Anders Skovsted Buch
Pieri formulas for the quantum K-theory of cominuscule Grassmannians
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.
William Graham
Positivity in weighted flag varieties
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.
Reuven Hodges
Levi-spherical Schubert varieties
I will present a root-system uniform, combinatorial classification of Levi-spherical Schubert varieties for any generalized flag variety G/B of finite Lie type. This will be applied to the study of multiplicity-free decompositions of a Demazure module into irreducible representations of a Levi subgroup.
Dennin Hugh
Bijective proofs of derivative formulas for Schubert polynomials
Recently, Gaetz and Gao extended a lowering operator $\nabla$ on weak order, first introduced by Stanley, to an $\mathfrak{sl}_2$ poset representation, thus proving the strong Sperner property of weak order. Hamaker, Pechenik, Speyer, and Weigandt later showed that $\nabla$ can be realized as a certain differential operator on Schubert polynomials which, in particular, gives a short proof of the Macdonald reduced word identity. In this talk, we give bijective proofs of this and related derivative identities for Schubert polynomials and $\beta$-Grothendieck polynomials using the combinatorics of pipe dreams.
Minyoung Jeon
Mather classes of Schubert varieties via small resolutions
The Chern-Mather class is a characteristic class, generalizing the Chern classes of tangent bundles of nonsingular varieties to singular varieties. It uses the Nash-blowup for singular varieties instead of tangent bundles. In this talk, we consider Schubert varieties, known as singular varieties in most cases, in the even orthogonal Grassmannians and discuss the work computing the Chern-Mather classes of the Schubert varieties by the use of the small resolutions of Sankaran and Vanchinathan with Jones’ technique. We also describe the Kazhdan-Lusztig classes of Schubert varieties in Lagrangian Grassmannians, as analogous results if time permitted.
Nathan Lesnevich
Splines on Cayley Graphs of the Symmetric Group
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.
Leonardo Mihalcea
Presentations of the quantum K theory ring of Grassmannians
The quantum K theory ring of a complex projective manifold X is a deformation of the ordinary Grothendieck ring of vector bundles on X, defined in the early 2000’s by Givental and Lee. In this talk I will discuss two presentations of the quantum K ring of Grassmannians by generators and relations: a `Whitney presentation’, generalizing the presentation of Witten in quantum cohomology, and a physics inspired `Coulomb branch presentation’, arising as a Jacobi ring of a certain twisted superpotential. This is joint work with Wei Gu, Eric Sharpe, and Hao Zou. If time permits, I will state a conjectural presentation for partial flag manifolds, joint with the same collaborators along with Weihong Xu and Hao Zhang.
Jenna Rajchgot
Ladder determinantal ideals via Schubert varieties
Many classes of generalized determinantal ideals studied by the commutative algebra community are closely related to Schubert varieties. For example, classical determinantal ideals define patches of certain Schubert varieties in Grassmannians and one-sided ladder determinantal ideals are Schubert determinantal ideals associated to vexillary permutations.
In the first half of the talk, I’ll show how one can use combinatorial results in Schubert calculus to compute Castelnuovo-Mumford regularity for one-sided ladder determinantal ideals. This is joint work with C. Robichaux and A. Weigandt. Then I’ll discuss work in progress with L. Escobar, A. Fink, and A. Woo on how two-sided mixed ladder determinantal ideals in the ordinary, symmetric, and skew-symmetric settings can be interpreted in terms of Borel orbit closures in type A flag varieties, type C flag varieties, and certain symmetric varieties, respectively.
George Seelinger
K-theoretic Catalan functions
In 2008, Thomas Lam identified a family of symmetric functions known as k-Schur functions with the Schubert classes in the homology of the affine Grassmannian, in analogy with Schur functions serving as representatives for the (co)homology of the usual Grassmannian. Of additional interest, under an isomorphism between the quantum cohomology of the flag variety and the homology of the affine Grassmannian, known as the Peterson isomorphism, the quantum Schubert polynomials are sent to the k-Schur functions, up to suitable localization. Subsequently, much work had been done to carry out an analogous program in the K-theoretic generalization, but significant parts of the combinatorics of the symmetric function Schubert representatives remained elusive. In this talk, I will present how some new insights in the (co)homological setting enabled a K-theoretic refinement to give a direct understanding of some of the missing combinatorics surrounding the K-homology of the affine Grassmannian and the K-theoretic Peterson isomorphism.
Frank Sottile
A Murnaghan-Nakayama formula in quantum Schubert calculus
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.
Mihail Tarigradschi
Classifying cominuscule Schubert varieties up to isomorphism
Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are then indexed by posets of roots labeled long/short. These labeled posets generalize the Young diagrams that index Schubert varieties in Grassmannians. We discuss the question of how these posets determine the isomorphism class of a Schubert variety.
Rui Xiong
Automorphisms of the Quantum Cohomology of the Springer Resolution and Applications
In this talk, we introduce quantum Demazure-Lusztig operators acting on the equivariant quantum cohomology of the Springer resolution. Our main result is a presentation of the torus-equivariant quantum cohomology in terms of generators and relations. We also provide explicit descriptions for the classical types and recover earlier results for complete flag varieties.
Weihong Xu
A presentation for the quantum K ring of partial flag manifolds
We give 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 prove it for Fl(1, n-1, n). The 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. We also verify the specialization of this conjecture in quantum cohomology by comparing it with a presentation given by Gu and Kalashnikov using the abelian/non-abelian correspondence in mathematics. This is joint work with Gu, Mihalcea, Sharpe, Zhang, and Zou.
Changlong Zhong
Elliptic Schubert classes via the periodic Hecke module and its Langlands dual
Elliptic Schubert classes are defined by Rimanyi-Weber by introducing an extra parameter (dynamical parameter). They are generated by the so-called dynamical DL operators. These operators satisfy the braid relations, due to the independence of Bott-Samelson resolutions, a result followed from Borisov-Libgober. In this talk I will talk about a different approach in the study of these operators, that is, we use the polynomial representation and the periodic modules on which the dynamical DL operators act. They are vector bundles over the equivariant elliptic cohomology of a point. With that, I will talk about some interesting results about the elliptic classes and those of the Langlands dual flag variety. This is a work in progress of myself with C. Lenart and G. Zhao.