The generalized Dowling lattices

Hanlon, Phil

doi:10.1090/S0002-9947-1991-1014249-X

The generalized Dowling lattices
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Trans. Amer. Math. Soc. 325 (1991), 1-37 Request permission

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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