Irreducible factors of modular representations of mapping class groups arising in Integral TQFT

TL;DR

This paper investigates the structure of modular representations of mapping class groups derived from integral TQFT, revealing their decomposition series and calculating the dimensions of irreducible factors using Verlinde-type formulas.

Contribution

It provides a detailed analysis of the decomposition series of these representations and explicitly computes the dimensions of irreducible components in positive characteristic.

Findings

Decomposition series of length at most two for the representations.

Explicit Verlinde-type formulas for irreducible factor dimensions.

Insights into the structure of modular representations in TQFT contexts.

Abstract

We find decomposition series of length at most two for modular representations in positive characteristic of mapping class groups of surfaces induced by an integral version of the Witten-Reshetikhin-Turaev SO(3)-TQFT at the p-th root of unity, where p is an odd prime. The dimensions of the irreducible factors are given by Verlinde-type formulas.