{
  "schema_version": 1,
  "problem_number": "OWR-14299904-015",
  "title": "Optimizing over Nonzero Littlewood–Richardson Coefficients: The Quantum Max d-Cut of Complete Bipartite Graphs",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Konvalinka asked (Oberwolfach Report 2/2026, p. 150) for the maximum of Δ(λ,µ,ν) = η(λ) − η(µ) − η(ν), where η(λ) = |λ|² + d(d−1)(2d−1)/6 − Σ_{i=1}^{d} (λ_i − i + 1)², over the triples of partitions with |λ| = n, |µ| = k, |ν| = n − k, at most d parts and nonzero Littlewood–Richardson coefficient c^λ_{µν}. By the work of Klep, Štrekelj and Volčič this maximum is the quantum max d-cut of the complete bipartite graph K_{n−k,k}, that is, the largest eigenvalue of the sum of 2(I − SWAP_ij) over its edges acting on (C^d)^{⊗n}. They determined it for d ≤ 3 and for k ≤ 4 and conjectured that it is always attained at a triple in which at most one part of λ is the sum of a part of µ and a part of ν (their Conjecture 7.7). We determine the maximum for all n, k, d. An explicit Littlewood–Richardson tableau and a level-set argument reduce the problem to the maximization of an elementary function Ψ(µ,x) over partitions µ of k with at most d parts and vectors x ∈ Z^d_{≥0} with sum n − k, an optimal λ being the decreasing rearrangement of µ + x. Exchange and convexity arguments then give an optimum with at most one row in which both µ and x are nonzero and with balanced remaining rows. Consequently, for d ≥ 2 the maximum equals 2k(n−k) + 2 max{Ψ_A, Ψ_B} over an explicit list of at most (d−1)(1 + k(n−k)) integers built from balanced partitions; it is 0 for d = 1 and 4k(n−k) for d ≥ n. As a corollary, Conjecture 7.7 of Klep, Štrekelj and Volčič holds, with explicit optimal triples (λ,µ,ν) = (P ⊎ Z, P, Z) or (P ⊎ Z ⊎ (π+y), P ⊎ (π), Z ⊎ (y)), where P and Z are balanced partitions and ⊎ denotes the union of the parts. The answer is a maximum over a candidate list of polynomial size, not a single closed formula. The known cases (d = 2, which is the Lieb–Mattis theorem; d = 3; k ≤ 4; the star graph) are recovered, and the results agree with brute-force computations over all Littlewood–Richardson coefficients for n ≤ 22 and with the reduction for n ≤ 80. The problem is Konvalinka's; the reduction of the quantum max d-cut of K_{n−k,k} to this maximum, the cases d ≤ 3 and k ≤ 4 and the conjecture are due to Klep, Štrekelj and Volčič. New, to our knowledge, are the general solution and the proof of the conjecture. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "keywords": [
    "Littlewood–Richardson coefficients",
    "quantum max d-cut",
    "complete bipartite graphs",
    "swap operators",
    "Heisenberg model",
    "Lieb–Mattis theorem",
    "Werner states",
    "balanced partitions",
    "content sum",
    "Klep–Štrekelj–Volčič conjecture",
    "complete answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-14299904-015",
    "math.CO",
    "math.RT",
    "quant-ph",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299904-015/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299904-015/paper.pdf?v=a7d66d27fa0f",
  "doi": "10.5281/zenodo.23392319",
  "zenodo_record_url": "https://zenodo.org/records/23392319",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Konvalinka's problem (Oberwolfach Report 2/2026): the maximum of Delta over triples of partitions with nonzero Littlewood-Richardson coefficient, which is the quantum max d-cut of the complete bipartite graph K(n-k,k), is determined for all n, k, d as an explicit maximum over a polynomial-size candidate list, and Conjecture 7.7 of Klep, Strekelj and Volcic follows.",
  "files": {
    "paper.pdf": {
      "sha256": "a7d66d27fa0f889a4e46ceb2509ad5454bc3d0c74b5b55ca1aa03821135053a2"
    },
    "source.zip": {
      "sha256": "68b80e7edd8c7beb0c7bb5dfd423350b327e6e9ec73dc9f875243882fa156d0f"
    },
    "verification_report.md": {
      "sha256": "89a43e3c66551e60673c9ae34371ebb556484a367fb538dfcada228835c3d1fc"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}