Algebraic Obstructions and the Collapse of Elementary Structure in the Kronecker Problem
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911578686750720 |
|---|---|
| author | Lee, Soong Kyum |
| author_facet | Lee, Soong Kyum |
| contents | 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 $k \leq 4$, the formulas exhibit elementary structure: oscillation bounds follow the triangular-Hogben pattern, and polynomial expressions factor completely over $\mathbb{Z}$. 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 $n \geq 3$ -- 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22856 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic Obstructions and the Collapse of Elementary Structure in the Kronecker Problem Lee, Soong Kyum Combinatorics Computational Complexity Representation Theory Quantum Physics 05E10, 20C30, 68Q17, 05E05, 14L24 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 $k \leq 4$, the formulas exhibit elementary structure: oscillation bounds follow the triangular-Hogben pattern, and polynomial expressions factor completely over $\mathbb{Z}$. 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 $n \geq 3$ -- 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. |
| title | Algebraic Obstructions and the Collapse of Elementary Structure in the Kronecker Problem |
| topic | Combinatorics Computational Complexity Representation Theory Quantum Physics 05E10, 20C30, 68Q17, 05E05, 14L24 |
| url | https://arxiv.org/abs/2511.22856 |