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 ]
Mi−1 [les ] Mi = M
with quotients isomorphic to Δ(λ) for various λ. Knowing
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