University of Oregon Combinatorics Seminar

In this talk, we will begin with a quick review of group actions on posets, rank-selected homology of posets, representation stability, and a nice way of understanding the rank-selected homology of the Boolean lattice as a Specht module of ribbon shape.  This will allow us to prove a sharp representation stability bound for the rank-selected homology of the Boolean lattice.  We then describe a new ``ribbon'' basis for the rank-selected homology of any geometric lattice.  In the case of the partition lattice, we prove that this basis interacts with Young symmetrizers in a way that will allow us to prove that irreducible symmetric group representations with sufficiently large first row cannot appear in the ``essential part'' of the rank-selected homology of the partition lattice.  We thereby prove a sharp representation stability bound for the rank-selected homology of the partititon lattice that had  previously been conjectured by the speaker and Vic Reiner.   This is joint work with Sheila Sundaram.

The ordered set partitions $\Omega_n$ of $\{1,\ldots ,n\}$, with a unique minimal element adjoined, coincide with the face lattice of the permutohedron. This talk will present a study of the combinatorics and topology of two subposets of $\Omega_n$: the first where every block has size divisible by some fixed $d\ge 2$, and the second where every block size is congruent to $1$ module $d$. For the $d$-divisible case we derive formulas for the action of the symmetric group on the Whitney homology and the rank-selected homology groups. In particular we determine formulas for the multiplicity of the trivial representation. Our investigations lead to interesting enumerative invariants, giving rise to a new refinement of the factorial numbers. By contrast, the analogous numbers for the lattice of (unordered) set partitions are known to give two distinct refinements of the Euler numbers. This is joint work with Bruce Sagan.

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