Degenerate affine Hecke algebras and centralizer construction for the symmetric groups

TL;DR

This paper extends the centralizer construction to symmetric groups, connecting it with degenerate affine Hecke algebras and exploring stable properties and representations of the infinite symmetric group.

Contribution

It introduces a new limit centralizer algebra for symmetric groups and links it to degenerate affine Hecke algebras, expanding the algebraic framework.

Findings

Construction of a limit centralizer algebra A for symmetric groups

Identification of algebra A with degenerate affine Hecke algebras

Analysis of tame representations of the infinite symmetric group

Abstract

In our recent papers the centralizer construction was applied to the series of classical Lie algebras to produce the quantum algebras called (twisted) Yangians. Here we extend this construction to the series of the symmetric groups S(n). We study the `stable' properties of the centralizers of S(n-m) in the group algebra C[S(n)] as n increases with m fixed. We construct a limit centralizer algebra A and describe its algebraic structure. The algebra A turns out to be closely related with the degenerate affine Hecke algebras. We also show that the so-called tame representations of the infinite symmetric group yield a class of natural A-modules.