Symplectic keys and Demazure atoms in type C

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.Comment: Accepted in Electronic Journal of Combinatorics. Some notation has changed. 26 page

An analogue of the Robinson-Schensted-Knuth correspondence and non-symmetric Cauchy kernels for truncated staircases

We prove a restriction of an analogue of the Robinson-Schensted-Knuth correspondence for semi-skyline augmented fillings, due to Mason, to multisets of cells of a staircase possibly truncated by a smaller staircase at the upper left end corner, or at the bottom right end corner. The restriction to be imposed on the pairs of semi-skyline augmented fillings is that the pair of shapes, rearrangements of each other, satisfies an inequality in the Bruhat order, w.r.t. the symmetric group, where one shape is bounded by the reverse of the other. For semi-standard Young tableaux the inequality means that the pair of their right keys is such that one key is bounded by the Sch\"utzenberger evacuation of the other. This bijection is then used to obtain %recover an expansion formula of the non-symmetric Cauchy kernel, over staircases or truncated staircases, in the basis of Demazure characters of type $A$, and the basis of Demazure atoms. The expansion implies Lascoux expansion formula, when specialised to staircases or truncated staircases, and make explicit, in the latter, the Young tableaux in the Demazure crystal by interpreting Demazure operators via elementary bubble sorting operators acting on weak compositions