On some Grothendieck expansions

2024/12/25 by Eric Marberg, Jiayi Wen, Marberg, Eric +1
Mathematics · Engineering · #Mathematical and Theoretical Analysis #Advanced Numerical Analysis Techniques

paper · pdf · doi:10.1016/j.jalgebra.2026.06.025

Abstract

The orthogonal and symplectic groups act on the complete flag variety with finitely many orbits. The orthogonal Grothendieck polynomials \mathfrakG^O_z and symplectic Grothendieck polynomials \mathfrakG^Sp_z are distinguished representatives for the K-theory classes of the corresponding orbit closures. There is a simple formula to expand \mathfrakG^Sp_z as a linear combination of Grothendieck polynomials \mathfrakG^(β)_w, which represent the K-theory classes of Schubert varieties. Although the constructions of \mathfrakG^Sp_z and \mathfrakG^O_z are similar, finding the \mathfrakG^(β)-expansion of \mathfrakG^O_z or even computing \mathfrakG^O_z is much harder. If z is vexillary then \mathfrakG^O_z has a nonnegative \mathfrakG^(β)-expansion, but the associated coefficients are mostly unknown. This paper derives several new formulas for \mathfrakG^O_z and its \mathfrakG^(β)-expansion when z is vexillary. Among other applications, we prove that the latter expansion has a nontrivial stability property.

Citations

• Gröbner geometry of Schubert polynomials

Cited by

• Key and Lascoux polynomials for symmetric orbit closures

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