Infnite-dimensional Schur-Weyl duality and the Coxeter-Laplace operator

TL;DR

This paper generalizes Schur-Weyl duality to infinite groups, introduces new representations of the infinite symmetric group, and analyzes the spectral properties of the Coxeter-Laplace operator within this framework.

Contribution

It extends classical duality to the infinite case, constructing novel representations of the infinite symmetric group and studying their spectral types.

Findings

New class of representations of the infinite symmetric group.

Spectral types of these representations with respect to Gelfand-Tsetlin algebra.

Analysis of the Coxeter-Laplace operator in the constructed representations.

Abstract

We extend the classical Schur-Weyl duality between representations of the groups $SL(n,\C)$ and $\sN$ to the case of $SL(n,\C)$ and the infinite symmetric group $\sinf$. Our construction is based on a "dynamic," or inductive, scheme of Schur-Weyl dualities. It leads to a new class of representations of the infinite symmetric group, which have not appeared earlier. We describe these representations and, in particular, find their spectral types with respect to the Gelfand-Tsetlin algebra. The main example of such a representation acts in an incomplete infinite tensor product. As an important application, we consider the weak limit of the so-called Coxeter-Laplace operator, which is essentially the Hamiltonian of the XXX Heisenberg model, in these representations.