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

TL;DR

This paper introduces a super-analogue of the RSK correspondence for super tableaux over signed alphabets, providing a geometric interpretation and a combinatorial super Littlewood-Richardson rule.

Contribution

It develops a super version of the RSK correspondence with symmetry properties and a geometric matrix-ball interpretation, extending classical combinatorial algorithms.

Findings

Established a super RSK correspondence with symmetry property.

Provided a geometric matrix-ball interpretation of the super RSK.

Derived a super Littlewood-Richardson rule for super Schur functions.

Abstract

The 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 via a matrix-ball construction, showing the symmetry property in complete generality. Finally, we deduce a combinatorial version of the super Littlewood-Richardson rule for super Schur functions over a finite signed alphabet.