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The Gerstenhaber problem for three commuting matrices: finrank_adjoin_triple_le

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Statement

The Gerstenhaber problem: if AA, BB, and CC are pairwise commuting n×nn \times n matrices over a field KK, is the dimension of the unital KK-algebra K[A,B,C]K[A, B, C] they generate always at most nn?

Lean statement

theorem finrank_adjoin_triple_le :
    ∀ (K : Type*) [Field K] (n : ℕ) (A B C : Matrix (Fin n) (Fin n) K),
      ({A, B, C} : Set (Matrix (Fin n) (Fin n) K)).Pairwise Commute →
        Module.finrank K (Algebra.adjoin K {A, B, C}) ≤ n := by
  sorry

Sources(1)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/Other/GerstenhaberThreeMatrices.lean.

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