Some Hecke Algebra Products and Corresponding Random Walks

TL;DR

This paper derives explicit expansions for certain products in the Hecke algebra of the symmetric group and interprets these results through the lens of random walks on the group, linking algebraic structures to probabilistic processes.

Contribution

It provides a new explicit expansion formula for specific Hecke algebra products and connects these algebraic results to random walk interpretations on symmetric groups.

Findings

Explicit expansion formulas for Hecke algebra products.

Connection between algebraic products and random walks.

Simplification of understanding Hecke algebra structures.

Abstract

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