Multiplicity of the trivial representation in rank-selected homology of the partition lattice
| dc.creator | Hanlon, Phil | |
| dc.creator | Hersh, Patricia | |
| dc.date | 2003-11-16 | |
| dc.date.accessioned | 2026-07-07T05:02:57Z | |
| dc.date.available | 2026-07-07T05:02:57Z | |
| dc.description | 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) $1\not\in S$, (2) $S=1,..., i$ for $i\ge 1$ and (3) $S=1,..., i,j_1,..., j_l$ for $i,l\ge 1$, $j_1 > i+1$. 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 $1\not\in S$. On the other hand, we use the spectral sequence of a filtered complex to show $b_S(n)=0$ for $S=1,..., i,j_1,..., j_l$ 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 $S=1,..., i,j_1,..., j_l$ 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 $S_1\times S_{n-1}$-representations on the homology of the partition lattice. | |
| dc.identifier | https://arxiv.org/abs/math/0311264 | |
| dc.identifier | http://arxiv.org/abs/math/0311264 | |
| dc.identifier | J. Algebra 266 (2003), no. 2, 521-538 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/69215 | |
| dc.subject | Combinatorics | |
| dc.subject | Representation Theory | |
| dc.subject | 05E25, 03E10, 20C30 | |
| dc.title | Multiplicity of the trivial representation in rank-selected homology of the partition lattice | |
| dc.type | text |