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

Let \H_n be the Iwahori-Hecke algebra of the symmetric group $S_n$, and let Z(\H_n) denote its centre. Let $B=b_1,b_2,\mathrm{.}\mathrm{.}\mathrm{.},b_t$ 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 $n\ge 3$\mathrm{.}Inaddition,weprovethatthereexistexactlytwomultiplicativebasesfor$Z(\Z S_2)$andnonefor$Z(\H_2)$.