Free assosymmetric algebras as modules of groups

TL;DR

This paper investigates the module structures of free assosymmetric algebras under symmetric, alternating, and general linear groups, providing insights into their algebraic representations.

Contribution

It introduces the module structures of free assosymmetric algebras for various groups, expanding understanding of their algebraic and representation-theoretic properties.

Findings

Characterization of $S_n$-module structures

Analysis of $A_n$-module structures

Description of $GL_n$-module structures

Abstract

An algebra with identities $(a,b,c)=(a,c,b)=(b,a,c)$ is called {\it assosymmetric}, where $(x,y,z)=(xy)z-x(yz)$ is associator. We study $S_n$-module, $A_n$-module and $GL_n$-module structures of free assosymmetric algebra.