Schedule for: 22w5143 - Interactions between Hessenberg Varieties, Chromatic Functions, and LLT Polynomials

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

TBD

Hessenberg varieties are subvarieties of the flag variety which have rich connections to algebraic combinatorics and representation theory. Springer varieties, Peterson varieties, and the toric variety of the permutohedron are examples of Hessenberg varieties. This talk will serve as an introduction to the geometry of these varieties.

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 PDC front desk 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!

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

Groups and assigned rooms will be posted in TCPL 201.

We will discuss the history of Macdonald symmetric functions and the Shuffle conjecture to give inspiration for a more fundamental effort to develop symmetric function theory over the field $\mathbb Q(q,t)$. We will see how formulas for symmetric functions coming from infinite series have led us to solve a number of problems in this theory, as well as in quantum/affine Schubert calculus. We'll finish by showing some new combinatorial, representation theoretic, and geometric questions that naturally arise. Joint work with J. Blasiak, M. Haiman, A. Pun, and G. Seelinger

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

Lascoux, Leclerc, and Thibon defined a remarkable family of symmetric functions that are $q$-deformations of products of skew Schur functions. These LLT polynomials $G_\lambda(x;q)$ can be indexed by a tuple $\lambda$ of skew diagrams. When each skew diagram is a row, we define a weighted graph $\Pi(\lambda)$. We show that a horizontal-strip LLT polynomial is determined by this weighted graph. When $\Pi(\lambda)$ has no triangles, we establish a combinatorial Schur expansion of $G_\lambda(x;q)$. We also explore a connection to extended chromatic symmetric functions.

Discussion moderated by Per Alexandersson.

Virtual discussion on Zoom for graduate students and postdocs about professional development questions: related to finding academic jobs, work-life balance, and networking. Questions can be submitted before-hand (check your email for a survey).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

Classically, Schubert polynomials form a basis for the cohomology ring of the flag variety, and they can be defined using divided difference operators. In this talk, we generalize the divided difference operators to the larger cohomology rings of regular semisimple Hessenberg varieties (a family of Catalan-many sub-varieties of the flag variety). With these generalized operators, we can decompose the Hessenberg cohomology rings in a way which categorifies the modular relation of chromatic quasi-symmetric functions.

The e-positivity conjecture for the quasisymmetric Chromatic Symmetric Functions of Shareshian-Wachs has inspired many investigations toward understanding the underlying combinatorial structure. In this talk, I'll discuss some recent work on combinatorial identities involving q-rook numbers (joint with L. Colmenarejo and A. Morales), as well as proofs of e-positivity in some new cases (work of my student Shiyun Wang).

Discussion moderated by Laura Colmenarejo.

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

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

We show how sign-reversing involutions can be used to prove results about chromatic symmetric functions. In particular, one can make progress on the $({\bf3}+{\bf1})$-free Conjecture. This is joint work with Zachary Hamaker and Vincent Vatter.

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

There are at least four related objects that one can consider in general Lie type for a fixed Hessenberg space: (1) Poincare polynomials of nilpotent Hessenberg varieties; (2) Poincare polynomials of nilpotent "ideal" varieties; (3) the dot-action/monodromy representation of the Weyl group on the cohomology of the regular semisimple Hessenberg variety; (4) the associated LLT/Procesi representation. In the first part of this talk, I will survey these objects and how they are related to each other, highlighting the type A case. In the second part, I will present some tools for computing these objects, including the geometric modular law and a decomposition formula in the case of the Poincare polynomials. This is based on joint work with Martha Precup.

Discussion moderated by Martha Precup

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

To each permutation there is an associated element of the Kazhdan-Lusztig basis of the Hecke algebra of the symmetric group. When the permutation is codominant the corresponding characters recover the chromatic quasisymmetric function of the associated indifference graph. These characters also have geometric meaning, in the sense that they are encoded in the geometry of certain subvarieties of the flag variety, which were introduced almost 40 years ago by Lusztig. In this talk we plan to discuss parabolic analogues of such varieties. The decomposition theorem applied to the forgetful map from the Hessenberg variety to the projective space describes the cohomology of the Hessenberg variety as a sum of smaller pieces, involving parabolic varieties. We give a combinatorial description of the Frobenius character of each piece. As a consequence, we can prove that the coefficient of $e_{\lambda}$, where $\lambda$ is a partition of length $2$, in the $e$-expansion of the chromatic symmetric function of indifference graphs is non-negative. This is based on joint works with Alex Abreu.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.