Real Representations of $C_2$-Graded Groups: The Linear and Hermitian Theories

TL;DR

This paper explores the relationships between linear, antilinear, and hermitian representations of finite $C_2$-graded groups, establishing categorical equivalences that deepen understanding of their structures.

Contribution

It demonstrates categorical equivalences between linear, antilinear, and hermitian representations of finite $C_2$-graded groups, revealing new structural insights.

Findings

Linear representations are equivalent to antilinear representations as an $ppa$-categories.

Hermitian representations are equivalent to usual representations as an $ppa$-categories.

The results unify different types of group representations through categorical equivalences.

Abstract

We study linear and hermitian representations of finite $C_2$-graded groups. We prove that the category of linear representations is equivalent to a category of antilinear representations as an $\infty$-category. We also prove that the category hermitian representations, as an $\infty$-category, is equivalent to a category of usual representations.