Multiplicity of the trivial representation in rank-selected homology of the partition lattice

2003/11/16 by Phil Hanlon, Patricia Hersh · 1 citation
Mathematics · #math.CO #math.RT #msc:05E25 #msc:03E10 #msc:20C30

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published as J. Algebra 266 (2003), no. 2, 521-538

arxiv created 2003/11/16 · arxiv updated 2009/12/01

Abstract

We study the multiplicity b_S(n) of the trivial representation in the symmetric group representations β_S on the (top) homology of the rank-selected partition lattice Π_n^S. We break the possible rank sets S into three cases: (1) 1\not∈ S, (2) S=1,..., i for i≥ 1 and (3) S=1,..., i,j₁,..., j_l for i,l≥ 1, j₁ > i+1. It was previously shown by Hanlon that b_S(n)=0 for S=1,..., i. We use a partitioning for Δ(Π_n)/S_n due to Hersh to confirm a conjecture of Sundaram that b_S(n)>0 for 1\not∈ S. On the other hand, we use the spectral sequence of a filtered complex to show b_S(n)=0 for S=1,..., i,j₁,..., j_l unless a certain type of chain of support S exists. The partitioning for Δ(Π_n)/S_n allows us then to show that a large class of rank sets S=1,..., i,j₁,..., j_l for which such a chain exists do satisfy b_S(n)>0. We also generalize the partitioning for Δ(Π_n)/S_n to Δ(Π_n)/S_λ; when λ= (n-1,1), this partitioning leads to a proof of a conjecture of Sundaram about S₁× S_n-1-representations on the homology of the partition lattice.

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