Group algebras and semigroup algebras defined by permutation relations of fixed length

TL;DR

This paper studies the algebraic structures arising from permutation relations of fixed length, revealing properties like nilpotency of the Jacobson radical and conditions for semiprimitivity, with implications for growth and dimension.

Contribution

It introduces a new class of groups and algebras defined by permutation relations of fixed length, analyzing their radical, semiprimitivity, and growth properties.

Findings

The group has a free subgroup of finite index.

The Jacobson radical of the algebra is always nilpotent.

Conditions for the algebra to be semiprimitive are established.

Abstract

Let $H$ be a subgroup of $\text{Sym}_n$, the symmetric group of degree $n$. For a fixed integer $l \geq 2$, the group $G$ presented with generators $x_1, x_2, \ldots ,x_n$ and with relations $x_{i_1}x_{i_2}\cdots x_{i_l} =x_{\sigma (i_1)} x_{\sigma (i_2)} \cdots x_{\sigma (i_l)}$, where $\sigma$ runs through $H$, is considered. It is shown that $G$ has a free subgroup of finite index. For a field $K$, properties of the algebra $K[G]$ are derived. In particular, the Jacobson radical $\mathcal{J}(K[G])$ is always nilpotent, and in many cases the algebra $K[G]$ is semiprimitive. Results on the growth and the Gelfand-Kirillov dimension of $K[G]$ are given. Further properties of the semigroup $S$ and the semigroup algebra $K[S]$ with the same presentation are obtained, in case $S$ is cancellative. The Jacobson radical is nilpotent in this case as well, and sufficient conditions for the algebra to be semiprimitive are given.