Super RSK-algorithms and super plactic monoid

TL;DR

This paper introduces a super analogue of the plactic monoid for signed alphabets, extending classical combinatorial structures and invariants to a signed setting, with broad applicability.

Contribution

It develops a super version of the plactic monoid, generalizing Knuth's relations, Greene's invariants, and the Young-Pieri rule, including a symmetry theorem for signed alphabets.

Findings

Constructed the super plactic monoid for signed alphabets.

Generalized Greene's invariants and Young-Pieri rule.

Proved a symmetry theorem in the signed case.

Abstract

We construct the analogue of the plactic monoid for the super semistandard Young tableaux over a signed alphabet. This is done by developing a generalization of the Knuth's relations. Moreover we get generalizations of Greene's invariants and Young-Pieri rule. A generalization of the symmetry theorem in the signed case is also obtained. Except for this last result, all the other results are proved without restrictions on the orderings of the alphabets.