The Schur--Weyl graph and Thoma's theorem

TL;DR

This paper introduces the Schur--Weyl graph, a new combinatorial structure linking RSK algorithm and representation duality, providing a novel proof of the classification of indecomposable characters of the infinite symmetric group.

Contribution

It defines the Schur--Weyl graph and applies it to give a new proof of Thoma's theorem on characters of the infinite symmetric group.

Findings

Introduction of the Schur--Weyl graph

New proof of Thoma's theorem

Enhanced understanding of symmetric group representations

Abstract

We define a graded graph, called the Schur--Weyl graph, which arises naturally when one considers simultaneously the RSK algorithm and the classical duality between representations of the symmetric and general linear groups. As one of the first applications of this graph, we give a new proof of the completeness of the list of discrete indecomposable characters of the infinite symmetric group.