A combinatorial translation principle and diagram combinatorics for the symplectic group

TL;DR

This paper develops a combinatorial approach using diagram combinatorics to compute key representation-theoretic quantities for the symplectic group over fields of characteristic p>2, linking it to Brauer algebra decomposition numbers.

Contribution

It introduces a new combinatorial translation principle and diagrammatic method to determine Weyl filtration multiplicities and decomposition numbers for symplectic groups in positive characteristic.

Findings

Computed Weyl filtration multiplicities for symplectic groups.

Derived decomposition numbers for the Brauer algebra.

Connected diagram combinatorics with representation theory techniques.

Abstract

Let k be an algebraically closed field of characteristic p>2. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the symplectic group over k in terms of cap-curl diagrams under the assumption that p is bigger than the greatest hook length in the largest partition involved. As a corollary we obtain the decomposition numbers for the Brauer algebra under the same assumptions. Our work combines ideas from work of Cox and De Visscher and work of Shalile with techniques from the representation theory of reductive groups.