KU Combinatorics Seminar
Spring 2024

Friday 1/19
Organizational meeting

Friday 1/26
Jeremy Martin
Chromatic symmetric functions I: Basics and Crew's conjecture

Friday 2/2
Jeremy Martin
Chromatic symmetric functions II: Equivalence of the subtree and half-generalized degree polynomials

Abstract: Stanley introduced chromatic symmetric functions of graphs in a seminal 1995 paper and asked whether it is possible for two non-isomorphc trees to have equal CSFs. This problem remains open today and is considered very hard. I will discuss my recent work on deriving other graph invariants from the CSF, including Logan Crew's conjecture that the CSF of a tree determines its generalized degree sequence. (Don't worry, I will say what all these things mean and will not assume any prior knowledge about the CSF!) This is joint work with José Aliste-Prieto, Jennifer Wagner, and José Zamora.

Friday 2/9
Marge Bayer
Using discrete Morse theory for matching complexes

Friday 2/16
Reuven Hodges
Approximate Counting in Algebraic Combinatorics

Friday 2/23
No seminar

Friday 3/1
Jeremy Martin
The kernel of the chromatic symmetric function

Abstract: Stanley's problem about chromatic symmetric functions of trees can be reformulated linear-algebraically: does the kernel of the CSF, regarded as a linear transformation \(X\) from trees to symmetric functions, contain the difference of two trees in its kernel? The reformulation suggests trying to understand the kernel of \(X\). For general graphs, the map \(X\) is onto, and its kernel is generated by the modular relations of Guay-Paquet and Orellana-Scott, and for forests it is generated by the deletion/near-contraction relations of Aliste-Prieto, de Mier, Orellana and Zamora. For trees, \(X\) is not onto. We find the dimension of the cokernel of \(X\) (it's something nice) and have a partial description of a generating set for the kernel. This is joint work with José Aliste-Prieto, Jennifer Wagner, and José Zamora.

Friday 3/8
Shiliang Gao (U. Illinois, Urbana-Champaign)
Degrees of the stretched Kostka quasi-polynomials
Zoom coordinates: Meeting ID 995 1980 5587 (passcode 2024)

Abstract: The Kostka coefficient \(K_{\lambda,\mu}\) is the dimension of the weight space \(V^\lambda(\mu)\) in the irreducible representation \(V^\lambda\) of a complex semisimple Lie algebra. We provide a type-uniform formula for the degrees of the stretched Kostka quasi-polynomials \(K_{\lambda,\mu}(N):= K_{N\lambda,N\mu}\) in all classical types, improving and extending a previous result by McAllister in type A. Our proof relies on a combinatorial model for the weight multiplicity by Berenstein and Zelevinsky. This is based on joint work with Yibo Gao.

Friday 3/15
No seminar - Spring Break

Friday 3/22
Jianping Pan (North Carolina State U.) Pattern-avoiding polytopes and Cambrian lattices
Zoom coordinates: Meeting ID 992 9471 4793 (passcode 2024)

Abstract: In 2017, Davis and Sagan found that a pattern-avoiding Birkhoff subpolytope and an order polytope have the same normalized volume. They ask whether the two polytopes are unimodularly equivalent. We give an affirmative answer to a generalization of this question. For each Coxeter element \(c\) in the symmetric group, we define a pattern-avoiding Birkhoff subpolytope, and an order polytope of the heap poset of the \(c\)-sorting word of the longest permutation. We show the two polytopes are unimodularly equivalent. As a consequence, we show the normalized volume of the pattern-avoiding Birkhoff subpolytope is equal to the number of the longest chains in a corresponding Cambrian lattice. In particular, when \(c = s_1s_2\dots s_{n-1}\), this resolves the question by Davis and Sagan. This talk is based on ongoing joint work with Esther Banaian, Sunita Chepuri and Emily Gunawan.

Friday 3/29
No seminar

Friday 4/5

Friday 4/12
Aaron Ortiz
Defective Parking Functions and the Symmetric Group

Abstract: We recall that a parking function is a vector of parking preferences (spots) for cars that want to park in a one-way street. We can extend this definition to (\(m,n\)) parking functions, which has \(m\) cars parking in \(n\) spots (\(m < n\)). Although a lot of research has been done on these parking functions, we do not have a strong understanding of their "defective" counterparts, or cases where not all cars park. In this talk we will introduce defective parking functions and show results when we act under the symmetric group \(S_{n}\). We will then attempt to enumerate these objects and find a bijection with standard Young Tableaux.

Friday 4/19
No seminar

Friday 4/26
Reuven Hodges
Schubert varieties and sphericity

Abstract: The study of group orbits and their closures inside flag varieties has a long and storied history. The development of this topic touches on, and interweaves, fundamental ideas from Lie theory, algebraic geometry, representation theory, and algebraic combinatorics. The foundational objects of study in this area are the orbits of Borel subgroups and their closures, the Schubert varieties. In this talk we will study when a Schubert variety is a spherical variety under the action of certain reductive groups. Spherical varieties generalize several important classes of algebraic varieties including toric varieties, projective rational homogeneous spaces and symmetric varieties. I will discuss a root-system uniform, combinatorial classification of Levi-spherical Schubert varieties for any generalized flag variety \(G/B\) of finite Lie type. This will be applied to the study of multiplicity-free decompositions of a Demazure character.

Friday 5/3 (Stop Day)

For seminars from previous semesters, please see the KU Combinatorics Group page.

Last updated Fri 4/19/24