Inversions Tableaux

TL;DR

This paper introduces inversions tableaux, a new combinatorial model for Schubert polynomials and Stanley symmetric functions, connecting them to semi-standard Young tableaux and exploring their properties and transformations.

Contribution

It presents inversions tableaux as a novel combinatorial framework, extending existing models and setting the stage for proving Rubey's chute moves conjecture.

Findings

Explicit description of inversions tableaux for minimal and maximal monomials

Characterization of the unique inversions tableau for dominant permutations

Analysis of generalized chute moves on inversions tableaux

Abstract

We introduce inversions tableaux, a new combinatorial model for Schubert polynomials and Stanley symmetric functions that directly specializes to semi-standard Young tableaux in the Grassmannian case. They are a modification of the balanced staircase tableaux of Edelman and Greene. We explicitly describe inversions tableaux that correspond to the lexicographically minimal and maximal monomials in each Schubert polynomial and characterize the unique inversions tableau for dominant permutations. We also characterize the action of generalized chute moves on inversions tableaux, and establish related background that will be used to prove Rubey's chute moves conjecture in upcoming work.