# A symplectic refinement of shifted Hecke insertion

## TL;DR

This paper introduces a symplectic refinement of shifted Hecke insertion, providing a combinatorial rule for expanding certain K-theoretic Schur P-functions related to symplectic groups.

## Contribution

It generalizes shifted Hecke insertion to symplectic groups and establishes a combinatorial rule for related K-theoretic Schur P-functions.

## Key findings

- Developed a symplectic analogue of shifted Hecke insertion.
- Derived a combinatorial expansion rule for symplectic K-theoretic Schur P-functions.
- Connected combinatorial insertion algorithms with orbit closures of symplectic groups.

## Abstract

Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials $G_\pi$ indexed by permutations in the basis of stable Grothendieck polynomials $G_\lambda$ indexed by partitions. Patrias and Pylyavskyy introduced a shifted analogue of Hecke insertion whose natural domain is the set of maximal chains in a weak order on orbit closures of the orthogonal group acting on the complete flag variety. We construct a generalization of shifted Hecke insertion for maximal chains in an analogous weak order on orbit closures of the symplectic group. As an application, we identify a combinatorial rule for the expansion of "orthogonal" and "symplectic" shifted analogues of $G_\pi$ in Ikeda and Naruse's basis of $K$-theoretic Schur $P$-functions.

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Source: https://tomesphere.com/paper/1901.06771