Stratified Categories and a Geometric Approach to Representations of the Schur Algebra

Abstract

In this thesis we present three new results:

(i) We define a stratification of abelian categories as an iterated system of recollements of abelian categories. This definition generalises the definitions of categories of (equivariant) perverse sheaves as well as epsilon-stratified categories (and in particular highest weight categories) in the sense of Brundan-Stroppel (2018). We give necessary and sufficient conditions for a stratification of abelian categories to be equivalent to a category of finite dimensional modules of a finite dimensional algebra - this generalises the main result of Cipriani-Woolf (2022).

(ii) We define a product of Schur algebra modules that corresponds under Schur-Weyl duality to the Kronecker product of symmetric group modules. This new product is a Schur algebra module theoretic version of Krause’s (2013) internal product on the category of homogeneous strict polynomial functors.

(iii) We give a characteristic-free version of Ginzburg’s (1997) construction of the Schur algebra via the convolution product on the Borel-Moore homology of smooth varieties related to the nilpotent cone, N, in gl_n. As an application, we give a new proof of Mautner’s (2014) equivalence of categories between GL_n -equivariant perverse sheaves on N and a category of Schur algebra modules.