2025/12/06 by Han, Shaolong
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2512.06480
openalex publication_date 2025/12/06 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28
We derive Rogers--Ramanujan type partition identities at the fundamental weight Λ0 for the exceptional affine types G2(1), D4(3), F4(1), E6(2), E6(1), E7(1) and E8(1). Our starting point is the Dousse--Konan reformulation of the (KMN)2 crystal character formula, applied to the level-one perfect crystal B=B(θ)\sqcup B(0) of Benkart--Frenkel--Kang--Lee with ground element ϕ∈ B(0). This realizes the normalized character e-Λ0ch L(Λ0) as generating functions of grounded B-colored partitions governed locally by the crystal energy. After principal specialization, we obtain a colored partition model subject to explicit difference, congruence, and initial conditions. On the product side, under the same specialization, the Weyl--Kac character formula yields an explicit Euler-type product, equivalently the generating function for partitions with parts in a concrete allowed set. Comparing the two specializations gives coefficientwise equalities of generating functions. A key computational feature is that the difference matrix can be produced from the crystal data without explicitly computing the energy function. For each type we tabulate the congruence data, forbidden initial parts, and the full difference matrix, and we provide reproducible coefficient checks.