A new basis for the representation ring of a Weyl group, II

TL;DR

This paper introduces a new basis for the representation ring of a Weyl group, demonstrating its relation to the standard basis and providing a new parametrization of representations, with potential extensions to Chevalley groups.

Contribution

It establishes a connection between a newly defined basis and the standard basis via an upper triangular unipotent matrix, and offers a novel parametrization method for representations.

Findings

The new basis relates to the standard basis through an upper triangular unipotent matrix.

A new parametrization of representations using subgroup pairs is proposed.

Extensions to unipotent representations of finite Chevalley groups are outlined.

Abstract

In a previous paper I have defined a new basis for the representation ring of a Weyl group. In this paper we show that the new basis is related to the standard basis by an upper triangular unipotent matrix. We also give a new parametrization of representations in a fixed family by certain pairs of subgroups of a finite group attached to the family. We outline an extension to the case of unipotent representations of a finite Chevalley group.