RSK as a linear operator

TL;DR

This paper investigates the RSK correspondence as a linear operator on matrices, analyzing its algebraic properties such as diagonalizability, eigenvalues, trace, and determinant, with criteria linked to Dynkin diagram classifications.

Contribution

It introduces a novel perspective by studying RSK as a linear operator and provides criteria for its diagonalizability based on Dynkin diagram classifications.

Findings

RSK can be analyzed as a linear operator on the coordinate ring of matrices.

Diagonalizability of RSK relates to ADE classification of Dynkin diagrams.

Results include eigenvalues, trace, and determinant formulas for RSK as an operator.

Abstract

The Robinson-Schensted-Knuth correspondence (RSK) is a bijection between nonnegative integer matrices and pairs of Young tableaux. We study it as a linear operator on the coordinate ring of matrices, proving results about its diagonalizability, eigenvalues, trace, and determinant. Our criterion for diagonalizability involves the $ADE$ classification of Dynkin diagrams, as well as the diagram for $E_9$.