Diagonalizability criteria for individual blocks RSK_{ω,τ}

Determine necessary and sufficient conditions on weight vectors ω and τ under which the block matrix RSK_{ω,τ}, obtained by restricting the Robinson–Schensted–Knuth operator to the weight space R_{ω,τ}, is diagonalizable.

Background

Theorem 6.2 gives a complete classification of when the global matrix RSK_{m,n,d} is diagonalizable, and Proposition 6.1 shows that blocks corresponding to triangular weights are diagonalizable. However, beyond these cases, a general criterion for the diagonalizability of individual blocks RSK_{ω,τ} is unknown.