CHAPTER 1
Highest weight categories, q-Schur algebras, Hecke
algebras, and finite general linear groups
We brutally summarise the representation theory of the q-Schur algebra, of
Hecke algebras of type A, and of finite general linear groups in non-describing
characteristic.
Although in later chapters, we will invoke such theory over more general com-
mutative rings, for simplicity of presentation, in this chapter we only consider rep-
resentation theory over a field k, of characteristic l.
Highest weight categories
We state some of the principal definitions and results of E. Cline, B. Parshall,
and L. Scott’s paper, [17].
Definition 1 ([17], 3.1). Let C be a locally Artinian, Abelian category over
k, with enough injectives. Let Λ be a partially ordered set, such that every interval
[λ, µ] is finite, for λ, µ Λ. The category C is a highest weight category with respect
to Λ if,
(a) Λ indexes a complete collection {L(λ)}λ∈Λ of non-isomorphic simple objects
of C.
(b) Λ indexes a collection {∇(λ)}λ∈Λ of “costandard objects” of C, for each of
which there exists an embedding L(λ) ∇(λ), such that all composition factors
L(µ) of ∇(λ)/L(λ) satisfy µ λ.
(c) For λ, µ Λ, we have, dimkHom(∇(λ), ∇(µ)) ∞, and in addition,
[∇(λ) : L(µ)] ∞.
(d) An injective envelope I(λ) C of L(λ) possesses a filtration
0 = F0(λ) F1(λ) ...,
such that,
(i) F1(λ)

=
∇(λ).
(ii) For n 1, we have Fn(λ)/Fn−1(λ)

=
∇(µ), for some µ = µ(n) λ.
(iii) For µ Λ, we have µ(n) = µ for finitely many n.
(iv ) I(λ) =
i
Fi(λ).
Definition 2 ([17], 3.6). Let S be a finite dimensional algebra over k. Then
S is said to be quasi-hereditary if the category S mod, of finitely generated left
S-modules is a highest weight category.
For M C, and Γ Λ, let be the largest subobject of M, all of whose
composition factors L(γ) correspond to elements γ Γ.
7
Previous Page Next Page