RSK as a linear operator

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 $E_9$.