Abstracts for AMS Special Session The Modern Schubert Calculus 2004 Fall AMS Central Section Meeting Northwestern University, Evanston 23-24 October 2004.

We define a new ("smaller") product on the cohomology of projective homogenous spaces G/P based on a geometric degeneration and show that it this product which is relevant to Eigenvalue problems and the Geometric Horn problem. We exhibit a relation of this product to Lie algebra cohomology. We also obtain two (a priori) different sets of necessary recursive conditions on when a cohomology product of Schubert cycles in a G/P is non-zero (the Geometric Horn problem).
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We describe the T-equivariant cohomology of the Quot scheme compactifying the space of degree d maps from P¹ to the Grassmannian Gr(r,n), where T is the product of the natural torus acting on the Grassmannian with a c^* acting on P¹. The calculation is by equivariant localization. The one-dimensional orbits are not isolated, but we can describe explicitly the relations coming from each connected family of one-dimensional orbits, since the closure of each such family is a product of projective spaces.

This is joint work with Linda Chen and Frank Sottile.
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In this talk I will discuss new Littlewood-Richardson rules in terms of combinatorial objects called Mondrian tableaux. These rules are obtained using degeneration techniques.
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We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real whenever the data are real. For other similar problems we prove non-trivial lower bounds for the number of real solutions. Our arguments explore the connection between subspaces of codimension 2 and rational functions of one variable. Part of these results is our joint work with M. Shapiro and A. Vainshtein.
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Explicit computations of equivariant cohomology rings have many applications. In 1998, Goresky, Kottwitz, and MacPherson showed that for certain spaces with a torus action, the equivariant cohomology ring can be explicitly described by combinatorial data obtained from its orbit decomposition. We generalize their theorem to the (possibly infinite-dimensional) setting of cell complexes. These results include many new examples, including homogeneous spaces of a loop group LG.
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Mirkovic-Vilonen showed that certain subvarieties of the affine Grassmanian, called Mirkovic-Vilonen cycles, give bases for representations of complex semisimple groups. Anderson observed that to each MV cycle, it is possible to associate its moment map image, called a Mirkovic-Vilonen polytope. He showed that these polytopes can be used to count tensor product multiplicities.\par Here, we give a uniform description of the MV cycles and polytopes for all complex semisimple groups. Our description is in terms of the combinatorics developed by Berenstein-Zelevinsky in their tensor product multiplicities paper. However, our work does not rely on their results and it gives a new proof of their tensor product multiplicity formula.
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Misha Kogan selected a strange-looking subset of the ``Schubert polynomial times Schur polynomial'' class of problems: the Schubert polynomial shouldn't mention any of the variables the Schur doesn't. In particular, it includes the usual Littlewood-Richardson problem, in which the Schubert polynomial is another Schur polynomial, in the same number of variables.

We use the transition formula for Grothendieck polynomials (which is given a geometric explanation in Alex Yong's talk) to give a satisfying explanation of the naturality of Kogan's condition, while giving a simpler derivation, and one which extends to K-theory (but not equivariantly!).

Our rule is in terms of ``marching moves'' on the diagram of a permutation.
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We present new Chevalley-type and Pieri-type multiplication formulas in the T-equivariant K-theory of generalized flag varieties G/P. By these, we mean formulas for multiplying arbitrary Schubert classes in equivariant K-theory, on the one hand, with classes of certain line bundles, and Schubert classes indexed by simple reflections, on the other hand. The construction is given in terms of decompositions of a fixed affine Weyl group element, and saturated chains in the Bruhat order on the (nonaffine) Weyl group. Our model has certain advantages over the Littelmann path model, on which a Chevalley-type formula due to Pittie and Ram is based. As an application, we are able to give simple proofs of certain symmetries of the coefficients in the Chevalley-type formula, which are difficult to derive by other methods. This is a joint work with Alexander Postnikov. We also discuss the way in which our model leads to a more general multiplication formula (by certain Schubert classes pulled back from a Grassmannian projection) in the K-theory of the type A flag variety. The latter formula was obtained in collaboration with Frank Sottile.
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The (small) equivariant quantum cohomology (eq.q.coh.) of a (smooth) variety X is an algebra which is a deformation of both equivariant and quantum cohomology algebras of X.

In this talk I will present two properties of the equivariant quantum cohomology of the Grassmannian which extend from its equivariant restriction. One is that there is a certain recurrence relation, which is implied by the equivariant quantum Pieri rule (i.e. the multiplication with the divisor class), and which determines completely the eq.q. multiplication.

nd is a positivity property of the structure constants of the equivariant quantum cohomology, which are certain polynomials (the 3-point, genus 0, equivariant Gromov-Witten invariants, introduced by Givental and Kim). This positivity holds for any homogeneous space G/P and generalizes the equivariant positivity conjectured by Peterson and proved by Graham.
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We discuss the question of transversality of Schubert cycles which are not general, but are associated to osculating flags at general points of the rational normal curve. This question may be rephrased in terms of maps from the projective line to projective spaces with prescribed ramification, and in this context it is natural to ask the same question for higher-genus curves as well. We give a simple degeneration argument using the theory of limit linear series to reduce the problem to the case of three ramification points on the projective line. We also discuss the consequences of this argument for reality of the maps in question.
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We study a family of polynomials whose values express degrees of Schubert varieties in the generalized flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. We specialize the results to the classical (type A) flag manifold and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, Stanley-Pitman polytopes, parking functions, binary trees, and the inverse extended Kostka matrix. The talk is based on a joint work with Richard Stanley.
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Horn's conjecture as classically stated, is a statement about the possible eigenvalues of triples of Hermitian matrices (A,B,C) satisfying A+B+C=0. Restated however, it can be viewed as a recursive construction of the set of non-vanishing Littlewood-Richardson numbers. We prove a generalisation of Horn's conjecture, which recursively characterises the non-vanishing Schubert intersection numbers for all minuscule flag varieties.
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Giambelli's formula computes the class of an orbit closure in the GL(n) \times GL(p)-equivariant cohomology of HOM(Cⁿ,C^p). In this talk we will present some new results in two different natural generalizations. (1) Quiver representations occur when we consider direct sums of HOM spaces---instead of just one of them---arranged with respect to a diagram. We will show how a simplification of earlier work (by A. Buch, L. Feher, R.R.), gives the sought equivariant class formulas in the A_n quiver case (arbitrarily oriented) [joint work with A. Buch]. (2) Instead of linear maps we can consider holomorphic germs from Cⁿ to C^p. Then generalizations of the Giambelli formula are called the Thom polynomials of singularities. We will present a so far hidden structure of these Thom polynomials [joint work with L. Feher].
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The Shapiro conjecture in the real Schubert calculus fails to hold for flag manifolds, but in a very interesting way. We give a refinement of that conjecture for the flag manifold and present massive experimentation that supports this conjecture. We also prove many special cases using discriminants and establish relationships between the different cases of the conjecture.
    This is joint work with Jim Ruffo, Yuval Sivan, and Frank Sottile.
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The multiplicative version of Horn's problem asks what the possible critical values of three matrices A, B and C are, given ABC=1. A Vinnikov curve is a projective plane curve which can be written as det(xX+yY+zZ)=0 with X, Y and Z positive definite Hermitian matrices; Vinnikov has solved the problem of classifying such curves. We will relate these two problems to each other and pose tropical versions of them. This allows us to reprove Knutson and Tao's criterion for the solvability of Horn's problem in terms of honeycombs and illuminate the origins of honeycombs.
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Let G be a classical Lie group and P be a maximal parabolic subgroup of G. The homogeneous space X=G/P is a Grassmannian which parametrizes subspaces in affine space which are isotropic with respect to a nondegenerate symmetric or skew symmetric bilinear form. We will discuss certain aspects of our work on the structure of the classical and small quantum cohomology ring of X; this is a joint project with Anders Buch and Andrew Kresch.
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Subvarieties of the flag variety defined by certain linear conditions arise naturally in many areas, including geometric representation theory, number theory, and numerical analysis. These subvarieties, called Hessenberg varieties, are naturally paved by the intersection of a Bruhat decomposition with the Hessenberg variety. In many cases, the Bruhat decomposition can be chosen so that each Schubert cell intersects the Hessenberg variety in an affine cell. We discuss properties of this cell decomposition. In particular, we show that the Poincare polynomials of a family of Hessenberg varieties factor in a way that generalizes the exponents of the group. Part of these results are joint work with E. Sommers.
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Enumerative problems in geometry often have groups attached, which have different interpretations in geometry, arithmetic, and algebra. The geometric interpretation is as follows: if the problem has N solutions, and the conditions are moved about and returned to their starting positions, how can the N solutions be permuted? Schubert problems are the only known enumerative problems that can have unexpectedly small Galois groups; the first such example is due to Derksen. By solving Schubert problems over the rational numbers (and using Schubert induction) we can compute these monodromy groups explicitly; we give examples where it is the full symmetric group, and where it seems to be mysteriously small. This is part of a larger project with Sara Billey.
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The pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant, so that the resulting closed-loop system has a desired set of eigenvalues. In this talk, we first show the connection between dynamic output feedback pole placement and enumerative geometry. Then we will present the realization of the output of the Pieri Homotopies as a useful control feedback machine in the time domain, which will involve numerical calculations of greatest common divisor. An application will be presented to illustrate real feedback laws may be found with dynamic feedback method when all the static feedback laws have nonzero imaginary coefficients. We implemented a parallel version of Pieri Homotopy algorithm and observed the algorithm is suitable for parallel computing.
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We geometrically interpret Alain Lascoux's transition formula for Grothendieck polynomials. For a permutation \pi thought of as a permutation matrix, let X_\pi be the closure (in the set of complex n-by-n-matrices) of the double coset B_- \pi B₊. This is a matrix Schubert variety. We construct a flat family whose general fiber is isomorphic to X_\pi and whose special fiber is reduced and Cohen-Macaulay. The components of the limit are also matrix Schubert varieties (one of which is intersected with an ``irrelevant'' coordinate hyperplane). We relate this geometry to some subtraction-free formulae (both new and old) from the study of (K-theory) Schubert calculus and degeneracy loci; see Allen Knutson's talk. For example, we deduce another geometric interpretation of the classical Littlewood-Richardson rule, as counting components of a partially degenerated matrix Schubert variety.
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