Reading-word rule for arbitrary bitableau shapes

Determine a general method for extracting a reading word from lexicographic bitableaux of arbitrary shape that extends the exhibited examples and supports the desired commuting crystal structures.

Background

The examples with maximum top entry three impose 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 authors exhibit a reading order for shape (2,1)(2,1), but explicitly state that they cannot yet extend this construction to arbitrary shapes. This is a concrete unresolved subproblem in the search for the missing crystal structure.

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 Example 3, Section “Maximum entry three”