C^{\lambda}_{T,S}$ holds for all $\lambda \in \Lambda$ and $S,T\in M(\lambda)$.

(C3)

For all $a \in A, \lambda \in \Lambda$ and $S,T\in M(\lambda)$ the congruence relation

$$aC^{\lambda}_{S,T}\equiv \sum_{S'\in M(\lambda)} r_a(S',S)C^{\lambda}_{S',T} \; \;$$    mod $${A(<\lambda )}$$$$\mbox{\index{${A(<\lambda )}$}}$$$$,$$

holds, where the elements $r_a(S',S) \in R$ are independent of $T$ and $A(<\lambda)$ is defined as the $R$-linear span of basis elements $C^\mu _{U,V}$ where $\mu < \lambda$ and $U,V \in M(\mu )$.

Starting with these axioms the representation theory of $A$ is developed in [GL] along the following lines. To each $\lambda \in \Lambda$ a standard module $W(\lambda)$ is defined on a free $R$-basis $\{C^{\lambda}_S\vert\; S \in M(\lambda)\}$. An element $a \in A$ acts on it via $aC^{\lambda}_S=\sum_{S'\in M(\lambda)}r_a(S',S)C^{\lambda}_{S'}$. Each $W(\lambda)$ possesses a symmetric bilinear form $\phi_{\lambda}$ for which the formula $\phi_{\lambda}(a^*x,y)=\phi_{\lambda}(x,ay)$ is valid for all $a \in A$ and $x,y \in W(\lambda)$. In the case where $R$ is a field and $\phi_{\lambda}\neq 0$, the radical of $W(\lambda)$ is the same as the radical of the bilinear form $\phi_{\lambda}$. The simple head $L_{\lambda}$ of $W(\lambda)$ then is absolutely irreducible. In this way a complete set of pairwise non-isomorphic simple $A$-modules $\{L_{\lambda}\vert\;\lambda \in \Lambda_0\}$ can be obtained. Here we have set $\Lambda_0:=\{\lambda \in \Lambda\vert\; \phi_{\lambda}\neq 0\}$.

Denoting the multiplicity of $L_\mu$ in $W(\lambda)$ by $d_{\lambda\mu }$ to each $\lambda \in \Lambda$ and $\mu \in \Lambda_0$ Graham and Lehrer show that $d_{\lambda\mu }=0$ for $\lambda \leq \mu$ and $d_{\lambda\lambda}=1$. To each order refining the given partial order on $\Lambda$ the corresponding decomposition matrix $D=(d_{\lambda\mu })_{\lambda \in \Lambda , \mu \in \Lambda_0}$ is unitriangular. The Cartan-matrix $C$ can be calculated as $C=D^tD$. The theory also supplies a criterion to decide whether $A$ is semisimple or quasi-hereditary. In the first case we must have ${\rm rad}(\phi_{\lambda})=(0)$ for all $\lambda \in \Lambda$ whereas in the second case $\Lambda_0=\Lambda$ will do.

Examples of cellular algebras are the Brauer centralizer algebras ${\cal B}_{R, x,r}$, Ariki-Koike-Hecke-algebras, Temperley-Lieb and Jones algebras ([GL]). R.M. Green ([GR]) constructs a $q$-analogue of the codeterminant basis (in the sense of [Gr]) for the classical Schur algebra $S_{R}(n,r)$ which is cellular as well. The corresponding standard modules $W(\lambda)$ are precisely the $q$-Weyl modules in the sense of [DJ2] (see [GR], Proposition 5.3.6).

It should be remarked that the finiteness of $\Lambda$ is not postulated in the original definition. Since this property is valid in our example we impose this restriction to avoid unnecessary trouble (cf. discussion in [KX], section 3).

Since we have defined the symplectic $q$-Schur algebra as the dual module of a coalgebra we now translate the concept of cellular algebras to coalgebras:

Let $K$ be a coalgebra over a commutative unital ring $R$, together with a partially ordered finite set $\Lambda$ and finite sets ${M(\lambda)}$ for each $\lambda \in \Lambda$. We call $K$ a cellular coalgebra if the following properties hold:

(C1*)

$K$ possesses an $R$-basis $\{{D^{\lambda}_{S,T}}$$\vert\; \lambda \in \Lambda , \; S,T \in M(\lambda) \}$.

(C2*)

$K$ possesses an $R$-linear involution $^*$ which is an coalgebra anti-automorphism, such that ${D^{\lambda}_{S,T}}^*= D^{\lambda}_{T,S}$ holds for all $\lambda \in \Lambda$ and $S,T\in M(\lambda)$.

(C3*)

For all $\lambda \in \Lambda$ and $S,T\in M(\lambda)$ the congruence relation

$$\Delta (D^{\lambda}_{S,T})\equiv \sum_{S'\in M(\lambda)} h(S',S) \otimes D^{\lambda}_{S',T} \; \;$$    mod $$K \otimes {K(>\lambda )}$$$$\mbox{\index{${K(>\lambda )}$}}$$$$$$

holds, where the coalgebra elements $h(S',S) \in K$ are independent of $T$ and $K(>\lambda)$ is defined as the $R$-linear span of basis elements $D^\mu _{U,V}$ where $\mu > \lambda$ and $U,V \in M(\mu )$.

To an arbitrary $R$-coalgebra the dual algebra is well defined. The dual coalgebra of an algebra $A$ is well defined if the algebra is known to be projective as an $R$-module, since then ${(A\otimes A)}^* \simeq {A}^* \otimes {A}^*$. In the case of a cellular algebra this is obviously valid. The connection between the above two concepts is given by the following proposition which can be proved straightforwardly using structure constants with respect to the bases (cf. [O1, 4.2.3]).

Proposition 5.1   The dual algebra of a cellular coalgebra is a cellular algebra. The dual coalgebra of a cellular algebra is a cellular coalgebra. In both cases the corresponding bases and involution maps can be constructed dual to each other, i.e. in the former case $C^{\lambda}_{S,T}(D^{\mu}_{U,V})$ is 1 if $\lambda = \mu,\; S=U$ and $T=V$ but 0 otherwise and $C^{\lambda}_{S,T}({D^{\mu}_{U,V}}^*)={C^{\lambda}_{S,T}}^*(D^{\mu}_{U,V})$.

According to the proposition our next task is to find a cellular basis for the coalgebra $A^{{\rm s}}_{R,q}(n,r)$ together with an appropriate involution map such that the axioms of the cellular coalgebra hold. As soon as this is done the representation theory of $S^{\rm s}_{R,q}(n,r)$ is developed to the extent indicated above.