Alison Parker

On the Relation Between Finitistic and Good Filtration Dimensions

Abstract In this paper, we discuss generalizations of the concepts of good filtration dimension and Weyl filtration dimension, introduced by Friedlander and Parshall for algebraic groups, to properly stratified algebras. We introduce the notion of the finitistic Δ-filtration dimension for such algebras and show that the finitistic dimension for such an algebra is bounded by the sum of the finitistic Δ-filtration dimension and the -filtration dimension. In particular, the finitistic dimension must be finite. We also conjecture that this bound is exact when the algebra has a simple preserving duality. We give several examples of well-known algebras where this is the case, including many of the Schur algebras, and blocks of category 𝒪. We also give an explicit combinatorial formula for the global dimension in this case.

A PRESENTATION FOR THE SYMPLECTIC BLOB ALGEBRA

The symplectic blob algebra bn(n ∈ ℕ) is a finite-dimensional algebra defined by a multiplication rule on a basis of certain diagrams. The rank r(n) of bn is not known in general, but r(n)/n grows unboundedly with n. For each bn we define an algebra by presentation, such that the number of generators and relations grows linearly with n. We prove that these algebras are isomorphic.

HOMOMORPHISMS AND HIGHER EXTENSIONS FOR SCHUR ALGEBRAS AND SYMMETRIC GROUPS

This paper surveys, and in some cases generalizes, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit results have been determined.

D-filtered modules and nilpotent orbits of a parabolic subgroup in On

We study certain ∆-filtered modules for the Auslander algebra of k[T ]/Tn ⋊ C2 where C2 is the cyclic group of order two. The motivation of this lies in the problem of describing the P -orbit structure for the action of a parabolic subgroup P of an orthogonal group. For any parabolic subgroup of an orthogonal group we construct a map from parabolic orbits to ∆-filtered modules and show that in the case of the Richardson orbit, the resulting module has no self-extensions.

First cohomology groups for finite groups of Lie type in defining characteristic

Let G be a finite group of Lie type, defined over a field k of characteristic p > 0 . We find explicit bounds for the dimensions of the first cohomology groups for with coefficients in simple kG-modules. We proceed by bounding the number of composition factors of Weyl modules for simple algebraic groups independently of P and using this to deduce bounds for the 1-cohomology of simple algebraic groups. If γl denotes the (finite) maximum of the dimensions of the 1-cohomology groups over all Lie groups of rank l we find bounds for the growth rate of the sequence. {γl}We show that log γl is O(l3log l)