2026/07/23 by Shaolong Han
#math.RT #math.CO
For every l≥1, we construct explicit closed-form coordinate formulas for the local energy functions on the tensor products Bl⊗ Bl 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 An(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.