Schedule for: 19w5131 - Representation Theory Connections to (q,t)-Combinatorics

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.

I will first describe explicit (GL_k x S_n)-modules, in k sets on n variables, whose graded Frobenius correspond (conjecturally) to the symmetric functions that occur in the rectangular shuffle theorem. I will then discuss many properties of the associated character, and show how the k-variate version (in fact one can assume that k goes to infinity) sheds new light and simplifies many aspect of the problems that have been considered in the last 25 years in relation to spaces of diagonal harmonic polynomials. I will also show how some of the properties alluded to are entirely natural in view of the natural ties that the subject seems to have with the study of (m,n)-links on the torus. I will also explain how to directly relate this to the Delta-conjecture, opening a clear path to its generalization to the rectangular context.

In the $1980$ paper ``Une famille de Polynomes ayant Plusieurs Propriétés Enumeratives", Kreweras gives a bijection that shows that the polynomials $P_n(t)$ that enumerate $n$-labelled rooted trees by number of inversions also $t$-enumerates $n$-Parking Functions by the area statistic. In the $1993$ paper ``A Remarkable $q,t$-Catalan sequence and Lagrange Inversion" with Haiman we relate the Frobenius Characteristic of Diagonal Harmonics to Parking Functions. A recent search in the Encyclopedia of integer sequences connects these two papers in a surprising manner leading to a variety of beautiful conjectures. In this talk the focus will be on what we have proved.

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.

The phrase superpolynomial is often taken to mean ``graded dimension of Khovanov-Rozansky link homology’’. In this talk I will discuss a combinatorial technique for computing Khovanov-Rozansky homology, which in the past couple years has led to the computation of superpolynomials for torus links (and also new recursions for the rational q,t-Catalan), through various works of myself, Ben Elias, and Anton Mellit. My goal will be to communicate all the main ideas, and outline how the torus link computation is carried out.

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 the Vistas Dining Room, the top floor of the Sally Borden Building.

I will first describe explicit (GL_k x S_n)-modules, in k sets on n variables, whose graded Frobenius correspond (conjecturally) to the symmetric functions that occur in the rectangular shuffle theorem. I will then discuss many properties of the associated character, and show how the k-variate version (in fact one can assume that k goes to infinity) sheds new light and simplifies many aspect of the problems that have been considered in the last 25 years in relation to spaces of diagonal harmonic polynomials. I will also show how some of the properties alluded to are entirely natural in view of the natural ties that the subject seems to have with the study of (m,n)-links on the torus. I will also explain how to directly relate this to the Delta-conjecture, opening a clear path to its generalization to the rectangular context.

Recently James Martin introduced multiline queues, and used them to give a combinatorial formula for the stationary distribution of the multispecies asymmetric simple exclusion exclusion process (ASEP) on a circle. The ASEP is a model of particles hopping on a one-dimensional lattice, which was introduced around 1970, and has been extensively studied in statistical mechanics, probability, and combinatorics. In this article we give an independent proof of Martin's result, and we show that by introducing additional statistics on multiline queues, we can use them to give a new combinatorial formula for both the symmetric Macdonald polynomials P_{lambda}(x; q, t), and the nonsymmetric Macdonald polynomials E_{lambda}(x; q, t), where lambda is a partition. This formula is rather different from others that have appeared in the literature, such as the Haglund-Haiman-Loehr formula. Our proof uses results of Cantini-de Gier-Wheeler, who recently linked the multispecies ASEP on a circle to Macdonald polynomials. This is joint work with Sylvie Corteel and Olya Mandelshtam.

Nonsymmetric Macdonald polynomials are a polynomial generalization of their symmetric counterparts that exist for all root systems. The combinatorial formula for type A, due to Haglund, Haiman and Loehr, resembles the symmetric formula by the same authors, but with rational functions that complicate the combinatorics. By specializing one parameter to 0, the combinatorics simplifies and we are able to give an explicit formula for the expansion into Demazure characters, a basis for the polynomial ring that contains and generalizes the Schur basis for symmetric polynomials. The formula comes via an explicit Demazure crystal structure on semistandard key tabloids, constructed jointly with Nicolle Gonzalez. By taking stable limits, we return to the symmetric setting and obtain a new formula for the Schur expansion of Hall-Littlewood polynomials that uses a simple major index statistic computed from highest weights of the crystal.

An ordered tuple of 1-dimensional subspaces $(L_1, \dots, L_n)$ of a fixed vector space $V$ is a {\em spanning line configuration} if $L_1 + \cdots + L_n = V$. We discuss the combinatorics of spanning line configurations, describing enumerative results when $V$ is a vector space over the finite field $\mathbb{F}_q$, and presenting the cohomology ring of the moduli space of spanning line configurations when $V$ is a vector space over $\mathbb{C}$. We present some ideas about how to extend these results to tuples $(W_1, \dots, W_n)$ of potentially higher-dimensional subspaces $W_i$ of $V$. Joint with Brendan Pawlowski and Andy Wilson.

The Kostka--Foulkes polynomials, $K_{\lambda, \mu}(q)$, arise in many contexts in combinatorics and representation theory. Lascoux and Schutzenberger showed that they are generating functions over $\text{SSYT}(\lambda, \mu)$ (the set of semistandard Young tableaux of shape $\lambda$ and content $\mu$) with respect to a statistic called charge. In particular, they evaluate to the familiar Kostka number at $q = 1$. One might hope that the evaluation at $q = -1$ counts the number of fixed points of a natural involution on $\text{SSYT}(\lambda, \mu)$. When the content $\mu$ is palindromic (for instance, in the case of standard tableau) it follows from work of Stembridge and Lascoux--Leclerc--Thibon that $K_{\lambda, \mu}(-1)$ is equal, up to sign, to the number of elements of $\text{SSYT}(\lambda, \mu)$ that are fixed by evacuation (the Schutzenberger involution). This restriction on $\mu$ is necessary because evacuation is content-reversing. In recent joint work with Mike Chmutov, Dongkwan Kim, Joel Lewis, and Elena Yudovina, we showed that in general, $K_{\lambda, \mu}(-1)$ counts, up to sign, the number of fixed points of the involution obtained by composing evacuation with the action of the long element $w_0$ by the Lascoux--Schutzenberger (or crystal) symmetric group action on tableaux. When the content is palindromic, the action of $w_0$ is trivial, so our result reduces to the above-mentioned one. The proof relies on the theory of rigged configurations.

Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase $(n-1, ..., 1,0)$ (of which there are Catalan many) and a composition weight of length $n$. They include the Schur functions ,the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to GL-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott theorem. We have discovered various properties of Catalan functions, providing a new insight on the existing theorems and conjectures inspired by Macdonald positivity conjecture. A key discovery in our work is an elegant set of ideals of roots that the associated Catalan functions are $k$-Schur functions and proved that graded $k$-Schur functions are G-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded $k$-Schur functions and resolved the Schur positivity and $k$-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are $k$-Schur positive which strengthens the Schur positivity of Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems. This is joint work with Jonah Blasiak, Jennifer Morse and Daniel Summers.

We study non-symmetric Macdonald polynomials whose variables $x_{m+1},x_{m+2},...$ are symmetrized (using the Hecke symmetrization), which we call $m$-symmetric Macdonald polynomials (the case $m=0$ corresponds to the usual Macdonald polynomials). In the space of $m$-symmetric polynomials, we define $m$-symmetric Schur functions (now depending on the parameter $t$) by certain triangularity conditions. We conjecture that the $m$-symmetric Macdonald polynomials are positive (after a plethystic substitution) when expanded in the basis of $m$-symmetric Schur functions and that the corresponding $m-(q,t)$-Kostka coefficients embed naturally into the $m+1-(q,t$)-Kostka coefficients. When $m=1$, an analog of the nabla operator can be defined, which provides a refinement of the bigraded Frobenius series of the space of diagonal harmonics. When $m$ is larger, how to define such a nabla operator is still an open problem.

We consider oriented braids in the thickened torus $T^2 \times I$, together with a single fixed base string. The based skein $H_n(T^2,*)$ is defined to be $\mathbb Z [s^{\pm 1}, q^{\pm 1}]$-linear combinations of $n$-braids subject to the Homflypt skein relation $X_{+}- X_{-} = (s-s^{-1})X_0$. In addition a braid string is allowed to cross through the base string at the expense of multiplying by the parameter $q$. Composition of braids induces an algebra structure on $H_n(T^2,*)$. We show that this algebra satisfies the relations of the double affine Hecke algebra ${\tilde H}_n$, as defined by Cherednik. We discuss how to include closed curves in the thickened torus in the model in an attempt to incorporate earlier work with Peter Samuelson on the Homflypt skein of $T^2$ into the setting of the algebras ${\tilde H}_n$, with an eye on the elliptic Hall algebra and the work of Schiffman and Vasserot.

The problem session was split into two pieces. The first part the following participants submitted a problem during this session:
François Bergeron, Lauren Williams, Hugh Morton, Brendan Pawlowski, Peter Samuelson.

A video of this session is available here: https://www.birs.ca/workshops/2019/19w5131/files/19w5131-PS1-20190124-1406-1505.mp4

Each person was asked to provide a written summary and potentially provide references. The written version that accompanies this video is at: https://www.birs.ca/workshops/2019/19w5131/files/open_problem_session_summary.pdf

The problem session was split into two pieces. The first part the following participants submitted/summarized a problem: Marino Romero, Mikhail Mazin, Gabriel Frieden, François Bergeron, Mike Zabrocki.

A video of this session is available here: https://www.birs.ca/workshops/2019/19w5131/files/19w5131-PS2-20190124-1531-1641.mp4

Each person was asked to provide a written summary and potentially provide references. The written version that accompanies this video is at: https://www.birs.ca/workshops/2019/19w5131/files/open_problem_session_summary.pdf

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.