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RSK correspondence for King tableaux with Berele insertion

Masato Kobayashi, Tomoo Matsumura

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.

RSK correspondence for King tableaux with Berele insertion

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 -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.

Paper Structure

This paper contains 21 sections, 18 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Combinatorics of Young tableaux
  3. King (1976), Berele (1986), Sundaram (1986), S.J.Lee (2025)
  4. Main results
  5. Organization of the article
  6. type A theory
  7. Young tableaux
  8. row insertion
  9. RS correspondence
  10. Knuth equivalence
  11. type C theory
  12. King tableaux
  13. Berele insertion
  14. type C RS correspondence
  15. type C Knuth equivalence
  16. ...and 6 more sections

Key Result

Theorem 2.4

is a bijection.

Theorems & Definitions (67)

  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.4: RS correspondence of type A
  • Definition 2.5
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Definition 3.1
  • ...and 57 more