Some factorizations in the twisted group algebra of symmetric groups

TL;DR

This paper explores factorizations within the twisted group algebra of symmetric groups, extending previous matrix factorizations to twisted algebra computations involving symmetric group actions on polynomial rings.

Contribution

It introduces a new approach to factorization in the twisted group algebra of symmetric groups using group actions on polynomial rings, enabling calculation of annihilating constants.

Findings

Established factorizations in the twisted algebra setting.

Connected matrix factorizations to twisted algebra computations.

Provided methods to compute constants annihilated by multiparametric derivatives.

Abstract

In this paper we will give a similar factorization as in \cite{4}, \cite{5}, where the autors Svrtan and Meljanac examined certain matrix factorizations on Fock-like representation of a multiparametric quon algebra on the free associative algebra of noncommuting polynomials equiped with multiparametric partial derivatives. In order to replace these matrix factorizations (given from the right) by twisted algebra computation, we first consider the natural action of the symmetric group $S_{n}$ on the polynomial ring $R_{n}$ in $n^2$ commuting variables $X_{a\,b}$ and also introduce a twisted group algebra (defined by the action of $S_{n}$ on $R_{n}$) which we denote by ${\mathcal{A}(S_{n})}$. Here we consider some factorizations given from the left because they will be more suitable in calculating the constants (= the elements which are annihilated by all multiparametric partial derivatives) in the free algebra of noncommuting polynomials.