Eigenspace arrangements of reflection groups

Miller, Alexander

doi:10.1090/tran/6337

Eigenspace arrangements of reflection groups
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by Alexander R. Miller PDF

Trans. Amer. Math. Soc. 367 (2015), 8543-8578 Request permission

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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