Abstract

We describe a log-concavity property of the Littlewood--Richardson numbers and explain its connection to the theory of Lorentzian polynomials.

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Abstract

We will investigate in the simplest setting, how an``R-matrix'' (the building block of ``quantum integrable systems'') is attached to the equivariant cohomology of Grassmannians. We will compute the R-matrix in the case of CP^1 and discuss how the result generalizes to arbitrary Grassmannians. As an application, we shall derive the AJSBilley formula (restriction of Schubert classes to fixed points).

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Abstract

Springer fibers are a family of varieties that have remarkable connections to representation theory and combinatorics. Springer constructed an action of the symmetric group on the cohomology ring of a Springer fiber, and used it to geometrically construct the Specht modules (in type A), which are the irreducible representations of the symmetric group. In this talk, I will survey some of the many nice properties of Springer fibers. I will then introduce a new family of varieties generalizing the Springer fibers, and show how they are connected to the (recently proved) Delta Conjecture from algebraic combinatorics. We’ll then use these varieties to geometrically construct the induced Specht modules. This is joint work with Jake Levinson and Alexander Woo.

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Abstract

We say that two permutations w and v have separated descents at position k if w has no descents before k and v has no descents after k. We give a counting formula in terms of reduced word tableaux for computing the structure constants of products of Schubert polynomials indexed by permutations with separated descents. This generalizes previous results by Kogan '00, rediscovered using different methods by Knutson-Yong '04, Lenart '10, and Assaf '17, that solved special cases of this separated descent problem where one of the permutations is required to have a single descent. Our approach uses generalizations of Schutzenberger's jeu de taquin and the Edelman-Greene correspondence via bumpless pipe dreams.

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Abstract

Schubert varieties in the full flag variety of Kac-Moody type are indexed by elements of the corresponding Weyl group. In this talk, I will discuss recent work with William Slofstra where we give a practical criterion for when two such Schubert varieties (from potentially different flag varieties) are isomorphic, in terms of the Cartan matrix and reduced words for the indexing Weyl group elements. As a corollary, we show that two such Schubert varieties are isomorphic if and only if there is an isomorphism between their integral cohomology rings that preserves the Schubert basis. As an application, we show that the isomorphism classes of Schubert varieties in a given flag variety are controlled by graph automorphisms of the Dynkin diagram.

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I'll present a new result with A. Mellit, which gives a combinatorial formula for a diagonalizing operator for the modified Macdonald polynomials, known as the nabla operator. This formula was discovered by searching for a Schubert-type basis of a certain explicit module from Haiman's polygraph theory, which is identified with both the matrix elements of this operator, and the equivariant homology of the unramified affine Springer fiber studied by Goresky, Kottwitz, and Macpherson.

Abstract

I will present a definition and some important properties of the bosonic operators for back-stable Schubert polynomials. The operators act on the left weak Bruhat order (divided difference and Monk’s rule use the right side action on permutations in my notations). These operators with an extra condition give sufficiently enough linear equations for the structure constants of flag varieties. In particular, they provide a recurrent formula for the structure constants. In some special cases it is easy to check the positivity of the structure constants using this formula, examples will be presented. One of the advantages of our method is that we do not need to use formulas for Schubert polynomials and back-stable Schubert polynomials. Nevertheless if time permits, I will also show how to establish the pipe dreams formula using these operators.

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Abstract

The Abelian/non-Abelian correspondence is a powerful tool that can be used to study GIT quotients V//G, where V is a vector space. Such GIT quotients include type A flag varieties and quiver flag varieties. The principle of the Abelian/non-Abelian correspondence is that a GIT quotient V//G can be studied by considering the much simpler Abelian GIT quotient V//T, where T is maximal torus of G. I'll discuss applications of the Abelian/non-Abelian correspondence to quantum cohomology and mirror symmetry of type A flag varieties and quiver flag varieties, focusing on rim-hook removal rules and Plücker coordinate mirrors. Part of this talk will report on joint work with Wei Gu.

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Abstract

Peterson varieties are certain singular subvarieties of flag manifolds, naturally admitting one-dimensional torus action. Starting with a natural basis for the equivariant homology of a Peterson variety, we construct a dual basis in cohomology and show that the structure constants of the cohomology ring are positive with respect to this basis. We also discuss the sense in which the fundamental classes of the Peterson varieties exhibit a stability analogous to the stability of Schubert classes, and how this can be used to streamline various calculations in the Schubert calculus of Peterson varieties. This is joint work with Rebecca Goldin and Leonardo Mihalcea.

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