Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pinnacles for Complex Reflection Groups

Aaron Burnham-Schmidt, Nicolle González

TL;DR

The paper extends pinnacle-set theory from symmetric and signed symmetric groups to the broad family of nonexceptional complex reflection groups $G(m,p,n)$. It introduces canonical witnesses and a $\pi_i$-decomposition to characterize admissible pinnacle sets for the generalized wreath products $\mathbb{Z}_m \wr S_n$, establishing the sharp bound $d\le\left\lfloor\frac{n-1}{2}\right\rfloor$ and providing a constructive admissibility criterion. The authors derive four independent enumeration formulas for $\mathsf{APS}_d(m,n)$—two recurrences and two closed forms—and show that, for $d<\left\lceil\frac{n-1}{2}\right\rceil$, the pinnacle-set counts for $G(m,p,n)$ agree with those for $\mathbb{Z}_m \wr S_n$, thereby enabling comprehensive enumeration across most nonexceptional groups. They further reduce the remaining maximal case (odd $n$) to the $m=p$ setting, via a reduction map, and provide corollaries describing the interplay between different $G(m,p,n)$ inside the classification; collectively, this work significantly broadens the combinatorial toolkit for pinnacle sets and resolves conjectures in the signed-permutation setting.

Abstract

We study, characterize, and enumerate the admissible pinnacle sets of nonexceptional complex reflection groups $G(m,p,n)$, which include all generalized symmetric groups $\mathbb{Z}_m \wr S_n$ as special cases. This generalizes the work of Davis--Nelson--Petersen--Tenner for symmetric groups $S_n$ and González--Harris--Rojas Kirby--Smit Vega Garcia--Tenner for signed symmetric groups $\mathbb{Z}_2 \wr S_n$. As a consequence, we prove a conjecture of González--Harris--Rojas Kirby--Smit Vega Garcia--Tenner for pinnacles of signed permutations.

Pinnacles for Complex Reflection Groups

TL;DR

The paper extends pinnacle-set theory from symmetric and signed symmetric groups to the broad family of nonexceptional complex reflection groups . It introduces canonical witnesses and a -decomposition to characterize admissible pinnacle sets for the generalized wreath products , establishing the sharp bound and providing a constructive admissibility criterion. The authors derive four independent enumeration formulas for —two recurrences and two closed forms—and show that, for , the pinnacle-set counts for agree with those for , thereby enabling comprehensive enumeration across most nonexceptional groups. They further reduce the remaining maximal case (odd ) to the setting, via a reduction map, and provide corollaries describing the interplay between different inside the classification; collectively, this work significantly broadens the combinatorial toolkit for pinnacle sets and resolves conjectures in the signed-permutation setting.

Abstract

We study, characterize, and enumerate the admissible pinnacle sets of nonexceptional complex reflection groups , which include all generalized symmetric groups as special cases. This generalizes the work of Davis--Nelson--Petersen--Tenner for symmetric groups and González--Harris--Rojas Kirby--Smit Vega Garcia--Tenner for signed symmetric groups . As a consequence, we prove a conjecture of González--Harris--Rojas Kirby--Smit Vega Garcia--Tenner for pinnacles of signed permutations.

Paper Structure

This paper contains 5 sections, 25 theorems, 67 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Pinnacles for Generalized Symmetric Groups
  3. Characterizing Admissible Pinnacle sets for $\mathbb{Z}_m\wr S_n$
  4. Enumerating Pinnacle Sets for Generalized Symmetric Groups
  5. Pinnacles for Complex Reflection Groups

Key Result

Lemma 2.11

If $P \in \mathsf{APS}(m, n)$ with $P = \{ \xi^{a_1}(p_1) \prec \xi^{a_2}(p_2) \prec \dots \prec \xi^{a_d}(p_d)\}$ then $p_i \neq p_j$ for all $i \neq j$.

Figures (2)

  • Figure 1: The graph for the witness of $P =\{\xi^0(2),\xi^1(3),\xi^0(5)\}$ given by $w =\xi^0(1)\xi^0(4)\xi^0(2)\xi^1(7)\xi^2(9)\xi^1(3)\xi^2(10)\xi^1(8)\xi^0(5)\xi^1(6) \in \mathbb{Z}_3 \wr S_{10}$. The pinnacle values are circles in red.
  • Figure 2: The graph for the canonical witness permutation $\omega_P$ for $P =\{\xi^1(3)\prec \xi^0(5)\prec \xi^0(2)\} \in \mathsf{APS}_3(3,10)$ given by $\omega_P =\xi^2(10)\xi^1(3)\xi^2(9)\xi^0(5)\xi^2(8)\xi^0(2)\xi^2(7)\xi^2(6)\xi^2(4)\xi^2(1) \in \mathbb{Z}_3 \wr S_{10}$. The pinnacle values are circles in red.

Theorems & Definitions (69)

  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Example 2.10
  • ...and 59 more