Dimension formulas for some modular representations of the symplectic group in the natural characteristic

TL;DR

This paper derives a simplified trigonometric formula for the dimensions of certain irreducible symplectic group representations over fields of characteristic p, connecting Gow's modules with TQFT representations.

Contribution

It provides a new, simpler proof of a dimension formula for Gow's symplectic modules, linking algebraic and topological quantum field theory representations.

Findings

Derived a trigonometric dimension formula for Gow's modules

Established equivalence with a special case of Foulle's formula

Simplified the proof compared to existing methods

Abstract

We compare the dimensions of the irreducible Sp(2g,K)-modules over a field K of characteristic p constructed by Gow with the dimensions of the irreducible Sp(2g,F_p)-modules that appear in the first approximation to representations of mapping class groups of surfaces in Integral Topological Quantum Field Theory. For this purpose, we derive a trigonometric formula for the dimensions of Gow's representations. This formula is equivalent to a special case of a formula contained in unpublished work of Foulle. Our direct proof is simpler than the proof of Foulle's more general result, and is modeled on the proof of the Verlinde formula in TQFT.