Multiplicative Bases for the Centres of the Group Algebra and Iwahori-Hecke Algebra of the Symmetric Group

TL;DR

This paper investigates the existence of multiplicative bases in the centers of group algebras and Iwahori-Hecke algebras of symmetric groups, proving their non-existence for most cases and identifying specific exceptions.

Contribution

It establishes the non-existence of multiplicative bases for centers of these algebras when n ≥ 3 and characterizes the cases for n=2.

Findings

No multiplicative bases for Z(ℤ S_n) and Z(ℋ_n) when n ≥ 3.

Exactly two multiplicative bases for Z(ℤ S_2).

No multiplicative bases for Z(ℋ_2).

Abstract

Let $\H_n$ be the Iwahori-Hecke algebra of the symmetric group $S_n$, and let $Z(\H_n)$ denote its centre. Let $B={b_1,b_2,...,b_t}$ be a basis for $Z(\H_n)$ over $R=\Z[q,q^{-1}]$. Then $B$ is called \emph{multiplicative} if, for every $i$ and $j$, there exists $k$ such that $b_ib_j= b_k$. In this article we prove that there are no multiplicative bases for $Z(\Z S_n)$ and $Z(\H_n)$ when $n\ge 3$. In addition, we prove that there exist exactly two multiplicative bases for $Z(\Z S_2)$ and none for $Z(\H_2)$.