Lecture 1. Chromatic polynomials and permutohedral varieties

In an attempt to prove the four-color conjecture, G. Birkhoff introduced the chromatic polynomial in 1912. It is easy to calculate the chromatic polynomial of a graph by the deletion-contraction relation. Playing with examples, one can find interesting properties such as the unimodality and the log-concavity of the coefficients. In this colloquium style talk, I will outline June Huh's proofs of these properties by the geometry of permutohedral varieties.

Lecture 2. Geometry of Hessenberg varieties

The notion of the chromatic polynomial was refined by R. Stanley in 1995 to that of chromatic symmetric functions. Stanley observed that the chromatic symmetric function of the path graph coincides with the dual of the character of the cohomology of the permutohedral variety. As the permutohedral variety is a Hessenberg variety, this observation led Stanley and others to conjecture that the chromatic symmetric functions of indifference graphs should be related to the cohomology of Hessenberg varieties. In this talk, I will introduce Hessenberg varieties, their cohomology, and the Tymoczko action by the Goresky–Kottwitz–MacPherson theory.

Lecture 3. Shareshian–Wach conjecture and generalizations

The Shareshian–Wach conjecture in 2012 tells us that the chromatic quasi-symmetric function of an indifference graph is exactly the dual of the character of the cohomology of the corresponding Hessenberg variety. This conjecture was proved by Brosan–Chow and Guay-Paquet independently. In this talk, I will discuss the work of Donggun Lee who introduced generalized Hessenberg varieties and provided an elementary proof of the Shareshian-Wach conjecture by birational geometry. If time permits, I will further discuss Lusztig varieties and their birational geometry.

Lecture 1. Chromatic symmetric functions and Hessenberg varieties.

R. Stanley, motivated by joint work with J. Stembridge, introduced a symmetric function generalization of the chromatic polynomial, which was further generalized by J. Shareshian and M. Wachs. In this talk, after restricting to indifference graphs, we will describe a certain relation between such combinatorial objects and others coming from algebra and geometry.

Lecture 2. Characters of Iwahori–Hecke algebras and Lusztig varieties.

For each indifference graph, there is an associated Hessenberg variety, whose cohomology recovers the chromatic symmetric function of the graph. Since indifference graphs are in bijection with codominant permutations of the symmetric group, one may wonder if there is a natural generalization of the above result which works for any permutation. In this talk, we will present Lusztig varieties and describe how to realize such correspondence through characters of the Hecke algebra of the symmetric group.

Lecture 3. Example of computations of characters of Hecke algebras and further directions

In this talk, we wish to describe through an example how the Frobenius character of a natural action of the symmetric group on the cohomology of a Lusztig variety is precisely the symmetric function given by the characters of a corresponding Kazhdan–Lusztig basis element of the Hecke algebra. We will finally talk about a further generalization of Lusztig varieties to the parabolic setting and describe some open questions.

Lecture 1. Unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties

Hessenberg varieties are subvarieties of a flag variety, and they are defined by two data; one is an element of the corresponding Lie algebra and the other is some ``good’’ subset of the positive root system, which is called a lower ideal. The geometry and topology of Hessenberg varieties are closely related to combinatorics of lower ideals. For example, the symmetric group acts on the cohomology ring of a regular semisimple Hessenberg variety of type A, and then the involution of its Frobenius series coincides with the chromatic symmetric function of some graph defined by a lower ideal. This surprising result was shown by Brosnan and Chow and by Guay-Paquet independently.
In this lecture, I introduce some manifold called the twins of regular semisimple Hessenberg varieties defined by Ayzenberg and Buchstaber, and I show the Frobenius series of a twin coincides with the LLT polynomial of the above graph. Surprisingly, a regular semisimple Hessenberg variety is to its twin what the chromatic symmetric function is to the LLT polynomial. The contents of this lecture are based on the joint work with Mikiya Masuda.

Lecture 2. Hessenberg varieties and Hyperplane arrangements

A regular nilpotent Hessenberg variety has a paving obtained by the intersection of Bruhat cells of a flag variety and itself. The dimension of each cell is visualized by the hyperplane arrangement corresponding to the lower ideal, and then the Poincaré series of a regular nilpotent Hessenberg variety is known. By the way, Borel showed that the cohomology ring of a flag variety is the coinvariant ring of its Weyl group. This result was interpreted in terms of hyperplane arrangements corresponding to its root system by Saito.

In this lecture, I introduce Saito’s interpretation, and it also describes the cohomology ring of a regular nilpotent Hessenberg variety. In this point of view, the class of regular nilpotent Hessenberg varieties is one of the most important classes among subvarieties of a flag variety. The contents of this lecture are based on the joint work with Takuro Abe, Tatsuya Horiguchi, Mikiya Masuda, and Satoshi Murai.

Lecture 1. Crash course on symmetric functions: from scratch

We start with defining what symmetric functions are and introducing several symmetric function bases. By observing interesting features of Schur functions and to extend them, we add another q-parameter to define Hall–Littlewood polynomials, and (q,t)-parameters to define Macdonald polynomials. We also observe some related combinatorial formulas.

Lecture 2. Toward chromatic symmetric functions and unicellular LLT polynomials

We define LLT polynomials in original form and obtain Haiman's variation model by using the quotient map. Haglund–Haiman–Loehr expanded the Macdonald polynomials positively in terms of the LLT polynomials, thus the Schur positivity of two polynomials are closely related. We define unicellular LLT polynomials as a special case of LLT polynomials and introduce the Dyck path model to compute them. Also, we introduce chromatic quasisymmetric functions, and in certain graph classes, they can be encoded in the Dyck path model as well. We will see the explicit relation of these two polynomials encoded in the same Dyck path.

Face posets of regular semisimple Hessenberg varieties and graphicahedra

(based on joint work with Victor Buchstaber)

Assume that a torus, either compact or algebraical, acts on a smooth manifold with isolated fixed points. With any such action, one can associate a combinatorial structure called the face poset. For example, the face poset of a (smooth complete) toric variety is the underlying simplicial sphere of its fan; it is locally modeled by a Boolean lattice, the face poset of a simplex. For actions of higher complexity, the posets of faces are locally modeled by geometric lattices of matroids.
It is instructive to understand the face posets in particular examples. In our work, we concentrated on the manifolds of isospectral Hermitian matrices of the given sparsity shape. For Hessenberg matrices, these manifolds are very similar to the corresponding regular semisimple Hessenberg varieties, in particular, they have isomorphic face posets. We described face posets of these manifolds and related them to graphicahedra, - certain posets known in combinatorial theory. The results can be extended to regular semisimple Hessenberg varieties in partial flags.
If time permits, I show what makes Hessenberg varieties special, in terms of their face posets, among all manifolds of isospectral matrices.

Lollipop chromatic quasi-symmetric functions

Shareshian and Wachs introduced the chromatic quasi-symmetric function as a refinement of Stanley’s chromatic symmetric function by considering an extra parameter. Well known e-positivity conjecture on the chromatic symmetric functions is also refined to the chromatic quasi-symmetric functions. Lollipop graphs form an interesting class that contains both complete graphs and path graphs, and in this talk we present some combinatorial results on lollipop chromatic quasi-symmetric functions mainly in the relation to the e-positivity.

First, by using the fact that the chromatic quasi-symmetric functions and the unicellular LLT polynomials are related via plethystic substitution, we apply some known linear relations on the unicellular LLT polynomials to the chromatic quasi-symmetric functions to obtain an explicit formula of a lollipop chromatic quasi-symmetric function. We then give a combinatorial model for the e-coefficients of a lollipop chromatic quasi-symmetric function in terms of acyclic orientations of the corresponding graph.

Also, we consider the GKM graph of the Hessenberg varieties determined by the corresponding Hessenberg functions of the lollipop graphs. We give a criterion for the regularity of the induced subgraph of the GKM graph determined by a permutation w, which provides a condition for the (rationally) smoothness of the corresponding Hessenberg Schubert variety.

This talk is based on the joint work with Sun-Young Nam and Meesue Yoo, and the work with Soojin Cho and Seonjeong Park.

Chromatic quasisymmetric functions and noncommutative P-symmetric functions

Chromatic quasisymmetric functions of natural unit interval orders P lie on the intersection of combinatorics, algebra and geometry. In this talk, I will introduce a combinatorial operation, called a local flip, on proper colorings of the incomparability graph of P. This operation defines an equivalence relation on the proper colorings, which refines the ascent statistic introduced by Shareshian and Wachs. I will also define an analogue of noncommutative symmetric functions introduced by Fomin and Greene, with respect to P. After establishing a duality between the chromatic quasisymmetric function of P and these noncommutative symmetric functions, I will explain how to compute coefficients of the chromatic quasisymmetric function in the expansions in terms of several symmetric function bases. If time permits, I will show that Harada–Precups’s conjecture naturally follows from the noncommutative symmetric functions.

Chromatic quasisymmetric functions and linked rook placements

R. Stanley introduced the chromatic symmetric functions of a graph G. This definition was later refined by J. Shareshian and M. Wachs, where a parameter q is introduced in the definition of chromatic quasisymmetric functions. In this talk, we discuss two separate results and their connection: (1) a Hall-Littlewood expansion of the chromatic quasisymmetric functions (2) e-expansion of the chromatic quasisymmetric functions when a partition λ​ indexing the elementary symmetric function is a hook shape or two rows. To explain both results we introduce "linked" rook placements, where each column and row of a board contains at most one rook and some of rooks are linked. (1) is based on joint work with Meesue Yoo and (2) is with Jeong Hyun Sung.