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Article

Young, Matthew B.
Real representation theory of finite categorical groups. (English). Higher Structures, vol. 5 (2021), issue 1, pp. 18-70
MSC: 18D05, 20J99 | MR 4367217 | Zbl 1492.18021 | DOI: 10.21136/HS.2021.02
Keywords:
Categorical groups; Real representation theory; Bicategories
Summary:
We introduce and develop a categorification of the theory of Real representations of finite groups. In particular, we generalize the categorical character theory of Ganter--Kapranov and Bartlett to the Real setting. Given a Real representation of a finite group $G$, or more generally a categorical group, on a linear category, we associate a number, the modified secondary trace, to each graded commuting pair $(g,\omega) \in G \times \hat{G}$, where $\hat{G}$ is the background Real structure on $G$. This collection of numbers defines the Real 2-character of the Real representation. We also define various forms of induction for Real representations of categorical groups and compute their effect on Real 2-characters. We formulate our results geometrically using gerbes, vector bundles and functions on iterated unoriented loop spaces. This perspective leads to connections with the representation theory of unoriented versions of the twisted Drinfeld double of $G$ and with discrete torsion in $M$-theory with orientifolds. We speculate on the interpretation of our results as a Hopkins--Kuhn--Ravenel-type character theory in Real equivariant homotopy theory.
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