Abstracts

Lecture 1 (Sottile): The Geometry of Grassmannians (Sottile): We define the Grassmannian, Stiefel coordinates, the Pluecker embedding, and its Schubert cells/varieties.  We then sketch the basics of Schubert calculus: Schubert classes in cohomology and Schubert's basis and duality theorems, define Littlewood-Richardson coefficients and prove the Pieri Rule.

Lecture 2 (Weigandt): Schur Polynomials and the Littlewood-Richardson Rule

Lecture 3 (Weigandt): The Complete Flag Variety and Schubert Polynomials

Lecture 4 (Sottile): Flag Manifolds

Lecture 1: We will review the basics of graded modules, Hilbert functions, and degrees of varieties.  We will then describe Rothe diagrams of permutations and introduce matrix Schubert varieties via their defining equations.  We will give a quick introduction to Gröbner bases and exposit some results of Knutson and Miller on antidiagonal initial ideals of Schubert detrimental ideals.  Time permitting, we will give a very brief introduction to the combinatorics of pipe dreams.

Lecture 2: We will discuss multidegrees of varieties and view Schubert polynomials as multidegrees.  We will learn how to compute the Schubert polynomial of a matrix Schubert variety from its antidiagonal initial idea.  Next, we will (re)view the Stanley--Reisner correspondence and discuss vertex decomposition of a simplicial complex (or, if one prefers, of a squarefree monomial ideal).  We will then discuss a generalization of vertex decomposition called geometric vertex decomposition, introduced by Knutson, Miller, and Yong in their study of matrix Schubert varieties of vexillary permutations.

Lecture 3: We will continue our discussion of geometric vertex decomposition, which will lead us to alternating sign matrices and alternating sign matrix varieties, which generalize matrix Schubert varieties.  We will discuss what is known about diagonal initial ideals of matrix Schubert varieties.  Time permitting, we will end with some open questions involving varieties arising in the vicinity of Schubert calculus where geometric vertex decomposition might be able to be made to apply.

Some ingredients in #1. Discussion of Bruhat cells on Flags(3) in terms of intersection with a fixed flag. Pretty board pics thereof. First interesting calculation: S_132 * S_312 = S_231 + S_312 {this one doesn't require any move-to-transverse). In K-theory, it's also got -S_321. Oracle: there is a way of computing these products in terms of scattering diagrams / puzzles. But that will require switching from perms to their inverses, because physics is okay when particles have indistinguishable charge but not indistinguishable position. Then follow the oracle to discover the puzzle pieces for 1- and 2-step puzzles, and for K-theory in 1-step.

Some ingredients in #2. This'll be after Dave Anderson's eqvt H^* talk, which is about foundations, e.g. Borel mixing spaces. I'll talk about it as intersection theory, and point out that if there will be puzzles, then (1) they'll be less symmetric (2) they'll have y_i-y_j factors in them. Use the intersection theory to come up with some data, and discover the eqvt puzzles. If Dave (and other people) haven't done Kirwan injectivity, then I'll have to talk about that, and use it to suggest the ingredients in a proof of the eqvt puzzle rule we've discovered.

Some ingredients in #3. Yang-Baxter. AJS/Billey and the proof of the puzzle rule. Very sketchy description of where R-matrices come from. Rundown of all the extant puzzle rules. Hints about the cotangent bundle story.

Lecture 1: Introduction to equivariant cohomology.  We’ll go over the definition via fiber bundles and the Borel construction, then apply this to compute equivariant cohomology rings of projective spaces and flag varieties.  Finally we’ll see a construction of Schubert classes, Poincare duality, and Graham-positivity.

Lecture 2: Equivariant K-theory.  In parallel to cohomology, we’ll do a quick introduction to K-theory, especially for flag varieties.  We’ll use Demazure operators to define Grothendieck polynomials representing Schubert classes, and see a statement of AGM-positivity.

Lecture 3: Back stable (aka enriched) Schubert and Grothendieck polynomials.  We’ll define these polynomials using the geometry of infinite flag varieties, describe their coproduct structure, and discuss positivity of the equivariant coproduct.

Lecture 1: Intro to K-theory: K-theoretic Schubert classes and boundary Schubert classes; duality; Hirzerbruch lambda_y classes of tautological vector bundles and Whitney presentations for K theory rings; Brion's positivity in K theoretic Schubert Calculus (with proof in a special case).

Lecture 2: Intro to Quantum K-theory: the moduli space and the K-theoretic Gromov-Witten invariants; QK pairing and the QK multiplication; curve neighborhoods and calculations of the QK pairing; Kato's functoriality of QK rings; rational connectedness and the `quantum=classical' for Grassmannians.

Lecture 3: QK positivity statement and idea of proof; QK from physics: the Coulomb branch relations and the QK Whitney presentation; recent developments and open problems.

I’ll discuss ongoing joint work with Nantel Bergeron to find a combinatorial formula for Schubert structure constants as certain chains in Bruhat order by developing an insertion algorithm on Kohnert’s diagrammatic model for Schubert polynomials.

Fubini words are generalized permutations, allowing for repeated letters, and they are in one-to-one correspondence with ordered set partitions. Brendan Pawlowski and Brendon Rhoades extended permutation matrices to pattern matrices for Fubini words. Under a lower triangular action, these pattern matrices produce cells in projective space, specifically $(\mathbb{P}^{k-1})^n$. The containment of the cell closures in the Zariski topology gives rise to a poset which generalizes the Bruhat order for Schubert cells/varieties indexed by permutations. Unlike Bruhat order, containment is not equivalent to intersection of a cell with the closure of another cell. This allows for a refinement of the poset. It is additionally possible to define a weaker order, giving rise to a subposet still containing all the elements. We call these orders, in order of decreasing strength, the espresso, medium roast, and decaf Fubini-Bruhat orders.  The espresso and medium roast orders are not ranked in general. The decaf order is ranked by codimension of the corresponding cells. In fact, the decaf order has rank generating function given by a well-known $q$-analog of the Stirling numbers of the second kind. We give increasingly smaller sets of equations describing the cell closures, which lead to several different combinatorial descriptions for the relations in all three orders. We also describe a few classes of covering relations in each of the orders.   We will conclude with several interesting open problems in this area.

This talk is based on joint work with Stark Ryan.

The quantum K-theory ring QK(X) of a flag variety X encodes the K-theoretic Gromov-Witten invariants of X, defined as arithmetic genera of Gromov-Witten varieties parametrizing curves meeting fixed Schubert varieties. A Pieri formula means a formula for multiplication with a set of generators of QK(X). Such a formula makes it possible to compute efficiently in this ring. I will speak about a Pieri formula for QK(X) when X is a cominuscule Grassmannian, that is, an ordinary Grassmannian, a maximal orthogonal Grassmannian, or a Lagrangian Grassmannian. This formula has a simple statement in terms of order ideals in a partially ordered set that encodes the degree distance between opposite Schubert varieties. This set generalizes both Postnikov's cylinder and Proctor's description of the Bruhat order of X. This is joint work with P.-E. Chaput, L. Mihalcea, and N. Perrin.

A T-variety is a variety with an algebraic torus action. The complexity of this action can be interpreted as a measure of how far the variety is from being a toric variety. Many subvarieties of the flag variety are T-varieties, and their complexity has been studied. In this talk, we focus on the problem of classifying which Kazhdan--Lusztig varieties have a given complexity. We will see how this is related to the complexity of Richardson varieties. This is joint work with Maria Donten-Bury and Irem Portakal.

Abstract Weighted flag varieties are generalizations of weighted projective spaces to the setting of flag varieties.   We prove that the product in the torus equivariant cohomology of a weighted flag variety has a positivity property  analogous to the equivariant positivity for non-weighted flag varieties. This generalizes a result proved by Abe and Matsumura for weighted Grassmannians.   We also discuss some related results about weighted flag varieties.  This is joint work with Scott Larson and Arik Wilbert.

This talk will present an overview of shifted tableaux and the role they play in Schubert calculus. Most notably, shifted tableaux describe the combinatorics of the orthogonal and Lagrangian Grassmannians, as well as certain symmetric spaces. I will give an overview of this geometry, with a special emphasis on how these settings relate to each other, how they connect to matrix Schubert varieties and open problems. This will include joint work with Eric Marberg and Brendan Pawlowski.

This talk will explain the tool of folded alcove walks, which enjoy a wide range of applications throughout combinatorics, representation theory, number theory, and algebraic geometry.  We will survey the construction of both finite and affine flag varieties through this lens, focusing on the problem of understanding intersections of different kinds of Schubert cells.  We then highlight several applications, including computing localizations in GKM theory for flag varieties and R-polynomials in Kazhdan-Lusztig theory.

We introduce projective schemes that are analogues of the James reduced product construction from homotopy theory and begin to develop a Schubert calculus for such spaces. This machinery yields K-theoretic and T-equivariant analogues of classic quasisymmetric function theory. Based on joint work with Matt Satriano.

Schubert problems are, fundamentally, just systems of polynomial equations, and so they are usually studied over an algebraically closed field (e.g. C).  I will talk about how and why I became interested in Schubert problems over R,  as well as some of the many conjectures in this direction, and some new results (based on joint work with Steven Karp).

Since at least the 1980s, mathematicians have found connections between orbit closures in type A quiver representation varieties and Schubert varieties in type A flag varieties. For example, singularity types appearing in type A quiver orbit closures coincide with those appearing in Schubert varieties in type A flag varieties; combinatorics of type A quiver orbit closure containment is governed by Bruhat order on the symmetric group; and classes of type A quiver orbit closures in equivariant cohomology and K-theory can be expressed in terms of formulas involving Schubert polynomials, Grothendieck polynomials, and other objects from Schubert calculus.

After recalling some of this story, I will discuss the related setting of Derksen-Weyman's symmetric quivers and their representation varieties. I will show how one can adapt results from the ordinary type A setting to unify aspects of the equivariant geometry of type A symmetric quiver representation varieties with Borel orbit closures in corresponding symmetric varieties G/K (G = general linear group, K = orthogonal or symplectic group). This is joint work with Ryan Kinser and Martina Lanini.

In this talk, I will discuss two combinatorial criteria for determining when Schubert varieties are isomorphic. The first criterion is for Schubert varieties from the full flag variety G/B and is joint work with W. Slofstra. In this case, Schubert varieties are indexed by the Weyl group of G. The second criterion is for cominuscule Schubert varieties and is joint work with M. Tarigradschi and W. Xu. In this second case, Schubert varieties are indexed by lower order ideals in a certain distributive lattice. In both cases, the isomorphism criterion is given by an equivalence relation on the indexing sets, and the proofs rely heavily on the Schubert calculus of the corresponding flag varieties.

Castelnuovo-Mumford regularity is a measure of complexity for a graded module. For Cohen-Macaulay varieties such as Kazhdan-Lusztig varieties, their regularities may be computed in terms of the degrees of their K-polynomials. We give an explicit combinatorial formula for the regularities of certain Kazhdan-Lusztig varieties. This generalizes a previous result of J. Rajchgot-Y. Ren-C. Robichaux-A. St. Dizier-A. Weigandt for Grassmannian matrix Schubert varieties. We then relate these results to the work of S. Ghorpade-C. Krattenthaler which computes the related a-invariant for certain ladder determinantal ideals. This is joint work with Jenna Rajchgot and Anna Weigandt.

We show that the equivariant K-homology of the affine Grassmannian Gr_{SL_n} can be realized as the functions on a family over the torus whose special fiber is a principal unipotent. We express the matrix entry functions in terms of the Schubert basis. We expect this to generalize to arbitrary type as is the case in (co)homology. A future goal is to upgrade this to an explicit map sending double quantum Grothendieck polynomials to double K-theoretic analogue of k-Schur functions.

This ongoing joint work with Takeshi Ikeda, Shinsuke Iwao, and Thomas Lam.

The  Murnaghan-Nakayama formula  expresses  the  product of  a Schur function  with a  Newton power  sum in  the basis  of Schur functions.  An  important generalization  of Schur  functions are Schubert polynomials (both classical  and quantum).  For these, a Murnaghan-Nakayama  formula  is   geometrically  meaningful.   In previous work with Morrison,  we established a Murnaghan-Nakayama formula  for  Schubert  polynomials  and  conjectured  a  quantum version.  In this  talk, I will discuss some  background and then some recent work proving this  quantum conjecture.  This is joint work with Benedetti, Bergeron, Colmenarejo, and Saliola.

Schubert polynomials represent cohomology classes of Schubert cycles in flag varieties and comprise a distinguished basis of the coinvariant algebra. The permutahedral variety is a subvariety of the flag variety, and one can ask for the expansion for its class in terms of Schubert classes. 

As we hope to demonstrate in the talk, the road to a convincing combinatorial answer to this Schubert-positivity question goes through the quotient of the polynomial ring by the ideal of quasisymmetric polynomials. The underlying combinatorics is governed by a new basis of the polynomial ring that we call forest polynomials, and the eventual answer involves applying a novel `parking procedure’  to reduced words. Joint work with Philippe Nadeau (CNRS & Univ. Lyon).

Schubert structure coefficients describe the multiplicative structure of the cohomology rings of flag varieties. Much work has been done on the problem of giving combinatorial formulas for these coefficients in special cases, as well as on the related problem of identifying vanishing and nonvanishing conditions. We establish families of linear relations among Schubert structure coefficients, which allow one to discern properties of unknown coefficients from properties of others.  Building on work of Wyser, we give new formulas for certain Schubert structure coefficients in terms of backstable (p,q)-clans.  This is joint work with Oliver Pechenik.

Hessenberg varieties are certain subvarieties of the flag variety whose cohomology was recently connected to the Stanley-Stembridge conjecture about the chromatic symmetric functions of certain graphs. They have a decomposition into affine cells, giving a geometric basis for their cohomology, but unlike in more familiar examples such as Schubert varieties, the closure of a cell is not a union of cells.

For a family of Hessenberg varieties which includes the toric variety for the permutahedron and the Peterson variety, we give an explicit combinatorial description of the closure of a cell. In particular, this allows us to calculate the classes of the cell closures in the cohomology ring of the flag variety. We also determine which of the torus-fixed points (of the flag variety) in these Hessenberg varieties are singular.

This is joint work with Erik Insko (Florida Gulf Coast U.) and Martha Precup (Washington U. in St. Louis).

A symmetric space is a homogeneous space G/K where G is a reductive (complex) algebraic group and K is the fixed-point subgroup of an algebraic involution θ : G → G. As in the particular examples of flag varieties, Borel subgroup orbits in other symmetric spaces are combinatorially parametrized and equipped with a closure-containment (or Bruhat) order which possesses nice order-theoretic properties. One poset property is called shellability, which implies that a simplicial complex associated to the poset has the homotopy type of a wedge of spheres. It has been known since the 1980s that Bruhat orders of (parabolic quotients of) Coxeter groups are shellable, and these results have been extended to other symmetric spaces more recently. We will discuss shellability of the Bruhat order for the symmetric space GLp+q /GLp × GLq (joint work with Néstor Díaz).

Brenti, Fomin, and Postnikov introduced two related objects: the quantum Bruhat graph and the tilted Bruhat order in connection with quantum Schubert calculus. Using the former, we provide an explicit formula for efficiently computing the minimum quantum exponent in the case of GL(n)/B. This complements the work of Fulton-Woodward and Postnikov. Our solution gives rise to an Ehresmann-type comparison test for the tilted Bruhat order in type A. Furthermore, in ongoing work, we develop the first geometric interpretation of tilted Bruhat order by introducing tilted Richardson varieties.

This is a joint work with Jiyang Gao (Harvard) and Yibo Gao (Michigan).

Prym varieties, named after Friedrich Prym, are abelian varieties constructed from etale double covers of algebraic curves. In 1985, Welters equipped Prym varieties with Brill-Noether loci. The Prym-Brill-Noether loci can be considered as certain Schubert varieties of type D. In this talk, we give formulas for the Euler characteristics of the Brill-Noether locus in Prym varieties with special vanishing at one point.

A spline is an assignment of polynomials to the vertices of a polynomial-edge-labeled graph, where the difference of two vertex polynomials along an edge must be divisible by the edge label. The ring of splines is a combinatorial generalization of the GKM construction for equivariant cohomoloy rings of flag, Schubert, Hessenberg, and permutohedral varieties. We consider spline rings where the underlying graph is the Cayley graph of a symmetric group generated by an arbitrary collection of transpositions. In this talk, we will give an example of when this ring is not a free module over the polynomial ring, and give a connectivity condition that precisely describes when particular graded pieces are generated by equivariant Schubert classes.

We compute the leading monomial of the top-degree components of Grothendieck and Lascoux polynomials using a uniform combinatorial construction we call snow diagrams.

Lascoux polynomials are K-theoretic generalizations of the key polynomials. The top-degree components of Grothendieck polynomials, also known as Castelnuovo-Mumford polynomials, have been studied by Pechenik, Speyer, and Weigandt. For any permutation w, they define a statistic rajcode(w), which gives the leading monomial of these top Grothendieck polynomials. We introduce a rajcode statistic on any diagram D through its snow diagram, which augments and decorates D. When D is the Rothe diagram of a permutation w, rajcode(D) agrees with the aforementioned rajcode(w). When D is the key diagram of a weak composition, rajcode(D) yields the leading monomial of the top Lascoux polynomial. We also describe the Hilbert series of the space spanned by these "top polynomials," which leads to a natural algebraic interpretation of a classical q-analog of Bell numbers after applying rook-theoretic results of Garsia and Remmel. This is joint work with Yu in arXiv:2302.03643.

I will discuss a conjectured generalization of the Whitney presentation for the (equivariant) quantum K ring of Grassmannians by Gu, Mihalcea, Sharpe, and Zou to all partial flag manifolds  and report on progress in proving various specializations of it. This presentation arises from realizations of partial flag manifolds as Gauged Linear Sigma Models and highlights the structure of these manifolds as towers of Grassmann bundles. This is joint work with Gu, Mihalcea, Sharpe, Zhang, and Zou.