On non-conjugate Coxeter elements in well-generated reflection groups

Reiner, Victor; Ripoll, Vivien; Stump, Christian

doi:10.1007/s00209-016-1736-4

Cited by 24 publications

(24 citation statements)

References 30 publications

(36 reference statements)

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“…For real reflection groups, all Coxeter elements are

W

‐conjugate. For well‐generated complex reflection groups, they are all related by what Marin and Michel call reflection automorphisms , and these give rise to the desired poset isomorphisms

false[e, c false] ≅ false[e, c^{'} false]

; see Reiner, Ripoll, and Stump . Remark In spite of Theorem , within some

{GL}_{n} false(F_{q} false)

there exist regular elliptic elements

c^{'}

and Singer cycles

c

for which

false[e, c^{'} false] ≇ false[e, c false]

.…”

Section: Final Remarks and Questionsmentioning

confidence: 99%

Absolute order in general linear groups

Huang

Lewis

Reiner

2017

J. London Math. Soc.

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Abstract. This paper studies a partial order on the general linear group GL(V ) called the absolute order, derived from viewing GL(V ) as a group generated by reflections, that is, elements whose fixed space has codimension one. The absolute order on GL(V ) is shown to have two equivalent descriptions, one via additivity of length for factorizations into reflections, the other via additivity of fixed space codimensions. Other general properties of the order are derived, including self-duality of its intervals.Working over a finite field F q , it is shown via a complex character computation that the poset interval from the identity to a Singer cycle (or any regular elliptic element) in GL n (F q ) has a strikingly simple formula for the number of chains passing through a prescribed set of ranks.

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“…For real reflection groups, all Coxeter elements are

W

‐conjugate. For well‐generated complex reflection groups, they are all related by what Marin and Michel call reflection automorphisms , and these give rise to the desired poset isomorphisms

false[e, c false] ≅ false[e, c^{'} false]

; see Reiner, Ripoll, and Stump . Remark In spite of Theorem , within some

{GL}_{n} false(F_{q} false)

there exist regular elliptic elements

c^{'}

and Singer cycles

c

for which

false[e, c^{'} false] ≇ false[e, c false]

.…”

Section: Final Remarks and Questionsmentioning

confidence: 99%

Absolute order in general linear groups

Huang

Lewis

Reiner

2017

J. London Math. Soc.

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“…The reference [44] uses a less general definition of a Coxeter element (it considers only those coming from the regular number e 2iπ/h ). However, the results in [43] guarantee that Proposition 2.5 also holds in the more general setting.…”

Section: Complex Reflection Groupsmentioning

confidence: 84%

“…For further background on partially ordered sets we recommend [19], an excellent introduction to the Sperner property and related subjects is [1]. An extensive textbook on complex reflection groups is [33], and a recent exposition on Coxeter elements is [43]. Throughout the paper we use the abbreviation [n] = {1, 2, .…”

Section: Preliminariesmentioning

confidence: 99%

Symmetric decompositions and the strong Sperner property for noncrossing partition lattices

Mühle

2016

J Algebr Comb

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We prove that the noncrossing partition lattices associated with the complex reflection groups G (d, d, n) for d, n ≥ 2 admit symmetric decompositions into Boolean subposets. As a result, these lattices have the strong Sperner property and their rank-generating polynomials are symmetric, unimodal, and γ-nonnegative. We use computer computations to complete the proof that every noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus answering affirmatively a question raised by D. Armstrong.2010 Mathematics Subject Classification. 06A07 (primary), and 20F55 (secondary).

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“…A regular element of W is an element with an eigenvector that lies in the complement of the reflecting hyperplanes. Following [RRS17], we define a Coxeter element of W to be a regular element of order h. For Weyl groups, Coxeter elements coincide with conjugates of standard Coxeter elements. More generally, a reflection automorphism of W is a group automorphism that preserves the set of reflections of W .…”

Section: Preliminariesmentioning

confidence: 99%

Coxeter Pop-Tsack Torsing

Defant¹,

Williams²

2021

Preprint

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Given a finite irreducible Coxeter group W with a fixed Coxeter element c, we define the Coxeter pop-tsack torsing operator Pop T : W → W by Pop T (w) = w • π T (w) −1 , where π T (w) is the join in the noncrossing partition lattice NC(w, c) of the set of reflections lying weakly below w in the absolute order. This definition serves as a "Bessis dual" version of the first author's notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack-sorting map on symmetric groups. We show that if W is coincidental or of type D, then the identity element of W is the unique periodic point of Pop T and the maximum size of a forward orbit of Pop T is the Coxeter number h of W . In each of these types, we obtain a natural lift from W to the dual braid monoid of W . We also prove that W is coincidental if and only if it has a unique forward orbit of size h. For arbitrary W , we show that the forward orbit of c −1 under Pop T has size h and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit.

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On non-conjugate Coxeter elements in well-generated reflection groups

Cited by 24 publications

References 30 publications

Absolute order in general linear groups

Absolute order in general linear groups

Symmetric decompositions and the strong Sperner property for noncrossing partition lattices

Coxeter Pop-Tsack Torsing

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