MAAGC 2019 ABSTRACTSDave Anderson (Ohio State University):Title: Old formulas for degeneracy loci, with a new twistAbstract: A basic problem from the 19th century asks for the degree of the locus of symmetric matrices of bounded rank; answers were given by Schubert and Giambelli. More recently, many extensions of this problem have been considered, including versions coming from symmetric maps of vector bundles, or for vector bundles equipped with a nondegenerate bilinear form. I will discuss ongoing work with William Fulton, in which we allow the bilinear form to be “twisted”, so that it takes values in a nontrivial line bundle. The formulas we obtain extend those of Billey-Haiman, Ikeda-Mihalcea-Naruse, and others, and exhibit new connections with algebraic combinatorics. Karola Meszaros (Cornell and IAS):Title: The many aspects of Schubert polynomialsAbstract: Schubert polynomials, introduced by Lascoux and Sch ützenberger in 1982, represent cohomology classes of Schubert cycles in flag varieties. While there are a number of combinatorial formulas for Schubert polynomials, their supports have only recently been established and the values of their coefficients are not well understood. We show that the Newton polytope of a Schubert polynomial is a generalized permutahedron and explain how to obtain certain Schubert polynomials as projections of integer point transforms of polytopes. The latter generalizes the well-known relationship between Schur functions and Gelfand-Tsetlin polytopes. We will then turn to the study of the coefficients of Schubert polynomials and show that Schubert polynomials with all coefficients at most k, for any positive integer k, are closed under pattern containment. We also characterize zero-one Schubert polynomials by a list of twelve avoided patterns. This talk is based on joint works with Alex Fink, Ricky Liu and Avery St. Dizier. Vic Reiner (University of Minnesota):Title: On the "coincidental" reflection groupsAbstract: Much combinatorics these days involves finite reflection groups, both real and complex. In this talk, we will focus on results that work out especially nicely for what have been called the "coincidental" reflection groups. These are the groups generated by n reflections acting on n-dimensional space whose exponents form an arithmetic sequence. They include the real reflection groups of types A, B, H_3, dihedral groups, and all of the non-real reflection groups that are symmetries of regular complex polytopes, which are known as Shephard groups. In particular, the coincidental groups have some extra elegant invariant theory, leading to product formulas for the face numbers and h-vectors of their associated cluster complexes, and a q-analogue of the transformation taking the h-vector to f-vector. This is joint work with Anne Shepler and Eric Sommers. Michelle Wachs (Miami):Title: Combinatorial aspects of the homogenized Linial arrangementAbstract: The homogenized Linial arrangement is a hyperplane arrangement recently introduced by Gabor Hetyei, who used the finite field method of Athanasiadis to show that the number of regions of the arrangement is a median Genocchi number. (These numbers count a class of permutations known as Dumont derangements.) Here, we take a different approach, which makes direct use of Zaslavsky's formula relating the intersection lattice of an arrangement to its number of regions. This approach enables us to refine Hetyei's result by providing a combinatorial interpretation of the M\”obius function of this lattice in terms of variants of the Dumont permutations. This interpretation leads to a formula for the generating function of the characteristic polynomial of the arrangement, which reduces to formulas of Barsky and Dumont for the Genocchi numbers and for the median Genocchi numbers. Our techniques also yield type B analogs of these results, and Dowling arrangement generalizations. This talk is based on joint work with Alex Lazar. |