Completion of the \(\mathfrak{gl}_2\)-crystal for two-part partitions

Complete the partially defined \(\mathfrak{gl}_2\)-crystal structure on the relevant highest-weight bitableaux so as to obtain a combinatorial interpretation of Kronecker coefficients when one indexing partition has two parts.

Background

The paper studies Kronecker tableaux associated with coefficients having a two-part partition and defines a partial lowering operation $\varphi$ on bitableaux that are highest weight for the $\mathfrak{gl}_m$-structure. The operation identifies certain tableaux as candidates for lowest-weight elements.

The authors conjecture that this partial operation can be completed to a full $\mathfrak{gl}_2$-crystal structure. Completing it would distinguish the tableaux that contribute to the relevant Kronecker coefficients and would finish the proposed interpretation in the two-part case.