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Hook formula for Coxeter groups via the twisted group ring

Leonardo C. Mihalcea, Hiroshi Naruse, Changjian Su

TL;DR

The paper addresses the problem of generalizing Nakada's colored hook formula to arbitrary Coxeter groups and provides a concise algebraic proof using the Kostant--Kumar twisted group ring $H_Q$, its $\mathcal{L}_w$-basis and dual $\eta^w$, and a Molev--Sagan recursion. It further connects to Yang--Baxter elements in $Q(V)[W]$ through a left $Q(V)$-module isomorphism, yielding a streamlined derivation of Shi's Yang--Baxter relations and a parabolic refinement framework via $\eta^v_J$ and structure constants $d^{w,J}_{u,v}$. The main result is a hook-type formula for Coxeter groups that specializes to Nakada's colored hook formula in the $\\lambda$-minuscule case and has a geometric interpretation in finite Weyl groups through Chern--Schwartz--MacPherson and Demazure--Lusztig theory, suggesting further connections to K-theory. Overall, the work provides a unified algebraic approach to hook formulas on Coxeter groups with potential geometric and K-theoretic extensions.

Abstract

We use Kostant and Kumar's twisted group ring and its dual to formulate and prove a generalization of Nakada's colored hook formula for any Coxeter groups. For dominant minuscule elements of the Weyl group of a Kac--Moody algebra, this provides another short proof of Nakada's colored hook formula.

Hook formula for Coxeter groups via the twisted group ring

TL;DR

The paper addresses the problem of generalizing Nakada's colored hook formula to arbitrary Coxeter groups and provides a concise algebraic proof using the Kostant--Kumar twisted group ring , its -basis and dual , and a Molev--Sagan recursion. It further connects to Yang--Baxter elements in through a left -module isomorphism, yielding a streamlined derivation of Shi's Yang--Baxter relations and a parabolic refinement framework via and structure constants . The main result is a hook-type formula for Coxeter groups that specializes to Nakada's colored hook formula in the -minuscule case and has a geometric interpretation in finite Weyl groups through Chern--Schwartz--MacPherson and Demazure--Lusztig theory, suggesting further connections to K-theory. Overall, the work provides a unified algebraic approach to hook formulas on Coxeter groups with potential geometric and K-theoretic extensions.

Abstract

We use Kostant and Kumar's twisted group ring and its dual to formulate and prove a generalization of Nakada's colored hook formula for any Coxeter groups. For dominant minuscule elements of the Weyl group of a Kac--Moody algebra, this provides another short proof of Nakada's colored hook formula.
Paper Structure (10 sections, 16 theorems, 44 equations)

This paper contains 10 sections, 16 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Coxeter group and root system
  3. Coxeter group and root system
  4. Some lemmas
  5. Twisted group ring and its dual
  6. Kostant--Kumar twisted group ring $H_Q$
  7. Dual basis $\{\eta^w\}_{w\in W}$ of $\{\mathcal{L}_w\}_{w\in W}$
  8. Yang--Baxter elements in the group algebra $Q(V)[W]$
  9. Main Theorem
  10. Appendix: Geometric interpretation

Key Result

Lemma 2.1

For a reduced expression $w=s_{i_1} s_{i_2}\cdots s_{i_\ell}$, let Then $|S(w)|=\ell \text{ and } S(w)=\{ \beta_1,\beta_2,\ldots, \beta_{\ell}\}.$

Theorems & Definitions (32)

  • Lemma 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 1
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • ...and 22 more