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 $R=\Z[q,q^{-1}]$. 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).Comment: 6 pages. To appear in Proceedings of the Southeastern Lie Theory Workshop Series, Proceedings of Symposia in Pure Mathematic