Construction of the full bicrystal on lexicographic bitableaux

Construct a \(\mathfrak{gl}_n \times \mathfrak{gl}_m\)-crystal structure on the set of lexicographic bitableaux of shape \(\lambda\) with entries in \([n] \times [m]\), so that the crystal structure provides a combinatorial interpretation of the Kronecker coefficients.

Background

The paper constructs a $\mathfrak{gl}_m$-crystal on bitableaux that preserves the $\mathfrak{gl}_n$-weights, yielding a positive combinatorial formula for a monomial expansion of a Kronecker product. The unresolved task is to add a commuting $\mathfrak{gl}_n$-crystal structure on the same set.

A full bicrystal would identify highest-weight elements of weight $(\mu,\nu)$ with the Kronecker coefficient $g(\lambda,\mu,\nu)$, thereby completing the proposed crystal-theoretic program.