Construction of a bicrystal structure on bitableaux

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

Background

The paper explains that a crystal structure on bitableaux would encode the decomposition of the relevant representation under gln×glm\mathfrak{gl}_n \times \mathfrak{gl}_m. In such a structure, Kronecker coefficients would be obtained by counting highest-weight elements of prescribed gln\mathfrak{gl}_n- and glm\mathfrak{gl}_m-weights.

The paper constructs a glm\mathfrak{gl}_m-crystal that preserves gln\mathfrak{gl}_n-weights, but the full commuting bicrystal structure remains an explicitly stated open problem.

References

Open Problem: Construct a $\mathfrak{gl_n \times \mathfrak{gl}_m$ crystal structure on the set of bitableaux of shape $\lambda$ with entries in $[n] \times [m]$.

— Kronecker Coefficients, Crystals, and Bitableaux  (2507.14026 - Harman et al., 18 Jul 2025) in Open Problem, Section 2.3, “A Crystal Approach to the Kronecker Coefficient Problem”