A note on a cyclotomic-friendly application of RSK

TL;DR

This paper presents a combinatorial realization of a level-$$ Robinson-Schensted-Knuth correspondence for cyclotomic Schur categories, extending classical RSK to a new algebraic setting with natural restrictions.

Contribution

It introduces a novel combinatorial framework that generalizes RSK to cyclotomic Schur categories, confirming a conjecture by Song and Wang.

Findings

Provides a canonical reorganization of cyclotomic basis elements into flagged block matrices.

Shows the level-$ RSK correspondence as an iteration of classical RSK.

Demonstrates the natural behavior of the correspondence under level restrictions.

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.