On Ext² between Weyl modules for quantum GL_n

Abstract

In the study of highest weight categories, the class of Weyl modules Δ(λ) and their duals ∇(λ) are of central interest; this is for example motivated by the problem of finding the characters of the simple modules. Weyl modules form the building blocks for the category [Fscr ](Δ), whose objects have a filtration 0 = M₀ [les ] M₁ [les ] … [les ] M_i−1 [les ] M_i = M with quotients isomorphic to Δ(λ) for various λ. Knowing Ext^r(Δ(λ), Δ(μ)) is essential for the understanding of this category.

In [3, 13], we determined Ext¹ for Weyl modules of SL(2, k) and q-GL(2, k) over an infinite field k of characteristic p > 0. Here we are able to extend these results to determine Ext² for Weyl modules in both these cases (see Theorem 4 · 6). Moreover, this also gives Ext² between any pair of Weyl modules Δ(λ), Δ(μ) for q-GL(n, k) (where n [ges ] 2) such that both λ and μ have at most two rows or two columns, or where they differ by some multiple of a simple root (see Section 7).

Consider (for simplicity) polynomial representations of degree d for GL(n, k). A partition of d which has at most two rows is uniquely determined by the difference in the row lengths, which we use as a label for the partition. In this case our main result is