Symplectic keys and Demazure atoms in type C

TL;DR

This paper introduces symplectic keys for Kashiwara-Nakashima tableaux in type C, providing combinatorial tools for Bruhat order, Demazure atoms, and characters, extending classical type A methods.

Contribution

It develops symplectic keys for type C tableaux, analogous to type A, and connects them to Bruhat order, Demazure atoms, and Lusztig involution, with a new evacuation operation.

Findings

Symplectic keys characterize Bruhat order on hyperoctahedral group.

They describe Demazure atoms and characters in type C.

A type C Schützenberger evacuation realizes the Lusztig involution.

Abstract

We compute, mimicking the Lascoux-Sch\"utzenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Sch\"utzenberger evacuation is defined to realize that involution.