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.

Abstract to be announced.

The extended weak order of a Coxeter group W is the collection of "biclosed sets" of roots in the root system of W, ordered by containment. For a finite Coxeter group, the extended weak order can be naturally identified with the weak order on the elements of the group, and for general Coxeter groups there is an embedding of weak order into extended weak order. Biclosed sets were studied by Matthew Dyer for their applications to Kazhdan-Lusztig theory, and are the subject of many open conjectures. For instance, it is conjectured that the extended weak order is a complete lattice, generalizing the fact that the weak order on a finite Coxeter group is a lattice. It is also conjectured that the extended weak order is a quotient of the lattice of torsion classes of an appropriate preprojective algebra. We will discuss progress on these and other conjectures, including applications to Kac-Moody groups and cluster algebras.

We introduce and study extremal ideals, which are a class of square-free monomial ideals that dominate and determine many algebraic invariants of powers of all square-free monomial ideals. In particular, we look at: free resolutions; integral closure; and symbolic powers. For extremal ideals and their powers, these invariants can be described combinatorially by discrete geometry and integer programming. This is joint work with Trung Chau, Sara Faridi, Thiago Holleben, Susan Morey, and Liana Şega.

In his 1901 thesis, Issai Schur discovered a finite dimensional algebra that connects the representation theory of the symmetric group and general linear group.  Motivated by a geometric description the Schur algebra, Tom Braden and I discovered a family of Schur-like algebras associated to any matroid.  Our original description of these algebras was frustratingly opaque.  In this talk I will define a new, extended version of these matroidal Schur algebras with a simple presentation by generators and relations.  I will highlight two special cases that can be defined purely in terms of more familiar objects, like vector spaces and set partitions, and are related to well-known representation theory and combinatorics.

Reconstructing simplicial complexes from partial information has been a problem of interest for decades. A triangulation of a d-dimensional sphere is obtained by gluing a collection of d-dimensional simplices along their faces, such that the resulting simplicial complex is homeomorphic to a topological d-sphere. For such a triangulation, if we only know the number of simplices in the triangulation and which pairs of simplices are glued along a (d-1)-face, can we uniquely recover the entire combinatorial structure of the triangulation? In this talk, I will show the answer is yes for shellable spheres, generalizing the result from the 1980s on reconstructing simple polytopes from their 1-skeleta.

A well-known conjecture of Stanley posits that the $h$-vector of any matroid is a pure O-sequence. Merino established this conjecture for cographic matroids using the theory of $G$-parking functions and chip-firing on graphs. In this talk, I will describe a generalization of Merino's approach to a broader class of matroids. The key construction is a "cycle system" for a matroid M, a collection of cycles satisfying an overlap condition that generalizes the structure of vertex edge-cuts in a graph. A cycle system defines a set of integer sequences called "coparking functions," which generalize $G$-parking functions. We show that the coparking functions are in explicit bijection with the bases of $M$ and form a pure multicomplex whose degree vector is the $h$-vector of $M$, establishing Stanley's conjecture for all matroids admitting a cycle system. This class includes, for instance, graphic matroids of all $K_{3,3}$-free graphs. I hope to make the entire talk accessible to first-year graduate students. This is joint work with Scott Corry, Anton Dochtermann, Solís McClain, and Lixing Yi.

Graded Ehrhart theory is a new $q$-analogue of Ehrhart theory introduced by Reiner and Rhoades. Roughly, it is the study of how a canonical, graded lattice point count of a polytope behaves under dilations. The grading in this count is constructed via the orbit harmonics method. In this talk, I will discuss the graded Ehrhart theory of unimodular zonotopes and its connections to matroid theory. In particular, I will explain why graded lattice point counts of unimodular zonotopes are $q$-integer evaluations of Tutte polynomials and how arrangement Schubert varieties can be used to study the harmonic algebras of unimodular zonotopes. This talk is based on joint work with Colin Crowley.

One rich problem in algebraic combinatorics is to deduce the Schur function expansion of a symmetric function whose expansion in terms of Gessel's fundamental quasisymmetric functions is known. Toward this goal, Maas-Gariépy introduced crystal skeletons—graphs obtained by contracting quasicrystal components of a crystal graph. In this talk I will discuss the rich combinatorial structure of crystal skeletons, including branching rules, symmetry under the Lusztig involution, and interesting subgraphs, including dual equivalence graphs and smaller crystals. Ongoing work is joint with Sarah Brauner, Sylvie Corteel, Sarah Mason, and Anne Schilling.

There are many ways to shuffle a deck of cards. In this talk, I will discuss a strange one introduced by Reiner-Saliola-Welker in 2014 called dyadic shuffling. Many mysteries about this shuffling process have endured, especially related to its eigenvalues. I will present recent progress understanding these eigenvalues using the representation theory of the symmetric group, thereby proving several conjectures from Nadia Lafrenière’s thesis and partially resolving a question by Reiner, Saliola, and Welker. Joint work with Patty Commins, Darij Grinberg, Trevor Karn, Nadia Lafrenière and Franco Saliola

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