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Factorization problems in complex reflection groups

Joel Brewster Lewis, Alejandro H. Morales

Abstract

We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of $W$-noncrossing partitions.

Factorization problems in complex reflection groups

Abstract

We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of -noncrossing partitions.

Paper Structure

This paper contains 36 sections, 29 theorems, 117 equations, 6 figures, 23 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Known factorization results in $\mathfrak{S}_n$
  4. Complex reflection groups
  5. Basic definitions
  6. Key examples
  7. Fixed space dimension
  8. Coxeter elements
  9. Degrees and coexponents
  10. Counting factorizations with representation theory
  11. Factoring an $(n - 1)$-cycle in $\mathfrak{S}_n$
  12. Factorization results for the group $G(d,1,n)$
  13. Factorization results for the subgroup $G(d,d,n)$
  14. Transitive factorizations
  15. Nontransitive factorizations
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $W$ be an irreducible well generated complex reflection group of rank $n$. Let $c$ be a Coxeter element in $W$, let $R$ and $R^*$ be the set of all reflections and all reflecting hyperplanes in $W$, and for $\ell \geq 1$ let $N_{\ell}(W) := \#\{ (\tau_1,\ldots,\tau_{\ell}) \in R^{\ell} \colon \t

Figures (6)

  • Figure 1: The matrix representation of the generalized permutations $[(1234); (0, 0, 0, 1)]$ in $G(d, 1, 4)$ (left) and $c_{(d, d, 4)} := [(123)(4); (0, 0, 1, -1)]$ in $G(d, d, 4) \subset G(d, 1, 4)$ (right). Here $d > 1$ and $\omega = \exp(2\pi i/d)$ is a primitive complex $d$th root of unity. In $c_{(d, d, 4)}$, the cycle $(123)$ has weight $1$ and the cycle $(4)$ has weight $-1$.
  • Figure 2: A diagonal reflection and a transposition-like reflection in $G(d, 1, 4)$. Here $a$ represents a $d$th root of $1$ other than $1$ itself, and $b$ represents an arbitrary $d$th root of $1$.
  • Figure 3: The transitive factorization of the $5$-cycle $\varsigma_{(5, 1)} = (12345)(6)$ in $\mathfrak{S}_6$. The thread involving the fixed point $6$ is highlighted.
  • Figure 4: (a) The genus-$1$ map of the factorization $c = [(1532)(4); (1, 2,2,0,1)] \cdot [(134)(25); (1, 2,0,2, 2)]$ in Example \ref{['example:map']}. (b) An "unfolding" of the map on the left, using the (blue) $v$-weights to determine which copies of the vertices to connect with each edge.
  • Figure 5: (a) The $C_2 = 2$ set partitions on $n = 2$ points are both noncrossing. (b, c) The $\binom{2 \cdot 2}{2} = 6$ noncrossing set partitions on $2d$ points with $d$-fold rotational symmetry, for $d = 2$ and $d = 3$.
  • ...and 1 more figures

Theorems & Definitions (65)

  • Theorem 1.1: Chapuy--Stump ChapuyStump
  • Theorem 1.2: Jackson Jackson, Schaeffer--Vassilieva SchaefferVassilieva
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • ...and 55 more