Resolutions of p-Modular TQFT's and Representations of Symmetric Groups

TL;DR

This paper constructs p-modular TQFTs over finite fields from integral TQFTs, linking them to symmetric group representations and providing new insights into 3-manifold invariants and group properties.

Contribution

It introduces resolutions of simple p-modular TQFTs and S_n-modules, connecting quantum invariants with modular representation theory.

Findings

Resolutions of simple p-modular TQFTs constructed using Lefschetz operators

Expressions for irreducible p-modular S_n-characters in terms of ordinary characters

Formulas for Alexander Polynomial at p-th roots of unity via traces in modular TQFTs

Abstract

We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to the p-modular representation theory of the symmetric groups S_n. We construct resolutions of simple p-modular TQFT's and S_n-modules over Z/pZ using powers of Lefschetz operators. These yield expansions of the irreducible p-modular S_n-characters in terms of the ordianry ones as well as expressions for the Alexander Polynomial of a 3-manifold evaluated at a p-th root of unity in terms of traces in the irreducible modular TQFT's over covering cobordisms. Together with identifications with the constant orders of quantum TQFT's this results, e.g., in non-trivial criteria for a group to be a 3-manifold group.