A combinatorial translation principle and diagram combinatorics for the general linear group

TL;DR

This paper develops a combinatorial approach using diagram techniques to compute decomposition numbers and filtration multiplicities for the general linear group and related algebras over fields of positive characteristic.

Contribution

It introduces a new combinatorial method involving cap diagrams for calculating decomposition numbers and extends the rational Schur functor to connect GL_n-modules with walled Brauer algebra modules.

Findings

Computed Weyl filtration multiplicities for tilting modules.

Determined decomposition numbers for the general linear group and walled Brauer algebra.

Established conditions under which the calculations hold, based on hook lengths.

Abstract

Let k be an algebraically closed field of characteristic p>0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. Then we introduce and study the rational Schur functor from a category of GL_n-modules to the category of modules for the walled Brauer algebra. As a corollary we obtain the decomposition numbers for the walled Brauer algebra when p is bigger than the greatest hook length in the partitions involved. This is a sequel to an earlier paper on the symplectic group and the Brauer algebra.