Quasi-Heredity of the Symplectic q-Schur Algebra

In [GL] Graham and Lehrer have presented a nice criterion for quasi-heridity of a cellular algebra which we will now verify in our case. This will prove Theorem 7.5.

To this aim we have to investigate the bilinear form $\phi_{\underline{\lambda}}$ on the standard modules $W(\underline{\lambda})$. We must show that they are not zero ([GL, 3.10]). Let us first calculate the Grammatrix of $\phi_{\underline{\lambda}}$ with respect to the basis $\{C^{\underline{\lambda}}_{{\bf i}}\vert\; {\bf i}\in M(\underline{\lambda})\}$ of $W(\underline{\lambda})$. We abbreviate its entries by $\phi_{{\bf i}{\bf j}}:= \phi_{\underline{\lambda}}(C^{\underline{\lambda}}_{{\bf i}}, C^{\underline{\lambda}}_{{\bf j}})\in R$. According to the definition in [GL, 2.3], these numbers satisfy

$$C^{\underline{\lambda}}_{{\bf i}, {\bf k}}C^{\underline{\lambda}}... ...}\equiv \phi_{{\bf k}{\bf l}}C^{\underline{\lambda}}_{{\bf i}, {\bf j}} \;\;\;$$    mod $$S^{\rm s}_{R,q}(n,r)(<\underline{\lambda}).$$

Such a congruence relation is valid with $\phi_{{\bf k}{\bf l}}$ being independent of ${\bf i}$ and ${\bf j}$ by the axioms of a cellular algebra (see [GL, 1.7]). Dualizing this congruence we obtain the following counterpart in the cellular coalgebra ${\cal K}=A^{{\rm s}}_{R,q}(n,r)$:

$$\Delta (D^{\underline{\lambda}}_{{\bf i}, {\bf j}})\equiv \sum_{... ...lambda}}_{{\bf i}, {\bf k}}\otimes D^{\underline{\lambda}}_{{\bf l}, {\bf j}}$$

modulo ${\cal K}(\geq \underline{\lambda})\otimes {\cal K}(>\underline{\lambda})+ {\cal K}(> \underline{\lambda})\otimes {\cal K}(\geq\underline{\lambda})$. According to the calculations for the verification of axiom $(C3^*)$ in the previous section we see using the notations from there that

$$\phi_{{\bf k}{\bf l}}=\sum_{{\bf h}\in I_{\lambda}^{<} }a_{{\bf h}{\bf k}} a_{{\bf h}{\bf l}}.$$ (30)

The bilinear form $\phi_{\underline{\lambda}}$ is different from zero if this is the case for a single entry $\phi_{{\bf k}{\bf l}}$. We calculate $\phi_{{\bf k}{\bf k}}$ where ${\bf k}$ is the $\lambda$-tableau $T^{\lambda}_{{\bf {\bf k}}}=T$ with constant rows $T(i,j):=m+i$ for all $1\leq i\leq m$ and $1\leq j \leq \lambda _j$. Obviously $T$ is a standard tableau with respect to both orders on n, namely $<$ as well as $\prec$. Furthermore the reverse symplectic condition holds because $T(i,j) = (m-i+1)'$. Consequently we have ${\bf k}\in M(\underline{\lambda})\cap I_{\lambda}^{<}$. Note that ${\bf k}$ does not contain any pair of associated indices $(i, i')$. The content $\eta:=\vert{\bf k}\vert$ of ${\bf k}$ is given by

$$\eta_i=\left\{\begin{array}{cl} 0 & i \leq m\\ \lambda _{i-m} & i>m. \end{array} \right.$$

Consider the endomorphism

$$\tau = \sum_{{\bf i} \in I(n,r), \vert {\bf i}\vert = \eta} e_{{\bf i}{\bf i}} \in {\rm End}_{R}{(V^{\otimes r})}.$$

It is easy to see that $\tau$ commutes with $\beta_i$ and $\gamma_i$ for all $i=1, \ldots, r-1$. Consequently it is an endomorphism of the $A^{{\rm s}}_{R,q}(n,r)$ comodul $V^{\otimes r}$ (in fact it is the idempotent of $S^{\rm s}_{R,q}(n,r)$ corresponding to the weightspace with weight $\eta$). There is an induced action of $\tau$ on $A^{{\rm s}}_{R,q}(n,r)$ from the right defined by

$$x_{{\bf i} {\bf j}}\tau := ( {\rm id}_{} \otimes \tau ) \Delta ( ... ...a, \\ x_{{\bf i} {\bf j}} & \vert{\bf j}\vert = \eta. \end{array} \right.$$

For a bideterminant we have

$$T^{\lambda}_q({\bf i}:{\bf j})\tau = \left\{\begin{array}{ll} ... ...\lambda}_q({\bf i}:{\bf j}) & \vert{\bf j}\vert = \eta. \end{array} \right.$$

Applying $\tau$ to the defining equation (30) of $a_{{\bf h}{\bf k}}$ we see that $a_{{\bf h}{\bf k}}=0$ if $\vert{\bf h}\vert \neq \eta$ by linear independence of $D^{\underline{\lambda}}_{{\bf k}, {\bf j}}$. Since ${\bf k}$ is the only element in $I_{\lambda}^{<}$ having content $\eta$ it follows $a_{{\bf h}{\bf k}}=0$ if ${\bf h} \neq {\bf k}$ and we conclude

$$\phi_{{\bf k}{\bf k}}=\sum_{{\bf h}\in I_{\lambda}^{<} }{a_{{\bf h}{\bf k}}}^2= {a_{{\bf k}{\bf k}}}^2=1.$$

By [GL, Remark 3.10], this finishes the proof of Theorem 7.5.