Determinantal conjecture

Does the determinant of the sum A + B of two n × n normal complex matrices A and B always lie in the convex hull of the n! points Π_i (λ(A)_i + λ(B)_σ(i))? Here the numbers λ(A)_i and λ(B)_i are the eigenvalues of A and B, and σ is an element of the symmetric group S_n.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-determinantal-conjecture,
  title        = {Determinantal conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/determinantal-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Does the determinant of the sum $A + B$ of two $n \times n$ normal complex matrices $A$ and $B$ always lie in the convex hull of the $n!$ points $\prod_i (λ (A)_i + λ (B)_σ (i))$ ? Here the numbers $λ (A)_i$ and $λ (B)_i$ are the eigenvalues of $A$ and $B$ , and $σ$ is an element of the symmetric group $S_n$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.DeterminantalConjecture.

theorem determinantal_conjecture
    (n : Type) [Fintype n] [DecidableEq n]
    (d1 d2 : n → ℂ) (U1 U2 : unitary (Matrix n n ℂ)) :
    (U1 * Matrix.diagonal d1 * star U1 + U2 * Matrix.diagonal d2 * star U2).det
      ∈ convexHull ℝ { ∏ i, (d1 i + d2 (σ i)) | σ : Equiv.Perm n }

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.