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Rank reduction of string C-group representations

Authors: Peter A. Brooksbank and Dimitri Leemans
Journal: Proc. Amer. Math. Soc. 147 (2019), 5421-5426
MSC (2010): Primary 52B11, 20D06
Published electronically: July 1, 2019
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Abstract: We show that a rank reduction technique for string C-group representations first used in [Adv. Math. 228 (2018), pp. 3207-3222] for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on $ d$-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks $ 3\leq n\leq d$. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group $ \operatorname {Alt}(11)$--the only known group having ``rank gaps''--is perhaps more unusual than previously thought.

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Peter A. Brooksbank
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Dimitri Leemans
Affiliation: Département de Mathématique, C.P. 216 Algèbre et Combinatoire, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 Bruxelles, Belgium

Keywords: Abstract regular polytope, string C-group, Coxeter group
Received by editor(s): December 3, 2018
Received by editor(s) in revised form: March 28, 2019
Published electronically: July 1, 2019
Additional Notes: This work was partially supported by a grant from the Simons Foundation (#281435 to the first author) and by the Hausdorff Research Institute for Mathematics
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2019 American Mathematical Society