A new approach to the representation theory of the symmetric groups. IV. $ \Bbb Z_{2}$-graded groups and algebras

TL;DR

This paper develops a new theoretical framework for $ ext{Z}_2$-graded algebras, focusing on inductive families and their representations, with applications to projective representations of symmetric groups.

Contribution

It introduces a novel approach to $ ext{Z}_2$-graded algebra representation theory, extending classical symmetric group theories to graded contexts.

Findings

Established a theory of inductive $ ext{Z}_2$-graded semisimple algebras

Connected the theory to projective representations of symmetric groups

Provided new insights into $ ext{Z}_2$-graded algebra structures

Abstract

We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the spirit of approach of the papers \cite{VO,OV} to representation theory of symmetric groups. The main example is the classical - theory of the projective representations of symmetric groups.