Real Representations of $C_2$-Graded Groups: The Antilinear Theory

TL;DR

This paper develops a new theory of antilinear representations for finite $C_2$-graded groups, combining algebraic structures with character theory and providing explicit examples like $A_n$ within $S_n$.

Contribution

It introduces the concept of antilinear blocks and extends representation theory to include antilinear actions in $C_2$-graded groups.

Findings

Defined antilinear blocks and their structure.

Analyzed characters and Frobenius-Schur indicators in this context.

Described antilinear representations of $A_n$ within $S_n$.

Abstract

We use the structure of finite-dimensional graded algebras to develop the theory of antilinear representations of finite $C_2$-graded groups. A finite $C_2$-graded group is a finite group with a subgroup of index 2. In this theory the subgroup acts linearly, while the other coset acts antilinearly. We introduce antilinear blocks, whose structure is a crucial component of the theory. Among other things, we study characters and Frobenius-Schur indicators. As an example, we describe the antilinear representations of the $C_2$-graded group $A_n \leq S_n$.