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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The generalized Dowling lattices

Author: Phil Hanlon
Journal: Trans. Amer. Math. Soc. 325 (1991), 1-37
MSC: Primary 06B05; Secondary 06B99, 20C15
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 $ n$, a finite group $ G$, and a meet sublattice $ K$ of the lattice of subgroups of $ G$. For an appropriate choice of $ K$ the generalized Dowling lattice $ {D_n}(G,K)$ agrees with the ordinary Dowling lattice $ {D_n}(G)$. For a different choice of $ K$, 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 $ G$, generalizes the thick diagonal in $ {\mathbb{C}^n}$.

We compute the Möbius function and characteristic polynomial of the lattice $ {D_n}(G,K)$ along with the homology of $ {D_n}(G,K)$ in terms of the homology of $ K$. We go on to compute the character of $ G$ wr $ {S_n}$ acting on the homology of $ {D_n}(G,K)$. This computation provides a nontrivial generalization of a result due to Stanley concerning the character of $ {S_n}$ acting on the top homology of the partition lattice.

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Article copyright: © Copyright 1991 American Mathematical Society