The minimal degree of a polynomial identity of all Matsuo algebras is at least six
Abstract
Matsuo algebras are the commutative non-associative algebras attached to groups of 3-transpositions; they are the axial algebras of Jordan type that come from a group, and for η ≠ 1/2 every algebra of Jordan type η is one of them or a quotient of one. A recent survey of Gorshkov and Shpectorov records a problem of Rowen: do all Matsuo algebras satisfy identities not implied by commutativity, and if they do, what is the minimal degree of such an identity? The survey states the published lower bound as 4, attributing it to a proposition of Chayet and Garibaldi through work of Osborn. We raise the bound to 6. The proof rests on a single small algebra: the six-dimensional Matsuo algebra Mat(2, S₄) over ℚ, built from the six transpositions of S₄ with η = 2, whose space of multilinear identities is zero in degrees 3, 4 and 5. The degree-5 statement is certified by an explicit 105 × 105 integer block of the evaluation matrix together with an explicit inverse of that block modulo 101. We also verify that Mat(2, S₄) is unital with 1 = 1/5 Σ_(a ∈ D) a, that the Frobenius form printed in the survey is associative on it with Gram determinant 5, and that it is not a Jordan algebra — exactly the hypotheses under which the published proposition applies, so that the bound the published literature really gives is 5 and the theorem below is one degree past it. A rank computation carried out outside the formal development gives degree 6 as well, which would raise the bound to 7; that computation is labelled as such wherever it appears and is not machine-checked. Everything else is machine-checked in Lean 4.
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Stated results
MI1candidate2026-09-03
No nonzero multilinear polynomial of degree 5 with rational coefficients vanishes on all 5-tuples of axes of the Matsuo algebra M₂(S₄); hence the minimal degree of an identity of all Matsuo algebras not implied by commutativity is at least 6
MI5known2026-09-03
The same certificate at degrees 3 and 4: no nonzero multilinear polynomial of degree 3 or 4 vanishes on all tuples of axes of M₂(S₄)
MI2known2026-09-03
The source's own example, reproduced with explicit witnesses: the 12-dimensional Matsuo algebra of Jordan type half of the Fischer space of W(D₄) (3-transposition group 2³:S₄, centrally equivalent to 2⁴:S₄) fails the multilinear Jordan identity at the axes e₀, e₁, e₂, e₆, and is not power associative – x²x² and x*x³ differ for x = e₀+e₁+e₂+e₆
MI3routine2026-09-03
Controls at the same degree: the multilinear Jordan identity holds at every one of the 1296 substitutions of axes of M_(1/2)(S₄) (so identities of degree 4 do exist for eta = 1/2), and the same polynomial fails in M₂(S₄)
MI4known2026-09-03
The source's printed Frobenius form is associative ((a.b, c) = (a, b.c)) on all triples of axes of M₂(S₄), M_(1/2)(S₄) and the 12-dimensional M_(1/2)(D₄); and M₂(S₄) is unital with 1 = (1/5) sum of the six axes and has a nondegenerate Frobenius form (Gram determinant 5)
MI6measurement2026-09-03
Frontier measurement: the degree-6 multilinear identity space of M₂(S₄) is also zero (the 945-column evaluation matrix has full rank over F_(2³1-1)), so the minimal degree is at least 7; the Lean certificate for that is priced and parked, and degree 8 is priced out
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
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- Source context
- A 3-transposition group is a pair (G, D) with G generated by a normal set D of involutions such that |de| ≤ 3 for all d, e ∈ D. For a field F with char F ≠ 2 and η ∈ F 0,1, the Matsuo algebra M_η(G,D) is the commutative non-associative F-algebra with basis D and
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- 2026-09-07 03:53 UTC
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