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Eigenspace arrangements of reflection groups


Author: Alexander R. Miller
Journal: Trans. Amer. Math. Soc. 367 (2015), 8543-8578
MSC (2010): Primary 20F55, 05E10, 05E18, 55P20, 14N20
DOI: https://doi.org/10.1090/tran/6337
Published electronically: April 23, 2015
MathSciNet review: 3403065
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Abstract: The lattice of intersections of reflecting hyperplanes of a complex reflection group $ W$ may be considered as the poset of $ 1$-eigenspaces of the elements of $ W$. In this paper we replace $ 1$ with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset. After posing the main question of whether this poset is shellable, we show that all its upper intervals are geometric lattices and then answer the question in the affirmative for the infinite family $ G(m,p,n)$ of complex reflection groups, and the first 31 of the 34 exceptional groups, by constructing CL-shellings. In addition, we completely determine when these eigenspaces of $ W$ form a $ K(\pi ,1)$ (resp. free) arrangement.

For the symmetric group, we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing balanced partition posets, presented as particular upper order ideals of Dowling lattices, study the representation afforded by the top (co)homology group, and give a simple map to the posets of pointed $ d$-divisible partitions.


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Additional Information

Alexander R. Miller
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Address at time of publication: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: arm@illinois.edu

DOI: https://doi.org/10.1090/tran/6337
Keywords: $d$-divisible partitions, Dowling lattices, Eilenberg--MacLane spaces, reflection groups, ribbon representations, subspace arrangements
Received by editor(s): March 16, 2013
Received by editor(s) in revised form: October 14, 2013, and October 15, 2013
Published electronically: April 23, 2015
Article copyright: © Copyright 2015 American Mathematical Society