A Homotopy Equivalence for Partitions related to Liftings of $S_{n-1}$-modules to $S_{n}$

Sheila Sundaram

December 2,
refreshments at 3:45pm


We give a homotopy equivalence to explain an $S_{n-1}$-module isomorphism which occurs frequently in the homology of subposets of the partition lattice $\Pi_n.$ The isomorphism in question is a necessary, but not sufficient, condition for the existence of a lifting to the symmetric group $S_n,$ of the $S_{n-1}$-module involved. It has also been observed in certain deformations of the free Lie algebra. The topological explanation allows us to generate further examples in subposets of $\Pi_n.$ We show that, in these two contexts, it is often accompanied by an $S_{n}$-action on the $S_{n-1}$-module occurring in the isomorphism. A well-known example is the Whitehouse lifting of the homology of $\Pi_{n-1}.$

Speaker's Contact Info: sheila(at-sign)claude.math.wesleyan.edu

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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

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