The Prism tableau model for Schubert polynomials

The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1,x2,...]. We suggest the "prism tableau model" for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Groebner geometry of matrix Schubert varieties.Comment: 23 page

Bumpless pipe dreams encode Gr\"obner geometry of Schubert polynomials

In their study of infinite flag varieties, Lam, Lee, and Shimozono (2021) introduced bumpless pipe dreams in a new combinatorial formula for double Schubert polynomials. These polynomials are the TxT-equivariant cohomology classes of matrix Schubert varieties and of their flat degenerations. We give diagonal term orders with respect to which bumpless pipe dreams index the irreducible components of diagonal Gr\"obner degenerations of matrix Schubert varieties, counted with scheme-theoretic multiplicity. This indexing was conjectured by Hamaker, Pechenik, and Weigandt (2022). We also give a generalization to equidimensional unions of matrix Schubert varieties. This result establishes that bumpless pipe dreams are dual to and as geometrically natural as classical pipe dreams, for which an analogous anti-diagonal theory was developed by Knutson and Miller (2005).Comment: In the original, we made an error in a claimed reduction from arbitrary diagonal term orders. We now only address term orders compatible with droop moves. The case of an arbitrary diagonal term order remains open. We are very grateful to Kuei-Nuan Lin and Yi-Huang Shen for alerting us to a flipped inequality in Lemma 5.10. The proof has been amended. Other more minor improvements throughou

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