Inner products of 2-representations

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.

💡 Research Summary

The paper develops a theory of inner products for 2‑representations of a finite group G. Starting from the categorical trace introduced by Ganter‑Kapranov, Tr(F)=Nat(id,F), the author treats a 2‑representation 𝔐 as a functor G→Cat_k and defines its character X_𝔐(g)=Tr(𝔐(g)). Each X_𝔐(g) is a finite‑dimensional k‑vector space equipped with conjugation isomorphisms ψ_h, making X_𝔐 a categorical class function. The central technical result (Theorem 2.5) shows that for linear categories V and W with endofunctors g and h, the natural map

 µ: Tr(g)⊗Tr(h) → Tr(g⊠h)

is an isomorphism provided the Hom‑spaces are finite‑dimensional and at least one of the traces is finite‑dimensional. Here ⊠ denotes the linear tensor product of categories defined in