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Weyl groups and the Kostant game

Juan Sebastián Cortés-Cruz

Abstract

This paper establishes a novel combinatorial framework at the intersection of Lie theory and algebraic combinatorics, based on a generalization of the Kostant game. We begin by reviewing the foundations of root systems, the classification of Dynkin diagrams, and the structure of Weyl groups. Subsequently, we analyze the original Kostant game as a tool for generating positive roots, demonstrating its unique termination on simply-laced diagrams and its role in an alternative classification thereof. The main contribution of this work -- which, to our knowledge, has not been studied before -- is a multi-vertex generalization of the game that allows for the simultaneous modification of multiple vertices of a Dynkin diagram. We prove that the resulting configurations of this new game establish a natural bijection with the elements of the quotient W/W_J of Weyl groups by parabolic subgroups. This formalism is applied to problems in algebraic geometry, specifically addressing cases of the Mukai conjecture via Hilbert polynomials, and is accompanied by a computational implementation in Java. These results offer new combinatorial perspectives for studying root counting problems, the regularity of reduced word languages, and the construction of Young Tableaux.

Weyl groups and the Kostant game

Abstract

This paper establishes a novel combinatorial framework at the intersection of Lie theory and algebraic combinatorics, based on a generalization of the Kostant game. We begin by reviewing the foundations of root systems, the classification of Dynkin diagrams, and the structure of Weyl groups. Subsequently, we analyze the original Kostant game as a tool for generating positive roots, demonstrating its unique termination on simply-laced diagrams and its role in an alternative classification thereof. The main contribution of this work -- which, to our knowledge, has not been studied before -- is a multi-vertex generalization of the game that allows for the simultaneous modification of multiple vertices of a Dynkin diagram. We prove that the resulting configurations of this new game establish a natural bijection with the elements of the quotient W/W_J of Weyl groups by parabolic subgroups. This formalism is applied to problems in algebraic geometry, specifically addressing cases of the Mukai conjecture via Hilbert polynomials, and is accompanied by a computational implementation in Java. These results offer new combinatorial perspectives for studying root counting problems, the regularity of reduced word languages, and the construction of Young Tableaux.
Paper Structure (28 sections, 39 theorems, 146 equations, 21 figures)

This paper contains 28 sections, 39 theorems, 146 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Root systems
  3. Abstract root systems
  4. Simple and positive roots
  5. Cartan matrix and Dynkin diagrams
  6. Classification of irreducible root systems
  7. Weyl groups
  8. The Weyl group as an automorphism group
  9. Length and reduced expressions
  10. Parabolic subgroups
  11. The Kostant game
  12. Definition of the game and examples
  13. Game dynamics and unique termination
  14. The Kostant game on multiply-laced diagrams
  15. Interpretation of the game in root systems
  16. ...and 13 more sections

Key Result

Theorem 2.6

Let $\alpha, \beta \in \Phi$ be roots such that neither is a multiple of the other, with $|\alpha| \geq |\beta|$, and let $\theta$ be the angle between them. Then one of the following possibilities must hold:

Figures (21)

  • Figure 2: Rank 2 root systems.
  • Figure 3: Dynkin diagrams associated with irreducible root systems.
  • Figure 4: A cycle of length 4 — cannot occur in a Dynkin diagram.
  • Figure 5: A vertex with four simple neighbors is impossible in Dynkin diagrams.
  • Figure 6: A graph with two branching points cannot be a Dynkin diagram.
  • ...and 16 more figures

Theorems & Definitions (133)

  • Definition 2.1: Root system
  • Remark 2.2: On the axioms
  • Definition 2.3: Co-root
  • Definition 2.4: Root system isomorphism
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Corollary 2.7: Rank 2 root systems
  • Example 2.8: A more general example
  • Definition 2.9: Reducible root systems
  • ...and 123 more