Cohomology of Dowling lattices and Lie superalgebras

Date of Award




Degree Name

Doctor of Philosophy (Ph.D.)



First Committee Member

Michelle Wachs - Committee Chair


It follows from the work of Hanlon (Ha1), Stanley (St1), Witt (Wi), and Brandt (Br) that the cohomology of $\Pi\sb{n}$, the lattice of partitions of $\{1,\...,n\}$, is isomorphic to sgn $\otimes$ Lie$\sb{n}$, the sign representation tensored with the multilinear component of the free Lie algebra on n letters, as representations of the symmetric group ${\cal S}\sb{n}$.Wachs (Wa2) gave a new proof of this result by describing generators for the cohomology of $\Pi\sb{n}$ and for Lie$\sb{n}$, using ordered binary trees whose leaves are labeled with elements of $\{1,\...,n\}$. She found relations for these generators and produced a map which respects the action of ${\cal S}\sb{n}$, and takes generators to generators and relations to relations up to sign.Barcelo (Ba) gave a combinatorial proof of the isomorphism by exhibiting a bijection between bases for the spaces and showing that the representation matrices relative to these bases are the same. Bergeron (Be) used her methods to show that the Whitney homology of the signed partition lattice $\bar\Pi\sb{n}$, is isomorphic to sgn tensored with a multilinear component of the tensor product of the enveloping algebra of the fixed point sub-Lie algebra of the free Lie algebra on ($n\rbrack \times \{{\pm}1\}$ with the exterior algebra of the same sub-Lie algebra as a representation of the hyperoctahedral group.The Dowling lattice $Q\sb{n}(G)$ is a generalization of $\Pi\sb{n}$, and $\bar\Pi\sb{n}$, that depends on afinite group G. By extending Wachs' method, we show that the cohomology of $Q\sb{n}(G)$ is isomorphic to a sign representation tensored with a multilinear component of the enveloping algebra of the fixed point sub-Lie superalgebra of the free Lie superalgebra on ($n\rbrack \times G$ as a representation of the wreath product ${\cal S}\sb{n}(G)$ of ${\cal S}\sb{n}$ with G. Bergeron's result follows from our work by taking G to be the group with two elements. Taking G to be trivial gives Stanley's result (St1) that the action of ${\cal S}\sb{n}$ on $\Pi\sb{n+1}$ is the regular representation.We study the Whitney cohomology of $Q\sb{n}(G)$ and use our results to understand the cohomology of the complement of the Dowling hyperplane arrangement. (Abstract shortened by UMI.)



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