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A note on a cyclotomic-friendly application of RSK

Holden Eriksson

TL;DR

This work constructs a concrete level-$\ell$ Robinson–Schensted–Knuth correspondence for cyclotomic Schur categories by recasting cyclotomic basis data as flagged block composition matrices and translating them into flagged biwords. The level-$\ell$ bijection is obtained by applying the classical RSK componentwise after an order-preserving relabeling of alphabets, yielding semistandard multitableaux of common shape that encode multipartition data. The construction extends the level-$\ell-1$ conjecture and specializes to the usual RSK when $\ell=1$, while remaining compatible with restrictions to lower levels. This provides a direct combinatorial link between diagrammatic cyclotomic bases and tableau-based realizations, clarifying how multipartition data organize under the RSK framework.

Abstract

We give a combinatorial realization of a level-$\ell$ Robinson-Schensted-Knuth correspondence conjectured to exist by Song and Wang for cyclotomic Schur categories. We show that cyclotomic basis elements can be canonically reorganized into flagged block composition matrices encoding families of biwords, so that the correspondence is obtained by applying the classical RSK correspondence componentwise. This perspective identifies the level-$\ell$ correspondence as an iteration of classical RSK, specializing to the usual correspondence when $\ell=1$ and behaving naturally under restriction to lower levels.

A note on a cyclotomic-friendly application of RSK

TL;DR

This work constructs a concrete level- Robinson–Schensted–Knuth correspondence for cyclotomic Schur categories by recasting cyclotomic basis data as flagged block composition matrices and translating them into flagged biwords. The level- bijection is obtained by applying the classical RSK componentwise after an order-preserving relabeling of alphabets, yielding semistandard multitableaux of common shape that encode multipartition data. The construction extends the level- conjecture and specializes to the usual RSK when , while remaining compatible with restrictions to lower levels. This provides a direct combinatorial link between diagrammatic cyclotomic bases and tableau-based realizations, clarifying how multipartition data organize under the RSK framework.

Abstract

We give a combinatorial realization of a level- Robinson-Schensted-Knuth correspondence conjectured to exist by Song and Wang for cyclotomic Schur categories. We show that cyclotomic basis elements can be canonically reorganized into flagged block composition matrices encoding families of biwords, so that the correspondence is obtained by applying the classical RSK correspondence componentwise. This perspective identifies the level- correspondence as an iteration of classical RSK, specializing to the usual correspondence when and behaving naturally under restriction to lower levels.
Paper Structure (8 sections, 5 theorems, 37 equations)

This paper contains 8 sections, 5 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Classical RSK ingredients
  4. Semistandard multitableaux
  5. Chicken-foot matrices
  6. Flagged block composition matrices
  7. Flagged biwords
  8. The correspondence

Key Result

Theorem 2.1

[Robinson--Schensted--Knuth RSK] There is a bijection For biword $w$ corresponding to $(P,Q)$, $\mathrm{wt}(P)=\text{content of } w_{\mathrm{bot}}$ and $\mathrm{wt}(Q)=\text{content of } w_{\mathrm{top}}.$

Theorems & Definitions (23)

  • Conjecture 1.1: Song--Wang wang2
  • Theorem 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Example 2.9
  • ...and 13 more