Inner products of 2-representations

TL;DR

This paper introduces the concept of inner products for 2-representations, proving key properties and exploring applications in representation theory and Hochschild cohomology.

Contribution

It defines and computes inner products of 2-representations, establishes multiplicativity of the categorical trace, and identifies the center of the category V^G.

Findings

Categorical trace is multiplicative under tensor products.

Identified the center of the category V^G.

Derived applications to projective representations and Hochschild cohomology.

Abstract

We define and calculate inner products of 2-representations. Along the way, we prove that the categorical trace Tr(-) of [Ganter and Kapranov, Representation and character theory in 2-categories, Sec. 3] is multiplicative with respect to various notions of categorical tensor product, and we identify the center of the category V^G of [loc. cit., Sec. 4.2]. We discuss applications, ranging from Schur's result about the number of projective representations to a formula for the Hochschild cohomology of a global quotient orbifold.