The generalized Dowling lattices

Author:
Phil Hanlon

Journal:
Trans. Amer. Math. Soc. **325** (1991), 1-37

MSC:
Primary 06B05; Secondary 06B99, 20C15

DOI:
https://doi.org/10.1090/S0002-9947-1991-1014249-X

MathSciNet review:
1014249

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Abstract: In this paper we study a new class of lattices called the generalized Dowling lattices. These lattices are parametrized by a positive integer , a finite group , and a meet sublattice of the lattice of subgroups of . For an appropriate choice of the generalized Dowling lattice agrees with the ordinary Dowling lattice . For a different choice of , the generalized Dowling lattices are the lattice of intersections of a set of subspaces in complex space. The set of subspaces, defined in terms of a representation of , generalizes the thick diagonal in .

We compute the Möbius function and characteristic polynomial of the lattice along with the homology of in terms of the homology of . We go on to compute the character of wr acting on the homology of . This computation provides a nontrivial generalization of a result due to Stanley concerning the character of acting on the top homology of the partition lattice.

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DOI:
https://doi.org/10.1090/S0002-9947-1991-1014249-X

Article copyright:
© Copyright 1991
American Mathematical Society