The Prism tableau model for Schubert polynomials

TL;DR

This paper introduces the prism tableau model for Schubert polynomials, providing a new combinatorial framework that directly involves semistandard tableaux and is grounded in Groebner geometry of matrix Schubert varieties.

Contribution

It presents a novel prism tableau model that directly incorporates semistandard tableaux, offering an alternative perspective to existing models for Schubert polynomials.

Findings

Prism tableau model effectively represents Schubert polynomials.

In the Grassmannian case, prism tableaux reduce to semistandard Young tableaux.

Model is developed using Groebner geometry of matrix Schubert varieties.

Abstract

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.