Author: Anna Vanden Wyngaerd

In this short talk I will present preliminary work aimed at automatically discovering unknown combinatorial statistics using machine learning.

In this talk I will introduce a very special class of varieties of quiver representations, which I have been studying together with Alex Puetz and Evgeny Feigin. Such varieties arise as flat degenerations of classical Grassmannians, have many nice geometric properties, and exhibit pretty combinatorial features too, not all of which have been properly investigated. For example, they admit both a cyclic group action and an algebraic torus action, turning them into an interesting object also from a q,t-combinatorial perspective.

This talk presents a parabolic extension of Lusztig’s varieties, obtained by projecting from the full flag variety to partial flag varieties. These parabolic Lusztig varieties preserve key geometric and combinatorial features. Through concrete examples and an application of the Decomposition Theorem, we describe how cohomology decomposes across parabolic strata. The two-step flag case is treated in detail, leading to explicit character formulas for codominant permutations.

The notion of P-tableau was introduced by Gessel an Viennot. Given a Dyck path P, the definition is similar to the usual semistandard Young tableau, but the condition of i<j resp. i<=j is replaced by i,j being over P, resp. j,i under P. It turns out, many enumerative questions, reduce to P-tableau q-counting. In particular, we obtain a framework in which the compositional shuffle formula, the Loehr-Warrington’s formula, and the recent nabla operator formula by Carlsson and myself are all equivalent.

Catalanimals have been a central tool to a family of proofs of various “shuffle identities” such as the shuffle theorem, the extended Delta theorem, and the Loehr-Warrington theorem.
In this series of talks, we will attempt to give a self-contained treatment of the theory of Catalanimals and their various properties that enable these proofs.
The first half of the series will focus on the theory of nonsymmetric Hall-Littlewood polynomials and LLT series, culminating in a statement of a Cauchy identity that is useful for extracting combinatorial formulae from certain Catalanimals.
The second half of the series will focus on the realization of certain Catalanimals as elements of Negut’s Shuffle Algebra and how they correspond to elements in Schiffmann’s elliptic Hall algebra.

Based on the connection to one-dimensional interacting particle models–the ASEP and the TAZRP–and the combinatorial objects (multiline queues) that make these connections explicit, a tableau formula was found for the modified Macdonald polynomial in terms of the so-called `queue inversion’ statistic. This statistic naturally reflects the dynamics of the TAZRP. In this talk, we present a new tableau formula for the symmetric Macdonald polynomials, using the same queue inversion statistic on non-attacking tableaux. Central to our approach is a probabilistic bijection on non-attacking fillings that enables swapping of entries between columns. Our techniques extend to fillings of composition shapes, leading to symmetry results for permuted basement Macdonald polynomials.

The two well-known monomial bases of the (classical) coinvariant ring, the Artin basis and the descent basis, are indexed by permutations and correspond to the statistics inv and maj respectively. Both bases give combinatorial explanations for the Hilbert series of the coinvariant ring, which is $[n]_q!$.

We give a construction for a basis of the Garsia-Procesi ring $R_\mu$, a quotient of the coinvariant ring, using the catabolizability type of standard Young tableaux and the charge statistic. This basis is a subset of the descent basis and is compatible with the Hilbert series of $R_\mu$ in the same manner.  Our construction of the basis provides the first direct connection between the combinatorics of the basis with the combinatorial formula for the modified Hall-Littlewood polynomials $\tilde{H}_\mu[X;q]$, due to Lascoux.

Orellana and Zabrocki introduced families of inhomogeneous bases of symmetric functions whose evaluation at roots of unity are characters of the symmetric group. The introduction of these new bases led to generalizations of know combinatorial objects and connected well-known unsolved problems, such as the restriction problem, plethysm problem, and the Kronecker problem. The basis corresponding to the irreducible characters can be shown to be intimately connected to the Lie character. In recent work with Sundaram and Zabrocki, we describe a general construction of families of stable bases, including one connected to a variant of the Lie character introduced by Sundaram. In this talk I will discuss recent progress and problems related to these inhomogeneous bases.

We study the symmetric function that conjecturally gives the Frobenius characteristic of a diagonal coinvariant ring with one set of commuting and two sets of anti-commuting variables, of which we give a combinatorial interpretation in terms of segmented Smirnov words. Furthermore, this function is related to the Delta conjectures, and this work is a step towards a unified formulation of the two versions, as we prove a unified Delta theorem at t=0.

Cambrian lattices, introduced by Reading in 2006, are a generalization of the Tamari lattice to any choice of Coxeter element in any finite Coxeter group. The Tamari lattice corresponds to the “linear type A” case. These partial orders admit many descriptions, non trivially equivalent. These give rise to a common generalization by Stump, Thomas, and Williams, called the m-Cambrian lattices. We propose a new description of m-Cambrian lattices, with an easy and effective comparison criterion on simple objects called m-noncrossing partitions, hence a combinatorial model for their intervals. The “linear type A” m-Cambrian lattice was conjectured to have the same number of intervals as the m-Tamari lattice (another generalization of the Tamari lattice), whose intervals were enumerated by Bousquet-Mélou, Préville-Ratelle and Fusy. Their enumeration formula appeared conjecturally as the dimension of the alternating component of higher diagonal coinvariants in the trivariate case (Bergeron—Préville-Ratelle, 2011). This is based on joint work with Corentin Henriet and Wenjie Fang.

The m-symmetric Macdonald polynomials are extensions of the Macdonald polynomials, symmetrized only in the variables starting from x_{m+1}. They are conjectured to be positive when expanded into a basis of m-symmetric Schur functions. We will describe the properties satisfied by the corresponding (q,t)-Kostka coefficients, which contain the usual (q,t)-Kostka coefficients as special cases. We will focus in particular on the algebraic structure underlying certain Butler’s rules for the (q,t)-Kostka ceofficients, a structure that is provided by a family of idempotents in the double affine Hecke algebra which appear to split the (q,t)-Kostka coefficients into m! nonnegative pieces.

In this talk, I will explain how to obtain geometric interpretations for both the Rectangular Shuffle Theorem and the Rise/Fall Delta Theorem in terms of affine Springer fibers, generalizing the work of Hikita for the Shuffle Theorem. The key connection between the two constructions is a simple Schur skewing formula, which allows one to obtain the Delta Theorem from the Rectangular Shuffle Theorem. If there is time remaining, I will discuss a generalization of this geometric construction to the setting of triangular partitions/paths under any line, in the sense of the work of Blasiak-Haiman-Morse-Pun-Seelinger. Based on joint work with Maria Gillespie and Eugene Gorsky.

The modified Macdonald functions play an important role in the study of the K-theory of the Hilbert schemes of points in the plane. Haiman showed in his seminal work on the Macdonald positivity conjectures that the modified Macdonald functions correspond to the torus fixed point classes of the Hilbert schemes by the way of a derived equivalence. Carlsson-Gorsky-Mellit introduced a larger family of schemes called the parabolic flag Hilbert schemes related to Carlsson-Mellit’s proof of the Shuffle Theorem. In this talk, I will discuss the partially-symmetric generalization of modified Macdonald polynomials, their relation to the K-theory of the parabolic flag Hilbert schemes, and their interactions with the stable-limit double affine Hecke algebra of Ion-Wu. This is joint work with Daniel Orr.

Recently, Syu Kato found a formula for chromatic symmetric functions of unit interval graphs in terms of the affine Weyl groups of type A. This formula potentially contains another parameter q which is not related to the parameter t in the chromatic quasisymmetric functions. In this talk, we define a (q,t)-analogue of the chromatic symmetric functions by deforming Kato’s formula and study their basic properties.

In 1994 Haiman introduced the ring of diagonal coinvariants, which is a quotient of a polynomial ring in two sets of commutative variables by invariants of the diagonal action of the symmetric group. Recently, there has been much interest in studying a more general class of coinvariant rings with k sets of n commutative (bosonic) variables and j sets of n anticommutative (fermionic) variables; denote this ring by R_n^{(k,j)}. We will focus on the coinvariant ring R_n^{(1,2)}, with one set of bosonic and two sets of fermionic variables. By interpolating between the modified Motzkin path basis for R_n^{(0,2)} of Kim–Rhoades (2022) and the super-Artin basis for R_n^{(1,1)} conjectured by Sagan–Swanson (2024) and proven by Angarone et al. (2025), we propose a monomial basis for R_n^{(1,2)}. We use the proposed basis to give combinatorial formulas for its conjectural Hilbert series and Frobenius series. We will explain how our work on R_n^{(1,2)} relates to the Theta conjecture and recent work of Iraci, Nadeau, and Vanden Wyngaerd (2024).

The rank $n$ superspace ring consists of regular differential forms on $n$-space and carries a natural action of the symmetric group $\mathfrak{S}_n$. The superspace coinvariant ring $SR_n$ is the quotient of this action by the $\mathfrak{S}_n$-invariants with vanishing constant term. After a review and overview of the broader theory of coinvariant quotients, we will discuss the algebraic structure of $SR_n$.

We study the symmetric functions $g_{\mathbf{m},k}(x;q)$, introduced by Abreu and Nigro for a Hessenberg function $\mathbf{m}$ and a positive integer $k$, which refine the chromatic symmetric function. Building on Hikita’s recent breakthrough on the Stanley–Stembridge conjecture, we prove the $e$-positivity of $g_{\mathbf{m},k}(x;1)$, refining Hikita’s result. We also provide a Schur expansion of the sum $\sum_{k=1}^n e_k(x) g_{\mathbf{m},n-k}(x;q)$ in terms of $P$-tableaux with 1 in the upper-left corner. We introduce a restricted version of the modular law as our main tool. Then, we show that any function satisfying the restricted modular law is determined by its values on disjoint unions of path graphs. This is based on joint work with JiSun Huh, Byung-Hak Hwang, Donghyun Kim, Jang Soo Kim.

In recent work with Josh Wen, we study wreath Macdonald polynomials, producing a story analogous to the one found in the theory of modified Macdonald polynomials. We will start by introducing wreath Macdonald polynomials and plethystic substitution for multisymmetric functions. In the end, we will present an analogue of Tesler’s Identity, Macdonald-Koornwinder Reciprocity, and other new identities for wreath Macdonald polynomials, some or all of which have been conjectured by Haiman and Shimozono.

We will discuss a (totally unbiased) overview of the history of Macdonald polynomials and the development of the rich framework around the modified version of these symmetric functions. We’ll then confront a long-standing mystery in nonsymmetric Macdonald theory and talk about some recent joint work with Blasiak, Haiman, Pun, and Seelinger that relates to this puzzle.