Stability of the Hecke algebra of wreath products

TL;DR

This paper proves that the structure constants of Hecke algebras associated with wreath products are stable and polynomial in size, enabling the construction of a universal algebra that encapsulates their structure.

Contribution

It establishes the stability and polynomial nature of structure constants for Hecke algebras of wreath products, generalizing previous results for specific cases.

Findings

Structure constants are independent of n.

Structure constants are polynomial in n.

Construction of a universal algebra for these structures.

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.