Abstract
We compute, mimicking the Lascoux-Schützenberger 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 Schützenberger evacuation is defined to realize that involution.
Highlights
The irreducible characters of the general linear group GLn, over C, the Schur polynomials, are combinatorially expressed as sums on semistandard Young tableaux with entries n [35]
King showed that the irreducible symplectic characters, the symplectic Schur polynomials, can be seen as a sum on a family of tableaux that are known as King tableaux [16], and De Concini has proposed the ones known as De Concini tableaux [8]
Using the right key map, we describe the tableaux that contribute to a Demazure atom and to a Demazure crystal, which is our main result, Theorem 52
Summary
The irreducible characters of the general linear group GLn, over C, the Schur polynomials, are combinatorially expressed as sums on semistandard Young tableaux with entries n [35]. C, the signed permutation group Bn, the Bruhat order on Bn and on its cosets, modulo the stabilizer of λ, the Kashiwara-Nakashima tableaux and the symplectic key tableaux. This is an explicit realization of Lusztig involution using insertion and sliding operations in type C.
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