Spring 2024

- The Combinatorics Seminar meets on
**Fridays, 3-4pm**, in**Snow 306**(including talks on Zoom). There will be cookies! - Organizers: Jeremy Martin, Mark Denker, Reuven Hodges, and Marge Bayer.
- Good general seminar-attending advice, especially for graduate students: The "Three Things" Exercise for getting things out of talks by Ravi Vakil
- Also consider attending the Graduate Online Combinatorics Colloquium.

**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)**

TBA

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

Last updated Fri 4/19/24