Schedule for: 18w5130 - Hessenberg Varieties in Combinatorics, Geometry and Representation Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this talk, I will give a brief survey of recent developments on Hessenberg varieties. The goal of this talk is to share the idea that Hessenberg varieties can be studied from various perspectives.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Motivated by a 1993 conjecture of Stanley and Stembridge, Shareshian and Wachs conjectured that the characteristic map takes the dot action of the symmetric group on the cohomology of a regular semisimple Hessenberg variety to $\omega XG(t)$, where XG(t) is the chromatic quasisymmetric function of the incomparability graph G of the corresponding natural unit interval order, and $\omega$ is the usual involution on symmetric functions. Our main result is a proof of the Shareshian-Wachs conjecture. The proof makes essential use of both geometric arguments and combinatorial arguments; the talk will focus on the combinatorics, but we also briefly describe the geometric ingredients. This is joint work with Patrick Brosnan.

The Brosnan--Chow--Shareshian--Wachs theorem gives a deep connection between the cohomology of regular semisimple Hessenberg varieties on one hand, and the $q$-chromatic symmetric functions of unit interval orders ($q$-CSFs) from combinatorics. Through this connection, a very interesting interplay of combinatorial and geometric results and techniques becomes possible, leading to progress on the long-standing $e$-positivity conjecture of Stanley--Stembridge and its generalizations. This talk is about recent combinatorial results on $q$-chromatic symmetric functions: (1) a complete description of the linear relations between $q$-CSFs; (2) a description of a large class of such linear relations which express a $q$-CSF as a positive combinations of other $q$-CSFs; and (3) a slight generalization of the $e$-positivity conjecture suggested by these results.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk will begin with a survey of results describing the Betti numbers of Hessenberg varieties. There are many geometric and combinatorial applications of these formulas. In the second half of the talk, I will report on recent joint work with M. Harada in which we prove an inductive formula for the Betti numbers of certain regular Hessenberg varieties called abelian Hessenberg varieties. Using a theorem of Brosnan and Chow, this formula yields a proof of the Stanley-Stembridge conjecture for this special case.

Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. The local geometric structure of Hessenberg varieties can often be studied using cell decompositions, group actions, patch ideals, and the combinatorics of the symmetric group. In this talk, we will give a survey of results regarding the singularities of Hessenberg varieties. This is based on joint works with Alex Yong and Martha Precup.

I will talk about an explicit ring presentation of the cohomology ring of the regular semisimple Hessenberg variety in the flag variety $Fl(\mathbb{C}^n)$ with Hessenberg function $h=(h(1),n,\dots,n)$ where $h(1)$ is an arbitrary integer between $2$ and $n$. This is joint work with Hiraku Abe and Tatsuya Horiguchi.

A regular nilpotent Hessenberg variety Hess(N,h) is a subvariety of a flag variety determined by a Hessenberg function h. I will explain a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials $f_{i,j}$ were introduced by Abe-Harada-Horiguchi-Masuda. In this talk, I will show that every polynomial $f_{i,j}$ is an alternating sum of certain Schubert polynomials $\mathfrak{S}_{w_k^{(i,j)}}$ (k=1,2,...,i-j). Moreover, we can interpret the permutations $w_k^{(i,j)}$ from a geometric viewpoint under the circumstances of having a codimension one regular nilpotent Hessenberg variety Hess(N,h′) in the original regular nilpotent Hessenberg variety Hess(N,h) where the Hessenberg function h' is obtained from the original Hessenberg function h and (i,j).

This is an expository talk on "regular $(\mathbb{G}_a,\mathbb{G}_m)$-actions" on projective varieties (over the complexes). A $(\mathbb{G}_a,\mathbb{G}_m)$-action is the same thing as an action of a Borel subgroup of $\mathrm{SL}(2)$. We call a $(\mathbb{G}_a,\mathbb{G}_m)$-action regular when the maximal unipotent subgroup $\mathbb{G}_a$ has a unique fixed point. There are lots of examples of varieties with a regular $(\mathbb{G}_a,\mathbb{G}_m)$-action such as Schubert varieties. Of particular interest are regular nilpotent Hessenberg varieties. The main point is that the cohomology algebra of a smooth projective variety X with $(\mathbb{G}_a,\mathbb{G}_m)$-action is isomorphic with the fixed point scheme of the $\mathbb{G}_a$ action, and this is also true of many singular $(\mathbb{G}_a,\mathbb{G}_m)$-stable subvarieties such as the two classes mentioned above. The $\mathbb{G}_m$-equivariant cohomology admits a nice description which I will also explain.

I will discuss how chromatic quasisymmetric functions can be viewed as modifications of LLT polynomials, certain building blocks from the combinatorial theory of Macdonald polynomials. I will also explain how Macdonald polynomials enter the picture, and what I hope can be gained from these relationships. Based on joint work with Jim Haglund.

LLT polynomials depend on a tuple of skew shapes and a parameter q. They occur in formulas for Macdonald polynomials, where the skew shapes are ribbons, and in character formulas for Sn modules such as the diagonal coinvariant ring. LLT polynomials are known to be Schur positive, but combinaotrial formulas for the Schur coefficients are known only for special cases. Several researchers including Alexandersson, Bergeron, and Garsia have noticed independently that when q is replaced by q+1, LLT polynomials often become e-positive, and it has been conjectured that for tuples of vertical strips, this is always the case. In this talk we discuss some partial results on the e-expansion of LLT polynomials corresponding to Dyck paths which rise in the character of the diagonal coinvariant ring; in particular we discuss a conjecture for the coefficient of e_n in their e-expansion.

When G is the indifference graph of a natural unit interval order, the Shareshian-Wachs chromatic quasisymmetric function is symmetric, and serves as a generating function for Hecke algebra trace evaluations. In particular, every coefficient appearing in every common symmetric function expansion is the evaluation of a common Hecke algebra trace at a modified Kazhdan-Lusztig basis element indexed by a 3412-, 4231- avoiding permutation. We will discuss various combinatorial interpretations of these, and related ideas in generating functions.

In this talk, I will show that there is an interesting connection between cohomology rings of regular nilpotent Hessenberg varieties and logarithmic derivation modules of hyperplane arrangements. In particular, I will explain that this connection gives an affirmative answer to a conjecture of Sommers and Tymoczko on Poincare polynomials of regular nilpotent Hessenberg varieties. If time allows, I also explain how this connection can be applied to compute relations of cohomology rings of regular nilpotent Hessenberg varieties explicitly.

A hyperplane arrangement is a finite set of linear hyperplanes in the complex vector space. From each hyperplane arrangement and arbitrary degree two homogeneous polynomial, we can define a finite dimensional algebra, called the Solomon-Terao algebra. If we choose an arrangement the ideal arrangement corresponding to the reflecting hyperplanes belonging to the lower ideal in the root poset (which corresponds to a Hessenberg space), and the lowest degree basic invariant polynomial which is always degree two, then their Solomon-Terao algebra coincides with the cohomology ring of the regular nilpotent Hessenberg variety, as shown in the talk by Murai. We discuss the origin and properties of the Solomon-Terao algebra coming from so called the mysterious Solomon-Terao formula and polynomials by Luis Solomon and Hiroaki Terao in 1986. We discuss also the possibility whether it could be a cohomology ring of some other varieties, e.g., Schubert varieties. This talk is based on the joint work with T. Maeno, S. Murai and Y. Numata.

Kostant's invariant-theoretic version of the Toda lattice has given rise to a powerful synergy of ideas from symplectic geometry, algebraic geometry, and representation theory. An example is the appearance of this Kostant-Toda lattice in calculations related to the quantum cohomology of the flag variety. It is in this setting that one compactifies the leaves of the Kostant-Toda lattice, thereby obtaining a certain class of Hessenberg varieties. One might then expect that the Kostant-Toda lattice can be defined on (the total space of) a family of Hessenberg varieties. I will show this to be the case, emphasizing the roles played by Slodowy slices and Mishchenko-Fomenko theory. This represents joint work with Hiraku Abe.

The Springer theory for reductive algebraic groups plays an important role in representation theory. It relates nilpotent orbits in the Lie algebra to irreducible representations of the Weyl group. We develop a Springer theory in the case of symmetric spaces using Fourier transform, which relates nilpotent orbits in this setting to irreducible representations of Hecke algebras with parameter -1. As an application we explain a general strategy for computing the cohomology of Hessenberg varieties. Examples of such varieties include classical objects in algebraic geometry: Jacobians, moduli spaces of vector bundles on curves with extra structure, and Fano varieties of linear subspaces in the intersection of two quadrics, etc. The talk is based on joint work with Tsao-Hsien Chen and Kari Vilonen.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.