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Complement Submodular Information Measures for Balanced and Robust Data Selection A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring An identity for second Eulerian numbers via lattice-point counting $t$-tone edge coloring of graphs Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Universality theorems for generalized splines Framing Triangulations for Arbitrary Integer Flow Polytopes On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs The complexity of frugal digraph homomorphisms Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux Enumerating Pattern Avoiding Parking Functions Incidence toric ideals and three-point functions Unique Winning Opening Move in Three-Row Chomp Strong majority colorings of graphs A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Spectral radius and edge-disjoint connected factors of graphs New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256 Flexible DP-4-coloring of planar graphs without 4-cycles and intersecting triangles Balanced intersection size distributions in projective planes List Reconstruction Problem with List Size Two Is Dimensionality a Barrier for Retrieval Models? The INIEP: Irreducible and Positive Realizations The number of Pfaffian orientations on punctured polygonally cellulated surfaces Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Finite-state enumeration of adjacency-constrained 132-avoiding permutations AMDS and quantum AMDS Constacyclic codes of length $4p^ς$ over $\mathbb{F}_{{p}^{m}}$
Rogers-Ramanujan type identities at $Λ_0$ from perfect cr...
Shaolong Han · 2025-12-07 · via math.CO updates on arXiv.org

We derive Rogers--Ramanujan type partition identities at the fundamental weight $Λ_0$ for the exceptional affine types $G_2^{(1)}$, $D_4^{(3)}$, $F_4^{(1)}$, $E_6^{(2)}$, $E_6^{(1)}$, $E_7^{(1)}$ and $E_8^{(1)}$. Our starting point is the Dousse--Konan reformulation of the $(\mathrm{KMN})^2$ crystal character formula, applied to the level-one perfect crystal $B=B(θ)\sqcup B(0)$ of Benkart--Frenkel--Kang--Lee with ground element $φ\in B(0)$. This realizes the normalized character $e^{-Λ_0}\mathrm{ch} L(Λ_0)$ 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.