RSK correspondence for King tableaux with Berele insertion

TL;DR

This paper introduces a new type C RSK correspondence for King tableaux using Berele insertion, utilizing semistandard oscillating tableaux, and reveals dualities and symmetries in related combinatorial identities.

Contribution

It establishes a bijective RSK correspondence of type C for King tableaux with Berele insertion, incorporating semistandard oscillating tableaux and proving symmetry of their generating functions.

Findings

Established a bijective RSK correspondence of type C for King tableaux.

Introduced semistandard oscillating tableaux (SSOT) as a new combinatorial object.

Proved the symmetry of the generating function of SSOT.

Abstract

We establish a bijective RSK correspondence of type C for King tableaux with Berele insertion as a reformulation of Sundaram's correspondence (1986). For its $Q$-symbol, we make use of semistandard oscillating tableaux (SSOT), a new object which Lee (2025) introduced. Further, we show hidden duality of Cauchy identity through RSK correspondences of type A and C. Finally, we prove that the generating function of SSOT is symmetric by constructing a new sort of Bender-Knuth involution.