Generalization of the second reading word

Generalize the reading methods \(u(T)\) and \(u'(T)\) from row and column shapes to all bitableau shapes so that the resulting reading method defines a \(\mathfrak{gl}_n\)-crystal structure commuting with the \(\mathfrak{gl}_m\)-crystal structure induced by \(w(T)\).

Background

For one-row and one-column shapes, the paper relates RSK, dual RSK, and Burge insertion to two compatible reading words. These special cases provide crystal structures for both factors of gln×glm\mathfrak{gl}_n \times \mathfrak{gl}_m.

Extending the second reading-word construction to arbitrary shapes is presented as a strategy for resolving the full Kronecker coefficient problem. The requested compatibility with the crystal arising from w(T)w(T) is part of the explicit unresolved question.

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 Open Problem, Section 3.1, “RSK and dual RSK crystals”