Séminaire Lotharingien de Combinatoire, 93B.46 (2025), 12 pp.
Eric Marberg and Jiayi Wen
On Some Grothendieck Expansions
Abstract.
The complete flag variety admits a natural action by both the orthogonal group and the symplectic group.
Wyser and Yong defined orthogonal Grothendieck polynomials GzO and symplectic Grothendieck polynomials GzSp as the K-theory classes of the corresponding orbit closures. There is an explicit formula to expand GzSp as a nonnegative sum of Grothendieck polynomials G, which represent the K-theory classes of Schubert varieties. Although the constructions of GzSp and GzO are similar, finding the G-expansion of GzO or even computing GzO is much harder.
When z is vexillary, it has been shown that GzO has a nonnegative G-expansion, but the G-coefficients are mostly unknown. This paper derives several new formulas for GzO and its G-expansion when z is vexillary. In particular, we prove that the latter expansion has a nontrivial stability property when z(1)=1.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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