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Algebraic Obstructions and the Collapse of Elementary Structure in the Kronecker Problem

Soong Kyum Lee

TL;DR

<3-5 sentence high-level summary>This work delivers the first explicit closed-form formulas for genuinely three-row Kronecker coefficients, notably g((n,n,1)^3) = 2 − (n mod 2), and uncovers a universal Five Threshold where elementary combinatorial structure collapses. It introduces integer forcing as a robust method to derive multiplicity formulas by reconciling continuous asymptotics with discrete integrality, and provides explicit staircase-hook formulas and Saxl's conjecture verifications for large three-row families. The results illuminate new asymptotic behavior (bounded oscillation) and reveal deep connections to quantum marginal problems and Geometric Complexity Theory, suggesting phase-transition-like phenomena in representation-theoretic multiplicities. Together, these findings deepen understanding of three-row Kronecker coefficients and open multiple avenues for rigorous generalization and applications in quantum information and complexity theory.

Abstract

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.

Algebraic Obstructions and the Collapse of Elementary Structure in the Kronecker Problem

TL;DR

<3-5 sentence high-level summary>This work delivers the first explicit closed-form formulas for genuinely three-row Kronecker coefficients, notably g((n,n,1)^3) = 2 − (n mod 2), and uncovers a universal Five Threshold where elementary combinatorial structure collapses. It introduces integer forcing as a robust method to derive multiplicity formulas by reconciling continuous asymptotics with discrete integrality, and provides explicit staircase-hook formulas and Saxl's conjecture verifications for large three-row families. The results illuminate new asymptotic behavior (bounded oscillation) and reveal deep connections to quantum marginal problems and Geometric Complexity Theory, suggesting phase-transition-like phenomena in representation-theoretic multiplicities. Together, these findings deepen understanding of three-row Kronecker coefficients and open multiple avenues for rigorous generalization and applications in quantum information and complexity theory.

Abstract

While Kronecker coefficients 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 , the formulas exhibit elementary structure: oscillation bounds follow the triangular-Hogben pattern, and polynomial expressions factor completely over . At the critical threshold , 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 for all -- 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.

Paper Structure

This paper contains 72 sections, 37 theorems, 96 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Historical context
  3. The computational dichotomy
  4. Recent breakthroughs for two-row cases
  5. Main results
  6. The integer forcing methodology
  7. Connections to major programs
  8. Geometric complexity theory
  9. Quantum information theory
  10. Organization
  11. Preliminaries
  12. Partitions and Young diagrams
  13. The hook-length formula for $(n,n,1)$
  14. Character theory and Kronecker coefficients
  15. The Murnaghan-Nakayama rule
  16. ...and 57 more sections

Key Result

Theorem 1.1

For all integers $n \geq 3$:

Theorems & Definitions (82)

  • Theorem 1.1: First Three-Row Formula
  • Remark 1.2: Novelty of bounded oscillation
  • Remark 1.4: Applicability of integer forcing
  • Corollary 1.5: Quantum compatibility
  • Definition 2.1
  • Definition 2.2: Hook length
  • Theorem 2.3: Hook-length formula, Frame-Robinson-Thrall
  • Example 2.4: Hooks for $(n,n,1)$
  • Lemma 2.5: Hook lengths for $(n,n,1)$
  • proof
  • ...and 72 more