A super Robinson-Schensted-Knuth correspondence with symmetry and the super Littlewood-Richardson rule

Abstract: Robinson-Schensted-Knuth (RSK) correspondence is a bijective correspondence between two-rowed arrays of non-negative integers and pairs of same-shape semistandard tableaux. This correspondence satisfies the symmetry property, that is, exchanging the rows of a two-rowed array is equivalent to exchanging the positions of the corresponding pair of semistandard tableaux. In this article, we introduce a super analogue of the RSK correspondence for super tableaux over a signed alphabet using a super version of Schensted's insertion algorithms. We give a geometrical interpretation of the super-RSK correspondence by a matrix-ball construction, showing the symmetry property in complete generality. We deduce a combinatorial version of the super Littlewood-Richardson rule on super Schur functions over a finite signed alphabet. Finally, we introduce the notion of super Littlewood-Richardson skew tableaux and we give another combinatorial interpretation of the super Littlewood-Richardson rule.