nLab
super q-Schur algebra

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

S m|n d=End U q(𝔰𝔩(m|n))(V d).S_{m|n}^d=End_{U_q(\mathfrak{sl}(m|n))}(V^{\otimes d}).

When qq 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=1m=n=1 the weight spaces of V dV^{\otimes d} are irreducible representations, with arm length of the hook corresponding to weight.

Ben Webster: Does S m|n dS_{m|n}^d inherit a basis from the Hecke algebra in the usual way? It certainly seems that the representation V dV^{\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=1m=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 nS_n

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