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Mathematics > Combinatorics

arXiv:1802.03854 (math)
[Submitted on 12 Feb 2018 (v1), last revised 24 Feb 2018 (this version, v2)]

Title:Steinberg's theorem for crystallographic complex reflection groups

Authors:Philip Puente, Anne V. Shepler
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Abstract:Popov classified crystallographic complex reflection groups by determining lattices they stabilize. These analogs of affine Weyl groups have infinite order and are generated by reflections about affine hyperplanes; most arise as the semi-direct product of a finite complex reflection group and a full rank lattice. Steinberg's fixed point theorem asserts that the regular orbits under the action of a reflection group are exactly the orbits lying off of reflecting hyperplanes. This theorem holds for finite reflection groups (real or complex) and also affine Weyl groups but fails for some crystallographic complex reflection groups. We determine when Steinberg's theorem holds for the infinite family of crystallographic complex reflection groups. We include crystallographic groups built on finite Coxeter groups.
Comments: 13 pages, 2 figures
Subjects: Combinatorics (math.CO); Representation Theory (math.RT)
MSC classes: 20F55, 52C35 (Primary) . 16W22, 52C07, 13A50 (Secondary)
Cite as: arXiv:1802.03854 [math.CO]
  (or arXiv:1802.03854v2 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.1802.03854
Journal reference: Journal of Algebra, Volume 522, 2019, pages 332--350
Related DOI: https://doi.org/10.1016/j.jalgebra.2018.11.032

Submission history

From: Anne V. Shepler [view email]
[v1] Mon, 12 Feb 2018 01:15:25 UTC (580 KB)
[v2] Sat, 24 Feb 2018 16:28:29 UTC (580 KB)
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