A new basis for the representation ring of a Weyl group

TL;DR

This paper introduces a novel basis for the representation ring of a Weyl group, combining special and cell representations, with a focus on bipositivity properties of these representations.

Contribution

It defines a new basis for the Grothendieck group of Weyl group representations that unifies special and cell representations with bipositivity features.

Findings

New basis includes special and cell representations

Representations in the basis exhibit bipositivity

Provides a structured framework for Weyl group representations

Abstract

Let W be a Weyl group. We define a new basis for the Grothendieck group of representations of W. This basis contains on the one hand the special representations of W and on the other hand the representations carried by the left cells of W. We show that the representations in the new basis have a certain bipositivity property.