2025/11/28 by Lee, Soong Kyum
#05E05 #05E10 #14L24 #20C30 #68Q17 #Combinatorics (math.CO) #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Quantum Physics (quant-ph) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2511.22856
While Kronecker coefficients g(λ,μ,ν) with bounded rows are polynomial-time computable via lattice-point methods, no explicit closed-form formulas have been obtained for genuinely three-row cases in the 87 years since Murnaghan's foundational work. This paper provides such formulas for the first time and identifies a universal structural boundary at parameter value 5 where elementary combinatorial patterns collapse. We analyze two independent families of genuinely three-row coefficients and establish that for k ≤ 4, the formulas exhibit elementary structure: oscillation bounds follow the triangular-Hogben pattern, and polynomial expressions factor completely over ℤ. At the critical threshold k=5, this structure collapses: the triangular pattern fails, and algebraic obstructions -- irreducible quadratic factors with negative discriminant -- emerge. We develop integer forcing, a proof technique exploiting the tension between continuous asymptotics and discrete integrality. As concrete results, we prove that g((n,n,1)3) = 2 - (n \mod 2) for all n ≥ 3 -- the first explicit formula for a genuinely three-row Kronecker coefficient -- derive five explicit polynomial formulas for staircase-hook coefficients, and verify Saxl's conjecture for 132 three-row partitions.