Stability of the Hecke algebra of wreath products

Abstract

The Hecke algebras ${\mathcal{H}}_{n,k}$ of the group pairs $(S_{kn},S_k\wr S_n)$ can be endowed with a filtration with respect to the orbit structures of the elements of $S_{kn}$ relative to the action of $S_{kn}$ on the set of $k$-partitions of $\{1,\dots ,kn\}$. We prove that the structure constants of the associated filtered algebra ${\mathcal{F}}_{n,k}$ is independent of $n$. The stability property enables the construction of a universal algebra $\mathcal{F}$ to govern the algebras ${\mathcal{F}}_{n,k}$. We also prove that the structure constants of the algebras ${\mathcal{H}}_{n,k}$ are polynomials in $n$. For $k=2$, when the algebras $({\mathcal{F}}_{n,2}{)}_{n\in \mathbb{N}}$ are commutative, these results were obtained by Aker and Can, by Can and Ozden, and by Tout.

21 pages, 4 figures