Reading-word method for arbitrary bitableau shapes

Generalize the proposed method of extracting a reading word from bitableaux to arbitrary shapes, subject to the stated structural requirements, so that it yields the missing \(\mathfrak{gl}_n\)-crystal structure compatible with the existing construction.

Background

The paper presents examples with maximum top entry three and imposes two additional assumptions on a prospective reading method: the order of top entries should depend only on bottom entries, and the resulting word should be the row-reading word of a semistandard Young tableau.

The examples produce compatible crystal structures under these assumptions, but the authors explicitly report that they cannot extend the method to arbitrary shapes. This is a concrete unresolved construction problem rather than a general aspiration.

References

So far we are unable to generalize this method of extracting a reading word for bitableaux of arbitrary shape, but we do think it is suggestive of what a general rule might look like.

— Kronecker Coefficients, Crystals, and Bitableaux  (2507.14026 - Harman et al., 18 Jul 2025) in Final paragraph of Example 4, Section 4.2, “Maximum entry three”