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On the cocenter of cyclotomic Hecke algebra of type $G(r,1,n)$

Jun Hu, Lei Shi

TL;DR

The paper generalizes Geck-Pfeiffer's cocenter-center framework from Iwahori-Hecke algebras to the cyclotomic Hecke algebra of the complex reflection group G(r,1,n). It constructs an integral basis for the cocenter and shows the center's rank is independent of characteristic and parameters, enabling base-change stability. A key advance is a robust treatment of minimal-length elements in conjugacy classes of W_n via BM and double coset normal forms, leading to canonical representatives used in cocenter and center bases. The authors apply these results to confirm Chavli-Pfeiffer's polynomial-coefficient conjecture for the G(r,1,n) case and to articulate a clear, representation-theoretic description of class polynomials and central elements. Overall, the work provides structural invariants for cyclotomic Hecke algebras that are intrinsic to the underlying complex reflection group, with concrete combinatorial and algebraic consequences.

Abstract

In this paper, we construct an integral basis for the cocenter of the cyclotomic Hecke algebra $\mathscr{H}_{n,K}$ of type $G(r,1,n)$ by generalizing Geck and Pfeiffer's work on the cocenters of the Iwahori-Hecke algebras associated to finite Weyl groups. We show that the dimensions of both the cocenter and the center of the cyclotomic Hecke algebra $\mathscr{H}_{n,K}$ are independent of the characteristic of the ground field, its Hecke parameter and cyclotomic parameters. As an application, we verify Chavli-Pfeiffer's conjecture on the polynomial coefficient $g_{w,C}$ for the complex reflection group of type $G(r,1,n)$.

On the cocenter of cyclotomic Hecke algebra of type $G(r,1,n)$

TL;DR

The paper generalizes Geck-Pfeiffer's cocenter-center framework from Iwahori-Hecke algebras to the cyclotomic Hecke algebra of the complex reflection group G(r,1,n). It constructs an integral basis for the cocenter and shows the center's rank is independent of characteristic and parameters, enabling base-change stability. A key advance is a robust treatment of minimal-length elements in conjugacy classes of W_n via BM and double coset normal forms, leading to canonical representatives used in cocenter and center bases. The authors apply these results to confirm Chavli-Pfeiffer's polynomial-coefficient conjecture for the G(r,1,n) case and to articulate a clear, representation-theoretic description of class polynomials and central elements. Overall, the work provides structural invariants for cyclotomic Hecke algebras that are intrinsic to the underlying complex reflection group, with concrete combinatorial and algebraic consequences.

Abstract

In this paper, we construct an integral basis for the cocenter of the cyclotomic Hecke algebra of type by generalizing Geck and Pfeiffer's work on the cocenters of the Iwahori-Hecke algebras associated to finite Weyl groups. We show that the dimensions of both the cocenter and the center of the cyclotomic Hecke algebra are independent of the characteristic of the ground field, its Hecke parameter and cyclotomic parameters. As an application, we verify Chavli-Pfeiffer's conjecture on the polynomial coefficient for the complex reflection group of type .
Paper Structure (7 sections, 29 theorems, 176 equations)

This paper contains 7 sections, 29 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Minimal length elements in each conjugacy class of $W_n$
  4. Normal forms and Double coset decomposition
  5. Some minimal length elements in conjugacy class
  6. Cocenters of cyclotomic Hecke algebra
  7. Class polynomials and integral polynomial coefficients

Key Result

Theorem 1.3

For any conjugacy class $C$ of $W_n$ and any $w\in C$, there exists an element $w'\in C_{\min}$, such that $w\rightarrow w'$, where $C_{\min}$ is the set of minimal length elements in $C$.

Theorems & Definitions (60)

  • Definition 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Definition 2.3
  • Lemma 2.5
  • Lemma 2.6: A1
  • Definition 2.7: cf. DJM, Ma
  • Definition 2.8: DJM, Ma, HW
  • Lemma 2.9: DJMMa
  • ...and 50 more