Prism tableaux and alternating sign matrices
A. Lascoux and M.-P. Schutzenberger introduced Schubert polynomials to study the cohomology ring of the complete flag variety Fl(C^n). Each Schubert polynomial corresponds to the class defined by a Schubert variety X_w in Fl(C^n). Schubert polynomials are indexed by elements of the symmetric group and form a basis of the ring Z[x1,x2,...]. The expansion of the product of two Schubert polynomials in the Schubert basis has been of particular interest. The structure coefficients are known to be nonnegative integers. As of yet, there are only combinatorial formulas for these coefficients in special cases, such as the Littlewood-Richardson rule for multiplying Schur polynomials. Schur polynomials form a basis of the ring of symmetric polynomials. They have a combinatorial formula as a weighted sum over semistandard tableaux. In joint work with A. Yong, the author introduced prism tableaux. A prism tableau consists of a tuple of tableaux, positioned within an ambient grid. With A. Yong, the author gave a formula for Schubert polynomials using prism tableaux. We continue the study of prism tableaux, detailing their connection to the poset of alternating sign matrices (ASMs). Schubert polynomials can be interpreted as multidegrees of the matrix Schubert varieties of Fulton. We study a more general class of determinantal varieties, indexed by ASMs. More generally, one can consider subvarieties of the space of n by n matrices cut out by imposing rank conditions on maximal northwest submatrices. We show that, up to an affine factor, such a variety is isomorphic to an ASM variety. The multidegrees of ASM varieties can be expressed as a sum over prism tableaux. In joint work with A. Yong and R. Rimanyi, the author studies representations of quivers and their connection to the dilogarithm identities of M. Reineke. We give a bijective proof to establish an identity of generating series. This bijection uses a generalization of Durfee squares. From this identity, we give a new proof of M. Reineke's identities in type A.
Schubert polynomials, Alternating sign matrices, Durfee, Prism tableaux
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign