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Coincidences between intervals in two partial orders on complex reflection groups

Joel Brewster Lewis, Jiayuan Wang

TL;DR

This work characterizes precisely when two natural posets on a complex reflection group coincide on intervals below a given element. By leveraging Shi’s cycle-weight framework for $G(m,p,n)$ and a general poset theory for subadditive functions, the authors derive necessary and sufficient conditions: the nonzero weights modulo $p$ must partition into pairs summing to zero, and any zero-sum subset must decompose into zero-mod-$p$ cycles and such pairs. The results unify and extend known cases (e.g., well-generated groups and regular/parabolic elements), provide a complete description for the infinite family $G(m,p,n)$, and explore implications for subadditive statistics on symmetric groups. The paper also discusses heritability, special element classes, and connections to permutation statistics, offering a framework for future investigations into posets arising from group-theoretic numerical invariants.

Abstract

In a finite real reflection group, the reflection length of each element is equal to the codimension of its fixed space, and the two coincident functions determine a partial order structure called the absolute order. In complex reflection groups, the reflection length is no longer always equal to the codimension of fixed space, and the two functions give rise to two different partial orders on the group. We characterize the elements $w$ in the combinatorial family $G(m, p, n)$ of complex reflection groups for which the intervals below $w$ in these two posets coincide. We also explore the relationship between this property and other natural properties of elements in complex reflection groups; some general theory of posets arising from subadditive functions on groups; and the particular case of subadditive functions on the symmetric group.

Coincidences between intervals in two partial orders on complex reflection groups

TL;DR

This work characterizes precisely when two natural posets on a complex reflection group coincide on intervals below a given element. By leveraging Shi’s cycle-weight framework for and a general poset theory for subadditive functions, the authors derive necessary and sufficient conditions: the nonzero weights modulo must partition into pairs summing to zero, and any zero-sum subset must decompose into zero-mod- cycles and such pairs. The results unify and extend known cases (e.g., well-generated groups and regular/parabolic elements), provide a complete description for the infinite family , and explore implications for subadditive statistics on symmetric groups. The paper also discusses heritability, special element classes, and connections to permutation statistics, offering a framework for future investigations into posets arising from group-theoretic numerical invariants.

Abstract

In a finite real reflection group, the reflection length of each element is equal to the codimension of its fixed space, and the two coincident functions determine a partial order structure called the absolute order. In complex reflection groups, the reflection length is no longer always equal to the codimension of fixed space, and the two functions give rise to two different partial orders on the group. We characterize the elements in the combinatorial family of complex reflection groups for which the intervals below in these two posets coincide. We also explore the relationship between this property and other natural properties of elements in complex reflection groups; some general theory of posets arising from subadditive functions on groups; and the particular case of subadditive functions on the symmetric group.
Paper Structure (14 sections, 21 theorems, 41 equations)

This paper contains 14 sections, 21 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The infinite family of complex reflection groups
  4. Cycle partitions and reflection length
  5. Posets from subadditive functions
  6. Necessity
  7. Sufficiency
  8. Further comments and questions
  9. Heritability
  10. Specializing to nice families of elements
  11. Regular elements
  12. Parabolic quasi-Coxeter elements
  13. The main theorem in the cases that $\ell_R = \operatorname{codim}\operatorname{fix}$ on $W$
  14. Subadditive functions on the symmetric group

Key Result

Theorem 1.1

An element $w \in G(m, p, n)$ satisfies $[\mathrm{id}, w]_{\ell_R} = [\mathrm{id}, w]_{\mathrm{cdf}}$ if and only if (1) the cycle weights of $w$ that are not $0 \pmod{p}$ can be partitioned into pairs that sum to $0$ (equivalently, $\ell_R(w) = \operatorname{codim}\operatorname{fix}(w)$), and (2) a

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 2.1: Shi Shi2007
  • Proposition 2.2: Shi2007
  • Corollary 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5: essentially HLR
  • Proposition 2.6
  • proof
  • Remark 2.7
  • ...and 37 more