On dominance and minuscule Weyl group elements

TL;DR

This paper investigates conditions under which a Weyl group element can be both p-minuscule and map a coweight to a dominant form, providing a general Coxeter group solution using a variant of Mozes's game.

Contribution

It generalizes the understanding of p-minuscule elements and dominance in Weyl groups to all Coxeter groups, introducing a new approach via Mozes's game.

Findings

Characterization of p-minuscule Weyl group elements

Conditions for existence of dominant images of coweights

Application of Mozes's game to Coxeter group problems

Abstract

Fix a Dynkin diagram and let p be a coweight. When does there exist an element w of the corresponding Weyl group such that w is p-minuscule and w(p) is dominant? We answer this question for general Coxeter groups. We express and prove these results using a variant of Mozes's game of numbers.