Positivity for certain Weyl group representations

TL;DR

This paper investigates the positivity of specific Weyl group modules using the asymptotic Iwahori-Hecke algebra framework, proving a conjecture for classical types and advancing understanding of representation positivity.

Contribution

It introduces a conjecture on the positivity of certain W-modules and proves it for classical Weyl groups, linking algebraic structures to positivity properties.

Findings

Conjecture on positivity of specific W-modules proposed.

Proved the conjecture for classical Weyl groups.

Established a connection between W-module positivity and Iwahori-Hecke algebra.

Abstract

Let W be a Weyl group. We can define the notion of positivity of a W-module in terms of the corresponding module over the asymptotic Iwahori-Hecke algebra. We state a conjecture which says that certain explicit W-modules are positive and we prove it in the case where W is of classical type.