Weyl Groups and the Modified Kostant Game

TL;DR

This paper generalizes the Kostant game to connect combinatorial configurations with minimal length representatives in Weyl groups, providing new insights into root systems, automata, and Young tableaux.

Contribution

It introduces a multi-vertex modification of the Kostant game, establishing bijections with minimal coset representatives and applying this to root counting, automata, and tableaux.

Findings

Bijection between game configurations and minimal coset representatives

New root counting identity derived from the framework

Automata-based proof of regularity of reduced word languages

Abstract

This paper presents a generalization of the Kostant game, a combinatorial framework originally for generating positive roots in Lie algebras. By introducing an arbitrary multi-vertex modification, we prove that the resulting game configurations naturally biject with the minimal length representatives of parabolic quotients W/W_J. This yields a dynamical and algorithmic perspective on reduced words. Finally, we apply this framework to derive a novel root counting identity, formalize the Coxeter-theoretic foundation for combinatorial approaches to the Mukai conjecture, establish the regularity of reduced word languages via finite state automata, and dynamically construct Standard Young Tableaux.