7 - Representation theory of S_q(n, r)

Summary

One task remaining from the previous chapter is to obtain a complete list of irreducible S_q(n, r)-modules, thus completing the job begun for characteristic zero in 6.5.3. Unfortunately the situation for finite characteristic is more problematic: whichever approach is adopted there seems to be no quick way of showing that the irreducible S_q(n, r)-modules are indexed by Λ⁺(n, r). One approach would be to bring out the quantum GL_n and use quantum group theory, a method employed by Parshall and Wang [1991, §8]. The Dipper and James [1991] approach is to define a certain q-analogue of the Weyl module. Over K the similarity with the classical case is striking: the q-Weyl module V_q(λ) has a unique maximal submodule and therefore a simple top factor F_q(λ). They show that, as λ runs through the dominant weights, the F_q(λ) will then give a complete set of simple S_q(n,r)-modules. The analogy with the situation studied in previous chapters (referred to henceforth as the ‘classical case’) is a persuasive one: for example in 7.5 we show that S_q(n,r) is quasi-hereditary having as standard modules the q-Weyl modules.

Returning to the q-universe constructed in Chapter 6, we already have at our disposal several q-analogues of structures in the classical situation. We have, for example, a qanalogue of the polynomial coalgebra, namely A_q(n, r), and of the Schur algebra, namely S_q(n, r), but as yet there is nothing even remotely resembling a q-version of GL_n.