Generalization of the RSK reading methods

Generalize the reading methods \(u(T)\) and \(u'(T)\) from one-row and one-column bitableaux to all shapes in a way that yields a \(\mathfrak{gl}_n\)-crystal structure commuting with the \(\mathfrak{gl}_m\)-crystal structure defined by the reading word \(w(T)\).

Background

For row-shaped bitableaux, the RSK correspondence uses the reading words w(T)w(T) and u(T)u(T) to produce two tableaux and a commuting gln×glm\mathfrak{gl}_n\times\mathfrak{gl}_m-crystal. For column-shaped bitableaux, the analogous role is played by u′(T)u'(T) through Burge and dual RSK insertion.

The paper does not identify a corresponding second reading method for arbitrary shapes. Such a method would supply the missing gln\mathfrak{gl}_n-crystal structure compatible with the already constructed glm\mathfrak{gl}_m-crystal.

References

Open Problem: Generalize $u(T)$ and $u'(T)$ to a reading method for all shapes that gives a $\mathfrak{gl_n$-crystal structure commuting with the $\mathfrak{gl_m$-crystal structure afforded by $w(T)$.

— Kronecker Coefficients, Crystals, and Bitableaux  (2507.14026 - Harman et al., 18 Jul 2025) in Section “RSK and dual RSK crystals,” immediately before Section “An Insertion Version of the Kronecker Coefficient Problem”