An application of TQFT to modular representation theory

TL;DR

This paper applies Topological Quantum Field Theory to analyze modular representations of symplectic groups, providing explicit formulas for dimensions and characters of certain modules in characteristic p.

Contribution

It introduces a novel application of TQFT to compute explicit properties of highest weight modules for symplectic groups in characteristic p.

Findings

Explicit formulas for module dimensions

Explicit formulas for module characters

Enhanced understanding of modular representation structure

Abstract

For p>3 a prime, and g>2 an integer, we use Topological Quantum Field Theory (TQFT) to study a family of p-1 highest weight modules L_p(lambda) for the symplectic group Sp(2g,K) where K is an algebraically closed field of characteristic p. This permits explicit formulae for the dimension and the formal character of L_p(lambda) for these highest weights.