Tableau formulas for skew Schubert polynomials

TL;DR

This paper develops tableau formulas for skew Schubert polynomials across all classical Lie types, including symplectic and orthogonal cases, extending known formulas for type A and other specializations.

Contribution

It provides the first tableau formulas for symplectic and orthogonal skew Schubert polynomials, broadening the combinatorial understanding of these polynomials.

Findings

First tableau formulas for symplectic and orthogonal skew Schubert polynomials.

Derived tableau formulas for double Schur, theta, and eta polynomials.

Generalized classical tableau formulas to all four classical Lie types.

Abstract

The skew Schubert polynomials are those which are indexed by skew elements of the Weyl group, in the sense of arXiv:0812.0639. We obtain tableau formulas for the double versions of these polynomials in all four classical Lie types, where the tableaux used are fillings of the associated skew Young diagram. These are the first such theorems for symplectic and orthogonal Schubert polynomials, even in the single case. We also deduce tableau formulas for double Schur, double theta, and double eta polynomials, in their specializations as double Grassmannian Schubert polynomials. The latter results generalize the tableau formulas for symmetric (and single) Schubert polynomials due to Littlewood (in type A) and the author (in types B, C, and D).