Some Hecke Algebra Products and Corresponding Random Walks

Abstract

Let $\mathbf{i}=1+q+...+q^{i-1}$. For certain sequences $(r_1,...,r_l)$ of positive integers, we show that in the Hecke algebra ${\mathcal{H}}_n(q)$ of the symmetric group ${\mathfrak{S}}_n$, the product $(1+{\mathbf{r}}_1{T}_{r_1})...(1+{\mathbf{r}}_{\mathbf{l}}{T}_{r_l})$ has a simple explicit expansion in terms of the standard basis $\{T_w\}$. An interpretation is given in terms of random walks on ${\mathfrak{S}}_n$.

8 pages