Lectures on the Stanley-Stembridge Conjecture

First lecture: The 1992 paper “Some Conjectures for Immanants” by John Stembridge contains a conjecture about immanants of Jacobi-Trudi matrices that has generated a lot of interesting work. In the 1993 paper “On Immanants of Jacobi-Trudi Matrices and Permutations with Restricted Position”, Richard Stanley and Stembridge showed that Stembridge’s Immanant Conjecture follows from two further conjectures. One of these conjectures is equivalent to the claim that chromatic symmetric functions of certain graphs are non-negative linear combinations of elementary symmetric functions. This is what is usually called the Stanley-Stembridge Conjecture. (Chromatic symmetric functions were introduced in the 1995 paper “A symmetric function generalization of the chromatic polynomial of a graph” by Stanley.) A proof of the Stanley-Stembridge Conjecture appears in the preprint ”A proof of the Stanley-Stembridge Conjecture” by Tatsuyuki Hikita. The second of the two above-mentioned conjectures remains open. I plan to introduce immanants, review basic facts about symmetric functions, and explain how Stembridge’s Immanant Conjecture follows from the two conjectures of Stanley-Stembridge. If time permits, I will discuss some work on these two conjectures prior to Hikita’s proof of the first one.

Second lecture: In the 2016 paper “Chromatic quasisymmetric functions”, Michelle Wachs and I introduced a refinement of Stanley’s chromatic symmetric function. We conjectured a refined analogue of the first Stanley-Stembridge Conjecture, which remains open. We conjectured also a relation between certain refined chromatic symmetric functions and representations of symmetric groups cohomology groups of regular semismiple Hessenberg varieties. This conjecture was proved first in the 2018 paper “Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties”, by Patrick Brosnan and Tim Chow. There are subsequent additional proofs in the literature. I will introduce the objects of interest and give some discussion of the relations between them. If time permits, I will discuss possible future directions.

Background talk 1: The type-A Hecke algebra and symmetric functions

We review the Hecke algebra H = H(Sn)  of the symmetric group, and its natural and Kazhdan–Lusztig bases.  Its characters span a module of linear functionals called traces on H. Traces correspond by the Frobenius map to homogeneous degree-n symmetric functions, while the Kazhdan–Lusztig basis elements indexed by 312-avoiding permutations correspond to n-element unit interval orders. Building on these facts, we arrive at a theorem linking trace evaluations, colorings of indifference graphs, and actions of Sn  on the cohomology rings of Hessenberg varieties.

Background talk 2: The type-BC Hecke algebra and symmetric functions

We describe BC-analogs of the algebras described in the previous background lecture, beginning with the Hecke algebra H = H(Bn​​) of the hyperoctahedral group, its characters which are formed from pairs of type-A characters, and the trace space spanned by these.  Two distinct analogs of the Frobenius map now lead to two different correspondences between BC-traces and the module of BC-symmetric functions.

A proof of the Stanley-Stembridge conjecture

We give a probability theoretic formula for elementary symmetric function expansion of chromatic quasisymmetric function for any unit interval graph. As a corollary, we prove the Stanley-Stembridge conjecture on the e-positivity of chromatic symmetric function for any (3+1)-free graphs.

Braid varieties, R-polynomials, and rational Catalan number

Braid varieties can be viewed as generalizations of Bott–Samelson and Richardson varieties. Recent works by Casals–Gorsky–Gorsky–Simental, Galashin–Lam, and Galashin–Lam–Trinh–Williams have uncovered intriguing combinatorial structures within braid varieties. In particular, they showed that Kazhdan–Lusztig R-polynomials encode information about braid varieties, building on Deodhar's work.

In this talk, I will introduce the recent findings on braid varieties and present our work establishing a concrete connection with rational Catalan numbers. In addition, I will highlight several novel combinatorial relationships we have identified between objects associated with the study of R-polynomials and braid varieties.

This work is joint with Jihyeug Jang and Minho Song.

Motivic classes of isotropic Grassmannian degeneracy loci and orbit closures

The Motivic Hirzebruch classes are polynomials unifying three characteristic classes of singular varieties: the Chern–Schwartz–MacPherson (CSM) class, the K-theoretic Todd class, and the Cappell–Shaneson L-class. While some of these invariants have been studied for certain degeneracy loci of type A and general Schubert varieties and cells, research on degeneracy loci of other types is less extensive. In this talk, we will provide formulas for the motivic classes of isotropic Grassmannian degeneracy loci in type C. Additionally, we will explore the orthogonal orbit closures associated with vexillary involutions and their motivic classes. These orbit closures arise from the action of complex orthogonal groups on the complete flag variety and are in bijection with the set of involutions.

Geometry of regular semisimple Lusztig varieties

Lusztig varieties are subvarieties in a flag variety that can be seen as variants of both Schubert varieties and Hessenberg varieties. For example, they naturally admit the Tymoczko-type action of the Weyl group on their intersection cohomology, and their singularities can be resolved via Bott-Samelson-type resolutions.

In this talk, I will review basics of Lusztig varieties and present various geometric results, including vanishing theorems for the cohomology of line bundles, their relationship with Hessenberg varieties, and their diffeomorphism types. Some of these results arise from our study of the representations on their intersection cohomology and how we interpret them through a trace map of the Hecke algebra. If time permits, I will also discuss the trace map we introduce.

This talk is based on ongoing joint work with Patrick Brosnan and Jaehyun Hong.

On Białynicki-Birula decompositions of regular semisimple Hessenberg

varieties

Hessenberg varieties are subvarieties of the flag variety which provide a fruitful connection between geometry, representation theory of finite groups, and combinatorics. Indeed, the symmetric group acts on the cohomology of a regular semisimple Hessenberg variety, and studying this representation is related to the Stanley--Stembridge and Shareshian--Wachs conjectures on chromatic symmetric functions. In this talk, we study a basis of the equivariant cohomology of a regular semisimple Hessenberg variety obtained by the Białynicki-Birula decomposition. The maximal torus acts on each cell and by analyzing this torus action we present a combinatorial description of the support of a basis element. Moreover, we will consider how this provides a geometric construction of permutation module decomposition for the equivariant cohomology of permutohedral varieties and the second cohomology of a regular semisimple Hessenberg variety. This talk is based on joint work with Soojin Cho and Jaehyun Hong.

Lusztig’s q-weight multiplicities and Kirillov-Reshetikhin crystals

A Kostka polynomial is a polynomial in q that computes the Schur coefficients of Hall-Littlewood polynomials. These polynomials have nonnegative coefficients, which can be computed using the Lascoux-Schenberger charge formula. In this talk, we explore Lusztig’s q-weight multiplicities, a generalization of Kostka polynomials to other Lie types. We present a combinatorial formula for Lusztig’s q-weight multiplicities for type C weights and type B spin weights, utilizing the theory of Kirillov-Reshetikhin crystals, which extend the Lascoux-Schenberger charge formula. Time permitting, we will also discuss another generalization of weight multiplicities and/or their relationship with Catalan functions. This is joint work with Hyunjae Choi and Donghyun Kim.

Automorphisms of GKM graphs and regular semisimple Hessenberg varieties 

I will introduce the automorphism group of a GKM graph and show that its analysis provides some information on the automorphism group of a smooth projective GKM manifold. Especially, we observe this for regular semisimple Hessenberg varieties which are smooth subvarieties of the full flag variety. This talk is based on joint work with Donghoon Jang, Shintaro Kuroki, Takashi Sato, and Haozhi Zeng.

About the smoothness of Hessenberg Schubert varieties

A Hessenberg Schubert variety Ωw,h​​ is the closure of an opposite Schubert cell Ωw∘​​  inside a given Hessenberg variety Hess(S, h) of type A. In general, the Hessenberg Schubert variety Ωw,h​​ is different from the intersection Ωw∩Hess(S, h)​​, and it is an irreducible component of Ωw∩Hess(S, h)​​. From the natural torus action on the intersection Ωw∩Hess(S, h), we get a graph Γw,h​​. In the first part of this talk, we show that Ωw∩Hess(S, h) is smooth if and only if Γw,h is regular. In the second part, we prove that the regularity of Γw,h is completely characterized by the avoidance of the patterns for the permutation w we found. The first part of this talk is based on joint work with Jaehyun Hong and Eunjeong Lee and the second part is jointly with Soojin Cho and JiSun Huh.

Components of Springer fibers equal to Richardson varieties

Springer fibers are subvarieties of the flag variety parameterized by partitions. They are central objects of study in geometric representation theory. Given a partition λ, one of the key conclusions of Springer theory is that the top dimensional components of the associated Springer fiber are indexed by standard tableaux of shape λ. In this talk, we study when components of a Springer fiber are also Richardson varieties. We give a precise characterization in terms of the associated tableaux. This yields an answer to a question raised by Springer regarding the cohomology class of each such component. This talk is based on joint work with Steven Karp.

Partitioned permutohedra and representations of Weyl groups on cohomology of permutohedral varieties

Given a Weyl group W acting on Euclidean space E by its natural reflection representation, the associated W-permutohedron P(W) is obtained by choosing a point x in E such that the stabilizer of x in W is trivial and taking the convex hull of the W-orbit of X. If x is a lattice point, one obtains a smooth projective toric variety X(W) with W-action from the fan dual to P(W). The associated representation of W on the cohomology of X(W) has been studied by various authors. I will discuss joint work with Tatsuya Horiguchi, Mikiya Masuda, and Jongbaek Song and further joint work with Horiguchi, Masuda, Takeshi Sato, and Song in which we relate this representation to polytopes obtained by cutting permutohedra with hyperplanes orthogonal to root vectors and to cohomology of certain regular Hessenberg varieties.

Evaluations of type-BC Hecke algebra traces at Kazhdan–Lusztig basis elements and a type-BC analog of graph coloring

We use pattern avoidance to characterize elements w of the hyperoctahedral group Bn for which type-B and type-C Schubert varieties indexed by w are simultaneously smooth. We show that these elements are counted by (type-A) Catalan numbers. We also present type-BC analogs of unit interval orders and indifference graphs and color these to combinatorially interpret q=1 specializations of trace evaluations at Kazhdan–Lusztig basis elements of ℤ​[Bn] indexed by the "simultaneously smooth" elements of Bn. This result and Macdonald's analog of the Frobenius map lead to a type-BC analog of Stanley's (q=1) chromatic symmetric function, and to several open questions.

A signed e-expansion of the chromatic quasisymmetric function

We prove a new signed elementary symmetric function expansion of the chromatic quasisymmetric function of any natural unit interval graph. We then use a sign-reversing involution to prove a new combinatorial formula for K-chains, which are graphs formed by joining cliques at single vertices. This formula immediately implies e-positivity and e-unimodality for K-chains. We also prove a version of our signed e-expansion for arbitrary graphs.