Institute for Computational and Experimental Research in Mathematics (ICERM)

Abstract

The space of diagonal harmonics $DH_n$ is an $S_n$-module in two sets of $n$ variables, which arose from a representation theoretic program to study Macdonald polynomials initiated by Garsia and Haiman. The doubly graded character of $DH_n$ has both a symmetric function and a combinatorial description: Haiman proved it is $\Delta'_{e_{n-1}}e_n$ for an eigenoperator $\Delta'$ of modified Macdonald polynomials, while Carlsson and Mellit established the combinatorial ``Shuffle Conjecture"" of Haiman, Haglund, Loehr, Remmel, and Ulyanov, expressing it as as a sum of LLT polynomials over Dyck paths. An expanded investigation led Haglund, Remmel and Wilson to the Extended Delta Conjecture, a combinatorial prediction for $\Delta_{h_l}\Delta'_{e_k}e_n$, allowing for any $0\leq l,k<n$. Bringing in new results about the action of the elliptic Hall algebra on symmetric functions, we reformulate the conjecture as the polynomial truncation of an identity of infinite series of $GL_l$ characters, expressed in terms of LLT series. The stronger infinite series identity is not difficult to prove using identities on non-symmetric Hall-Littlewood polynomials. This is joint work with Blasiak, Haiman, Pun, and Seelinger.

Abstract

Jenna Rajchgot observed that the Castelnuovo-Mumford regularity of matrix Schubert varieties is computed by the degrees of the corresponding Grothendieck polynomials. We give a formula for these degrees. Indeed, we compute the leading terms of the top degree pieces of Grothendieck polynomials and give a complete description of when two Grothendieck polynomials have the same top degree piece (up to scalars). Our formulas rely on some new facts about major index of permutations.

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In this talk I will introduce simple examples of sheaves of conformal blocks arising from representations of Lie algebras and discuss how they can be used to study certain moduli spaces. I will further discuss generalizations of these sheaves which go beyond the representation theory of Lie algebras, and related works in progress.

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There is a q-deformation of the reflection representation of the affine Weyl group in type A which leads to a q-deformation of many common constructions: Demazure operators, Soergel bimodules, geometric Satake, etcetera. When q is set to a root of unity, the action of the affine Weyl group factors through a finite quotient, the complex reflection group G(m,m,n), and a new kind of "Schubert calculus" appears. This talk will demonstrate this unusual construction in the well-understood case of affine A_1 and the mysterious case of affine A_2.

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Semi-infinite flag manifolds are variants of affine flag manifolds whose Schubert cells are simultaneously infinite-dimensional and infinite-codimensional. Such objects are introduced by Drinfeld and Lusztig around 1980 (as sets), and its finite-dimensional approximation model was presented by Finkelberg-Mirkovic in 1999. Recently, we described them as an explicit ind-scheme of ind-infinite type that satisfies a certain universal property. It lead us to a description of quantum $K$-groups of partial flag manifolds and some functorial relations between them. In this talk, I will start from a brief review of affine flag manifolds/varieties, define semi-infinite flag manifolds and their natural subschemes, and then explain their combinatorial and algebro-geometric properties (trying to stress the difference from affine flag manifolds/varieties). If time allows, then I will explain how some of these properties are derived as a shadow of the homological properties of affine Lie algebras. This talks is mainly based on arXiv:1810.07106.

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Combinatorics of quadratic spaces over finite fields
Semin Yoo, University of Rochester
Let $\mathbb{F}_q^n$ be the finite field with $q$ elements, where the char$F$ is not $2$. The size of the set of $k$-dimensional subspaces of $\mathbb{F}_q^n$ is called $q$-binomial coefficient, (or Gaussian binomial coefficient.) which has various interesting combinatorial descriptions and related works. Furthermore, the fact that makes $q$-binomial coefficients more interesting is that $q$-binomial coefficients have $q$-analogues. $\mathbf{q}$-\textbf{analogues} of quantities in mathematics involve perturbations of classical quantities using the parameter $q$ and revert to the original quantities when $q$ goes to $1$. In this talk, we add one more structure, called a quadratic form. We are mainly interested in the size of the set of $k$-dimensional subspaces that have orthonormal bases, which can be also written as analogues of binomial coefficients. Various combinatorial properties of this new binomial coefficient will be discussed. We will also show what will happen in the set side if we take the limit when $q$ goes to $1$.

Mixed Hodge structure on local cohomology with support in determinantal varieties
Michael Perlman, Queen's University
Given a closed subvariety Z in a smooth variety, the local cohomology sheaves with support in Z are functorially endowed with structures as mixed Hodge modules. This implies that they are equipped with two increasing filtrations: the Hodge filtration and the weight filtration. We will discuss new calculations of these filtrations in the case when Z is a generic determinantal variety. This talk includes joint work with Claudiu Raicu.

Geometric and Combinatorial aspects of Nonlinear Algebra
Papri Dey, University of Missouri-Columbia
Nonlinear algebra is an interdisciplinary area and I shall talk about some interaction among algebraic, geometric and combinatorial objects in this field.

Clans, sects, and symmetric space closure orders
Aram Bingham, Tulane University of Louisiana
We will give a combinatorial description of the closure order on Borel orbits in symmetric spaces of Hermitian type in terms of parameterizing objects called clans and their projections onto $G/P$ (sects). This description resolves part of a conjecture of Wyser on the restriction of Bruhat orders on these spaces.

A generating function for counting mutually annihilating matrices over a finite field
Yifeng Huang, University of Michigan
We count the number of pairs of n x n matrices (A, B) over a finite field such that AB=BA=0. We then give an explicit factorization of a generating function associated to this count, which in particular shows that the function can be meromorphically extended to the entire complex plane. This essentially says that the motivic formula about (the stack of) finite-length coherent sheaves on a nodal singular curve behaves as the geometry predicts.

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The geometric naturality of Schubert polynomials and the related combinatorics of pipe dreams was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of vexillary matrix Schubert varieties. We will discuss general diagonal degenerations, relating them to an older formula of Lascoux (2002) in terms of the 6-vertex ice model. Lascoux's formula was recently rediscovered by Lam, Lee, and Shimozono (2018), as "bumpless pipe dreams." We will explain this connection and discuss conjectures and progress towards understanding diagonal Gröbner degenerations of matrix Schubert varieties.

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Local systems are sheaves which describe the behavior of solutions of differential equations. A local system is rigid if local monodromy determines global monodromy. We give a construction which produces irreducible complex rigid local systems on a punctured Riemann sphere via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups SU(n) (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices (which can be inductively determined) of these polytopes give (all) possible unitary irreducible rigid local systems. We note that these polytopes are generalizations of the classical Littlewood-Richardson cones of algebraic combinatorics. Answering a question of Nick Katz, we show that there are no irreducible rigid local systems on a punctured Riemann sphere of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing n, when n is a prime number.

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Generating functions are a major theme in mathematics which unify many disciplines. It is remarkably common to find a combinatorial generating function which factors as a ratio of products of $q$-integers. Examples of such quotients arise from enumerative combinatorics (e.g. subset sums and $q$-binomials), representation theory (the $q$-Weyl dimension formula), geometry (iterated $\mathbb{P}^1$-bundles), probability (random variable decompositions), commutative algebra (homogeneous systems of parameters), and more. We call such quotients "cyclotomic generating functions" (CGFs) and initiate their general study. This talk will review some of the many known constructions of CGFs and give asymptotic estimates of their coefficients. We will also highlight a range of conjectures and accessible open problems. Joint work in progress with Sara Billey.

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Bumpless pipe dreams were introduced by Lam, Lee, and Shimozono in the context of back stable Schubert calculus. Like ordinary pipe dreams, they compute Schubert and double Schubert polynomials. In this talk, I will give a bijective proof of Monk's rule for Schubert polynomials, and show that the proof extends easily to the proof of Monk's rule for double Schubert polynomials. As an application, I will explain how to biject bumpless pipe dreams and ordinary pipe dreams using the transition and co-transition formulas, which are specializations of Monk's rule. If time permits I will also briefly discuss some work on how bumpless pipe dreams can be used to compute products of certain Schubert polynomials that generalize the Grassmannian case.

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Volumes of Root Zonotopes via the W-Laplacian
Theo Douvropoulos, University of Massachusetts, Amherst
The normalized volume of the classical permutahedron is given by the formula n^{n-2} which, ask any combinatorialist and they will tell you, agrees with the number of trees on n vertices. This coincidence is well understood; the classical permutahedron is a unimodular zonotope, on the set of positive roots of the Symmetric group S_n, and its bases are indexed by trees and hence enumerated by the determinant of a Laplacian matrix. This description of the permutahedron lends itself to a natural generalization: for a Weyl group W, the zonotope associated to the collection of positive roots of W will be called the root zonotope Z_W. These zonotopes Z_W are not unimodular, but their bases can be differentiated with respect to the reflection subgroup they generate (and its connection index). In joint work with Guillaume Chapuy (arxiv.2012.04519) we have introduced for any Weyl group W a (nxn) Laplacian matrix L_W whose spectrum encodes (nontrivially) many enumerative properties of W. In particular we will present new short formulas (ibid: Section 8.3) for the volumes of the root zonotopes Z_W involving only the Coxeter numbers of W and its reflection subgroups. Our approach is uniform and does not rely on the classification.

Effective point-counting for polygons in flag varieties
Brian Hwang, Cornell University
There are a number of seemingly similar "big" (e.g. affine, open) spaces that arise in the context of flag varieties. Certain classes of these spaces are known to exhibit special features. For example, open Richardson varieties have a decomposition into a disjoint union of simple components and the coordinate rings of double Bruhat cells are known to be cluster algebras. There are often combinatorial objects that govern their geometry, such as subword complexes for brick manifolds or various kinds of polytopes, especially in the context of toric varieties and related deformations and degenerations. The bestiary is rich and multifarious. Many of these spaces, however, turn out to admit a description as a polygon in a flag variety, that is, a (cyclic) tuple of flags where each flag is a prescribed distance from its immediate neighbors. Here, the distance is taken not with respect to the Euclidean metric, but with a non-symmetric Weyl-group-valued notion of "distance." Such polygons in flag varieties turn out to admit simple decompositions that can be phrased in terms of triangulations of the polygon into certain triangles with an "easy-exit" property. As an illustration of this organizing principle, we show how we can use this to easily count the points of open Richardson varieties over finite fields–––recovering a classic result of Deodhar, as well as its later enhancement by Marsh and Rietsch–––and how this naturally extends to spaces like double Bruhat cells and their amalgamations.

A 'Quantum Equals Classical' Theorem for n-pointed (K-theoretic) Gromov-Witten Invariants of Lines in Homogeneous Spaces
Weihong Xu, Rutgers University
'Quantum equals classical' refers to ways of identifying (K-theoretic) Gromov-Witten invariants on a flag variety G/P with (K-theoretic) classical invariants on a possibly different flag variety G/Q. It has been an active research topic in the last 20 years, but by far all the work has been for 3-pointed invariants. We obtain a 'quantum equals classical' theorem for n-pointed, genus 0 (T-equivariant K-theoretic) Gromov-Witten invariants of lines in G/P, which generalizes the 3-pointed result of Leonardo Mihalcea and Changzheng Li from 2013. This talk is based on joint work with Anders Buch, Linda Chen, Angela Gibney, Lauren Heller, Elana Kalashnikov, and Hannah Larson.

Matroids, Positroids, and Combinatorial Characterizations
Anastasia Chavez, UC Davis
Positroids are a special class of representable matroids introduced by Postnikov in the study of the nonnegative part of the Grassmannian. Postnikov defined several combinatorial objects that index positroids. In this talk, I'll briefly introduce some of these objects and how they can combinatorially characterize matroid properties.

Twisted quadratic foldings of root systems and combinatorial Schubert calculus
Maiko Serizawa, University of Ottawa
This thesis builds on the connection of two widely studied objects in the literature, that is, foldings of finite root systems and the structure algebras of moment graphs associated with finite root systems. Given a finite crystallographic root system <I> whose Dynkin diagram has a non­trivial automorphism, it yields a new root system <l>7 by a so-called classical folding. On the other hand, Lusztig's folding (1983) folds the root system of type Es to type H4 starting from an automorphism of the root lattice of type Es. Lanini-Zainoulline (2018) developed the notion of a twisted quadratic folding of a root system, which describes both the classical foldings and Lusztig's folding on the same footing. Our second key object of study is the structure algebra Z(Q) of the moment graph g associated with a finite root system and its reflection group W. The structure algebra Z(Q) is an algebra over a certain polynomial ring S whose underlying module is free with a distinguished basis { a( w) I w E W} called combinatorial Schubert classes. Each Schubert class a(w) is an S-valued function on W, whose value is explicitly known for any finite reflection group W. Lanini-Zainoulline (2018) showed that a twisted quadratic folding <I> -v--; <l>7 in­duces an embedding of the respective Coxeter groups E : W7 Y W and a ring homomorphism c:* : Z(Q) ➔ Z(Q7 ) between the corresponding structure algebras. This thesis investigates how the induced map c:* relates the Schubert classes of the original structure algebra Z(Q) to those of the folded structure algebra Z(Q7 ). In particular, we will provide a combinatorial criterion for a Schubert class a?u) of Z(Q7 ) to admit a Schubert class a(w) of Z(Q) such that the relation c:*(a(w) ) = c · a?u) holds for some nonzero scalar c. We will also prove that c:* is surjective after an appropriate extension of the coefficient ring.

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We study differential operators for Schur and back stable Schubert polynomials. Our operators are based on two operators of degree (-1), which satisfy Leibniz rule. For the case of Schur functions, these two operators fully determine the product of Schur functions, i.e., it is possible to define Littlewood-Richardson coefficients only using these operators. This new point of view on Schur functions gives us an elementary proof of The Giambelli identity and Jacobi-Trudi identities. For the case of Schubert polynomials, we construct a larger class of decreasing operators, which are indexed by Young diagrams. Operators from this family are related to Stanley symmetric functions. In particular, we extend bosonic operators from Schur to Schubert polynomials.

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We explain some properties of back-stable limits of double Grothendieck polynomials. We discuss the dual basis of equivariant K-homology of the infinite Grassmannian.

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There are pairs of seemingly unrelated spaces such that Schubert calculus on the two spaces `match’ in some concrete sense. This duality, called 3d mirror symmetry, is best observed in equivariant elliptic cohomology. For example, the dual of the 12-dimensional $T^*Gr(2,5)$ is a certain 4-dimensional Nakajima quiver variety. However, the set of cotangent bundles of homogeneous spaces, or even the set of quiver varieties are not closed for 3d mirror symmetry. In this talk, based on a joint work with Y. Shou, we present a larger pool of spaces: Cherkis bow varieties. Superstring theory predicts that bow varieties are closed for 3d mirror symmetry. The combinatorics necessary to play Schubert calculus on bow varieties includes binary contingency tables and tie diagrams. The existence of an operation (called Hanany-Witten transition) gives bow varieties extra flexibility. We will illustrate the combinatorics and geometry of bow varieties with examples, and we will calculate cohomological and elliptic Schubert classes (rather, `stable envelopes’) to explain the `matching Schubert calculus’ phenomenon.

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Let X be a Fano variety. Dubrovin’s conjecture predicts, among other things, an equivalence between the semi-simplicity of QH(X) the big quantum cohomology of X and the existence of a full exceptional collection in D(X) the bounded derived category of X. Recently, Kuznetsov and Smirnov proposed a refinement of this conjecture if qH(X) the small quantum cohomology is not semi-simple. I will present this conjecture, discuss it in the case of coadjoint varieties and make connections with roots systems and simple surface singularities. This is a joint work with Maxim Smirnov

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The motivic Chern class in K-theory is a generalization of the Chern-Schwartz-MacPherson class in homology. As the affine Hecke algebra, the motivic Chern classes of Schubert cells have a Langlands dual description. In joint works with P. Aluffi, L. Mihalcea, and J. Schurmann, we use this description to solve some problems about the Casselman basis for the p-adic dual group, and relate the Euler characteristic of the motivic Chern classes to the Iwahori-Whittaker functions. 

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Imposing conditions on the ranks of upper-left corners of a skew-symmetric matrix defines a skew-symmetric matrix Schubert variety. We give Gröbner bases for the prime ideals of these varieties, and identify the corresponding initial ideals as the Stanley-Reisner ideals of certain explicit shellable simplicial complexes. These results are analogous to results of Knutson and Miller in the setting of ordinary matrix Schubert varieties, but the techniques are new, and can be used to give new proofs of some of their results. Based on joint work with Eric Marberg.

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Coulomb branches of 4d N=2 gauge theories are a class of algebraic varieties which, together with their quantizations, appear in a variety of guises in geometry and representation theory. In this talk we outline a construction of canonical bases in the quantized coordinate rings of these varieties. These bases appear as byproducts of a more fundamental construction, a nonstandard t-structure on the dg category of coherent sheaves on the Braverman-Finkelberg-Nakajima space of triples. The heart of this t-structure is a tensor category which is not braided, but admits renormalized r-matrices abstracting those appearing in the finite dimensional representation theory of quantum affine algebras and KLR algebras. It is intended to provide a mathematical model of the category of half-BPS line defects in the relevant gauge theory, and is inspired by earlier work of Kapustin-Saulina and Gaiotto-Moore-Neitzke. On a combinatorial level, the resulting bases are expected to be controlled by specific cluster algebras, which we can confirm in simple examples. This is joint work with Sabin Cautis.

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Correspondences Z inside M x N induce maps on cohomology H^*(M) -> H^*(N), modulo smoothness and compactness issues. We soup up the diagonal inclusion Gr(k,n) -> Gr(k,n)^2 (whose induced cohomology map is multiplication) to a correspondence of cotangent bundles, which we then factor through T^*(a 2-step flag manifold). Grassmannian Schubert puzzles turn out to be a calculation on this intermediate manifold. As an application of this viewpoint we get a puzzle rule for the Euler characteristic of the intersection of three generically translated Bruhat cells (in up-to-4-step flag manifolds).

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I present a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the so-called quantum alcove model. One application is the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum K-theory of the flag manifold G/B; this has been a longstanding conjecture. I also discuss the Chevalley formula for partial flag manifolds G/P. Another application is that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivariant) quantum K-theory of the type A flag manifold. This is joint work with Satoshi Naito and Daisuke Sagaki.

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For a finite Coxeter system, we define “spherical elements” (extending Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Levi-spherical Schubert varieties in G/B. In type A, this connects to key polynomials (Demazure characters), multiplicity-freeness, and “split-symmetry” in algebraic combinatorics. This is joint work (some ongoing) with subsets of David Brewster (UIUC), Yibo Gao (MIT), and Reuven Hodges (UIUC). See arXiv:2007.09238, arXiv:2007.09229, and arXiv:2012.09749.

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We relate the cohomology of open positroid varieties and their point counts over finite fields to knot homology. In particular, we show that the bigraded Poincaré polynomials of top-dimensional open positroid varieties are given by rational q,t-Catalan numbers. As a consequence of the curious Lefschetz property, we obtain q,t-symmetry and unimodality statements for rational q,t-Catalan numbers. Joint work with Thomas Lam. No special background on the above objects will be assumed.

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We'll present several combinatorial rules (in terms of ``puzzles'') for the expansion of the product of motivic Segre classes (in equivariant K-theory) and Segre-Schwartz-MacPherson classes (in equivariant cohomology) in partial flag varieties. We'll show how to compute such classes and puzzles using the software Macaulay2.