super q-Schur algebra

The super $q$-Schur algebra $S_{m|n}^d$ is the analogue over the Lie superalgebra $\mathfrak{sl}(m|n)$ of the usual $q$-Schur algebra?. That is, if we let $V$ denote the deformation of the defining representation of $\mathfrak{sl}(m|n)$, then

$S_{m|n}^d=End_{U_q(\mathfrak{sl}(m|n))}(V^{\otimes d}).$

When $q$ is generic, this is a quotient of the Hecke algebra by the ideal generated by all representations whose corresponding representations are not hooks.

In fact, when $m=n=1$ the weight spaces of $V^{\otimes d}$ are irreducible representations, with arm length of the hook corresponding to weight.

Ben Webster: Does $S_{m|n}^d$ inherit a basis from the Hecke algebra in the usual way? It certainly seems that the representation $V^{\otimes d}$ has a basis on which positive integral combination of the Kazhdan-Lusztig basis vectors act with positive integral coefficients.

Also, does this picture have a categorification? One natural candidate to use in the $m=n=1$ case is the categorifications of the cell modules due to Stroppel and Mazorchuk. Also possibly of interest is their categorification of Wedderburn basis of $S_n$

- See also MathOverflow: Does the super Temperley-Lieb algebra have a Z-form?

Revised on October 30, 2009 21:31:29
by Ben Webster
(18.87.1.204)