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

Abstract

We compare the dimensions of the irreducible Sp(2g,K)-modules over a field K of characteristic p constructed by Gow (1998) [9] with the dimensions of the irreducible Sp(2g,Fp)-modules that appear in the first approximation to representations of mapping class groups of surfaces in integral topological quantum field theory [8]. 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 This formula is equivalent to a special case of a formula contained in unpublished work of Foulle (2004) [2,3]. 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.