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

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 $R=\mathbb{Z}[q,q^{-1}]$. Then $B$ is called \emph{multiplicative} if, for every $i$ and $j$, there exists $k$ such that $b_i{b}_j=b_k$. In this article we prove that there are no multiplicative bases for $Z(\mathbb{Z}{S}_n)$ and $Z(\H_n)$ when $n\ge 3$. In addition, we prove that there exist exactly two multiplicative bases for $Z(\mathbb{Z}{S}_2)$ and none for $Z(\H_2)$.

arXiv comment6 pages. To appear in Proceedings of the Southeastern Lie Theory Workshop Series, Proceedings of Symposia in Pure Mathematics