New interactions between Geometry and Combinatorics

Talks

Anderson, David

Title: Old and new formulas for degeneracy loci

Abstract: Loci defined by rank conditions on matrices are ubiquitous in algebraic geometry, and formulas for their degrees go back to work of 19th century geometers. These formulas evolved into Schubert polynomials, which represent classes of Schubert varieties in generalized flag varieties, and have been studied by many 21st century mathematicians.

Focusing on symmetric matrices and symplectic flag varieties, I will describe some new formulas for these loci, together with connections to combinatorics and infinite-dimensional flag varieties. Along the way, I will mention some of my contributions in joint work with W. Fulton, and with T. Ikeda, M. Jeon, and R. Kawago.

Harada, Megumi

Title: The cohomology rings of Hessenberg varieties

Abstract: Hessenberg varieties are a family of subvarieties of the flag variety which arise in the intersection of algebraic geometry, representation theory, and combinatorics. In this talk I will briefly introduce Hessenberg varieties, and survey recent progress on two questions concerning their cohomology rings, namely: (1) the Stanley-Stembridge conjecture from combinatorics, and (2) the construction of monomial bases. This is joint work with T. Horiguchi, S. Murai, M. Precup, and J. Tymoczko.

Hudson, Thomas

Title: The Bredon Z/2-equivariant cohomology of complex projective spaces

Abstract: Unlike Borel equivariant cohomology, which has been widely employed in connection to Schubert calculus, the study of the Bredon cohomology of homogeneous spaces is at a very rudimentary stage. It was only recently that this theory was endowed with Euler classes and pushforward maps by the work of Costenoble and Waner. In this talk I will give an overview of the different extensions of Bredon cohomology and discuss the case of finite and infinite complex projective spaces with a Z/2-action. (Joint with S. Costenoble and S. Tilson).

Marberg, Eric

Title: K-theory formulas for orthogonal and symplectic orbit closures

Abstract: The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. This talk will be about certain polynomials introduced by Wyser and Yong to represent the K-theory classes of the closures of these orbits. These polynomials have many interesting properties and are analogous to the Grothendieck polynomials representing K-classes of Schubert varieties. In special cases, one can use general degeneracy locus formulas of Anderson to derive explicit Pfaffian expressions for these K-theory representatives. Moreover, by taking an appropriate limit one recovers the K-theoretic Schur P- and Q-functions of Ikeda and Naruse. While the symplectic case is fairly well understood, there are several open problems of interest related to such "orthogonal" Grothendieck polynomials. The results in this talk are joint work with Brendan Pawlowski.

Nakagawa, Masaki

Title: Gysin Formulas for generalized Hall-Littlewood functions and related topics

Abstract: In this talk, I will make a survey of our recent work jointly with Prof. Hiroshi Naruse over the last five years: In 2014, P. Pragacz (arXiv:1403.0788) established a Gysin formula for Hall-Littlewood polynomials, generalizing some Gysin formulas for Schur S- and P-polynomials. This formula is the starting point of our current work. Our primary concern was to generalize the above formula in the ordinary cohomology theory, to the complex cobordism theory, which is universal among all complex-oriented generalized cohomology theories. More precisely, we have introduced a universal analogue of the ordinary Hall-Littlewood polynomials, called the universal Hall-Littlewood functions, and given analogous Gysin formulas for these functions in complex cobordism (arXiv:1604.00451). In the meantime, L. Darondeau and P. Pragacz (arXiv:1510.07852) have given remarkably compact and handy Gysin formulas for all flag bundles associated to vector bundles of types A, B, C and D. Their formulas are expressed in terms of the Segre classes of given vector bundles. Then our next goal was to generalize these “Darondeau-Pragacz formulas” in cohomology to complex cobordism. Segre classes for algebraic cobordism introduced by T. Hudson and T. Matsumura (arXiv:1602.05704) and the classical Quillen’s residue formula (1969) in complex cobordism made it possible for us to accomplish such a generalization (arXiv:1910.03649). In the course of our study, we have introduced another type of universal analogues of the ordinary Hall-Littlewood polynomials, called the universal factorial Hall-Littlewood P- and Q-functions. As an application of our Darondeau-Pragacz formulas, we gave generating functions for the universal factorial Hall-Littlewood P- and Q-functions (arXiv:1705.04791). Using generating functions, classical determinantal and Pfaffian formulas for Schur S- and Q-polynomials, and their K-theoretic analogues can be obtained in a simple and unified manner. Furthermore, as related topics, we shall discuss generalizations of the quadratic Schur functions due to Darondeau-Pragacz, the Pragacz-Ratajski theorem, the Gustafson-Milne type identity, and iterated residue formulas due to M. Zielenkiewicz.

Naruse, Hiroshi

Title: Generalized colored hook formula

Abstract: We consider a generalization of both Nakada's colored hook formula for a weighted sum of hook removal sequences and Gansner type hook formula for generating function of reverse plane partitions.For the Grassmannian case and finite Weyl group case we have a solution to this problem by using equivariant motivic Chern classes, and for general Kac-Moody case we can expect a similar formula.This is joint work in progress with L. Mihalcea and C. Su.

Okada, Soichi

Title: Generalization of Schur's Q-functions

Abstract: Schur's Q-functions are a family of symmetric functions introduced by Schur in 1911, and they appear in various situations parallel to Schur functions. In this talk, we give a generalization of Schur's P/Q-functions, and present several formulas and properties. This generalization can be viewed as "Macdonald's ninth variation" for P-/Q-functions, and includes as special cases Schur's P-/Q-functions, Ivanov's factorial P-/Q-functions and the t=-1 specialization of Hall-Littlewood functions associated to the classical root systems. We conclude with some discussion of a symplectic analogue of Schur's Q-functions.

Scrimshaw, Travis

Title: Crystallizing K-Theory

Abstract: A well studied object is the Grassmannian, which is known as Schubert calculus, and more modern approaches have looked at studying its K-theory ring. This can be modeled by Geothendieck polynomials, whose stable limits have positive Schur expansions. This suggests that they can be studied using Kashiwara's crystal theory. In this talk, we will survey some recent results of how crystals appear in Schubert calculus and discuss some open problems and questions.

Shimozono, Mark

Talk 1: Parabolic Hall-Littlewood polynomials and Kostka-Shoji polynomials.

Abstract: For any quiver we define parabolic Hall-Littlewood symmetric functions as equivariant K-classes of vector bundles on Lusztig's convolution diagram, which is a vector bundle over a product of partial flag varieties. Special cases yield Shoji's polynomials, wreath Macdonald polynomials with q set to zero appropriately, and a generalization of Kirillov-Reshetikhin characters of affine sl_n. This is joint with Dan Orr.

Talk 2: On the coproduct in affine Schubert calculus

Abstract: The cohomology of the affine flag variety of a complex reductive group is a comodule over the cohomology of the affine Grassmannian. We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite Schubert classes, in torus-equivariant cohomology and K-theory. As an application, we deduce monomial positivity for Seungjin Lee's affine Schubert polynomials. This is joint with Thomas Lam and Seungjin Lee.

Su, Changjian

Title: Maulik--Okounkov stable envelopes and Schubert calculus

Abstract: The stable envelopes are introduced by Maulik and Okounkov for symplectic resolutions. We will survey the recent developments about the stable envelopes for the cotangent bundle of the flag varieties, and their relations to Schubert calculus. In the first talk, we will talk about the stable envelopes in cohomology and relate them to Chern-Schwartz-MacPherson classes of Schubert cells. This leads to a proof of the positivity conjecture of Aluffi and Mihalcea. In the second talk, we will move to stable envelopes in K theory, and relate them to unramified principal series of the p-adic Langlands dual groups. In the last talk, we will introduce the motivic Chern classes, and prove conjectures of Bump, Nakasuji, and Naruse about the unramified principal series. This is based on various joint works with P. Aluffi, L. Mihalcea J. Schurmann, G. Zhao and C. Zhong.

Sugimoto, Shogo

Title: Factorial Flagged Grothendieck Polynomials

Abstract: Lascoux-Schutzenberger introduced double Grothendieck polynomials to represent the K-theory classes of the structure sheaves of Schubert varieties. Knutson-Miller-Yong expressed Grothendieck polynomials associated to vexillary permutations as the generating functions of flagged set-valued tableaux. On the other hand, Hudson-Matsumura proved the Jacobi-Trudi formula for double Grothendieck polynomials associated to vexillary permutations. In this talk I will talk about generalizations of these formulas. In particular, I will prove that the factorial flagged Grothendieck polynomials defined by set-valued tableaux can be expressed in a Jacobi-Trudi type determinant formula.