Closed formulas for energy functions on tensor squares of higher-level perfect crystals in classical affine types

Abstract

For every $l\geq1$, we construct explicit closed-form coordinate formulas for the local energy functions on the tensor products $B_l\otimes B_l$ of level-$l$ perfect crystals in classical affine types. A single finite-level max-linear formula covers all seven types: its two branches coincide in type $A_n^{(1)}$, yielding a cyclic maximum of partial sums, whereas in the remaining six types each branch is the maximum of finitely many explicit piecewise-linear expressions in barred coordinates, with type-dependent boundary data. We verify the defining local-energy recursion directly on the finite crystals and derive equivalent recursive forms, allowing the energy to be evaluated without applying the combinatorial $R$-matrix. Substitution into the KMN path character formula gives explicit positive coordinate path sums for the characters of all level-$l$ integrable highest weight modules. After principal specialization, the path exponent can be rewritten as a weighted sum of a position-independent adjacent-pair statistic; comparison with the specialized Weyl--Kac character formula yields a uniform family of level-$l$ Rogers--Ramanujan-type identities equating these sums with explicit infinite products. For a representative low-rank case at level two in each family, we display the complete adjacent-pair degree matrix and list the resulting identities.