Mathematics Department - Algebra Seminar - Spring 2017

# Algebra Seminar - Spring 2017

Charles Weibel

### Website

http://www.math.rutgers.edu/~weibel/algebra.seminar.html

## Wednesday, April 26th

### "Equivariant quantum K-theory of projective space"

Time: 2:00 PM
Location: Hill 705
Abstract: Recent developments of Buch-Chaput-Mihalcea-Perrin have allowed for a closer look at the quantum K-theory of cominuscule flag varieties. For example, their Chevalley formula allows one to compute quantum K-theoretic products involving Schubert divisor classes. In the special case of projective space, one can extend this Chevalley formula to describe products of arbitrary Schubert classes. We shall discuss this extension along with some of its potential combinatorial and representation-theoretic consequences.

## Wednesday, April 19th

### "Puzzles in quantum Schubert Calculus "

Time: 2:00 PM
Location: Hill 705
Abstract: The cohomology ring of a flag variety has a natural basis of Schubert classes. The multiplicative structure constants with respect to this basis count solutions to enumerative geometric problems; in particular they are non-negative. For example, the structure constants of a Grassmannian are the classical Littlewood-Richardson coefficients, which show up in numerous branches of mathematics. I will speak about a new puzzle-counting formula for the structure constants of 3-step partial flag varieties that describes products of classes that are pulled back from 2-step flag varieties. By using a relation between quantum cohomology of Grassmannians and classical cohomology of 2-step flag varieties, this produces a new combinatorial formula for the (3 point, genus zero) Gromov-Witten invariants of Grassmannians, which is in some sense more economical than earlier formulas.

## Wednesday, April 5th

### "Kirillov-Reshetikhin modules and Macdonald polynomials: a survey and applications "

Time: 2:00 PM
Location: Hill 705
Abstract: This talk is largely self-contained.

In a series of papers with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, we developed a uniform combinatorial model for (tensor products of one-column) Kirillov-Reshetikhin (KR) modules of affine Lie algebras. We also showed that their graded characters coincide with the specialization of symmetric Macdonald polynomials at t=0, and extended this result to non-symmetric Macdonald polynomials. I will present a survey of this work and of the recent applications, which include: computations related to KR crystals, crystal bases of level 0 extremal weight modules, Weyl modules (local, global, and generalized), q-Whittaker functions, and the quantum K-theory of flag varieties.

## Wednesday, March 22nd

### Ilya Kapovich, Rutgers University

Time: 2:00 PM
Location: Hill 705
Abstract: We develop a counterpart of the Thurston-Fried-McMullen "fibered face" theory in the setting of free-by-cyclic groups, that is, mapping tori groups of automorphisms of finite rank free groups. We obtain information about the BNS invariant of such groups, and construct a version of McMullen's "Teichmuller polynomial" in the free-by-cyclic context.

The talk is based on joint work with Chris Leininger and Spencer Dowdall.

## Wednesday, March 8th

### "Decompositions of Grothendieck polynomials"

Time: 2:00 PM
Location: Hill 705
Abstract: Finding a combinatorial rule for the Schubert structure constants in the K-theory of flag varieties is a long-standing problem. The Grothendieck polynomials of Lascoux and SchÃ¼tzenberger (1982) serve as polynomial representatives for K-theoretic Schubert classes, but no positive rule for their multiplication is known outside of the Grassmannian case. We contribute a new basis for polynomials, give a positive combinatorial formula for the expansion of Grothendieck polynomials in these "glide polynomials", and provide a positive combinatorial Littlewood-Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. A specialization of the glide basis recovers the fundamental slide polynomials of Assaf and Searles (2016), which play an analogous role with respect to the Chow ring of flag varieties. Additionally, the stable limits of another specialization of glide polynomials are Lam and Pylyavskyy's (2007) basis of multi-fundamental quasisymmetric functions, K-theoretic analogues of I. Gessel's (1984) fundamental quasisymmetric functions. Those glide polynomials that are themselves quasisymmetric are truncations of multi-fundamental quasisymmetric functions and form a basis of quasisymmetric polynomials.

(Joint work with D. Searles).

## Wednesday, March 1st

### "The Witt group of surfaces and 3-folds"

Time: 2:00 PM
Location: Hill 705
Abstract: If V is an algebraic variety, the Witt group is formed from vector bundles equipped with a nondegenerate symmetric bilinear form. When it has dimension $<4$, it embeds into the more classical Witt group of the function field (Witt 1934). When V is defined over the reals, versions of the discriminant and Hasse invariant enable us to determine W(V).

## Wednesday, February 22nd

### "Equivariant Quantum Cohomology of the Odd Symplectic Grassmannian "

Time: 2:00 PM
Location: Hill 705
Abstract: The odd symplectic Grassmannian IG:=IG(k, 2n+1) parametrizes k dimensional subspaces of C2n+1 which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on IG with two orbits, and IG is itself a smooth Schubert variety in the submaximal isotropic Grassmannian IG(k, 2n+2). We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of IG, i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case k=2, and it gives an algorithm to calculate any quantum multiplication in the equivariant quantum cohomology ring.

The current work is joint with L. Mihalcea.

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