Transition matrices between Young's natural and seminormal representations

TL;DR

This paper provides a recursive formula for the change-of-basis matrix entries between Young's natural and seminormal representations of the symmetric group, extending to affine Hecke and related algebras.

Contribution

It introduces a recursive computational method for the change-of-basis matrix entries, generalizing to various algebraic structures beyond the symmetric group.

Findings

Entries are sums over weighted paths in the weak Bruhat graph.

Entries can be computed recursively using at most two previous entries.

The results extend to affine Hecke, Ariki-Koike, Iwahori-Hecke algebras, and complex reflection groups.

Abstract

We derive a formula for the entries in the change-of-basis matrix between Young's seminormal and natural representations of the symmetric group. These entries are determined as sums over weighted paths in the weak Bruhat graph on standard tableaux, and we show that they can be computed recursively as the weighted sum of at most two previously-computed entries in the matrix. We generalize our results to work for affine Hecke algebras, Ariki-Koike algebras, Iwahori-Hecke algebras, and complex reflection groups given by the wreath product of a finite cyclic group with the symmetric group.