The 2-modular adjacency algebra of GQ(2,2): Wedderburn decomposition, Cartan matrix and Gabriel quiver
Abstract
Let X be the coherent configuration of type [3,2;3] attached to the generalized quadrangle GQ(2,2) and let 𝔽₂X be its 2-modular adjacency algebra, an algebra of dimension 10 over 𝔽₂. Shimabukuro determined dim_(𝔽₂)Rad(𝔽₂X)=4 and dim_(𝔽₂)Rad(𝔽₂X)²=2 by computer and asked for the Wedderburn decomposition of the semisimple quotient, the dimensions and Loewy series of the projective indecomposables, and the Gabriel quiver. We settle the first and the third. The semisimple quotient is M₂(𝔽₂) × 𝔽₂ × 𝔽₂; the two surviving types allowed by Shimabukuro's own constraints are separated by an idempotent count, 32 against 16. The Gabriel quiver has three vertices, with simple modules of dimensions 1,2,1, and exactly two arrows, running in both directions between the point-fibre and the block-fibre vertex; the vertex carrying the two-dimensional simple module is isolated. The Cartan matrix is ([2, 0, 1; 0, 1, 0; 1, 0, 2]), so the three projective indecomposables have dimensions 3,2,3. We also show Rad(𝔽₂X)³=0, which pins the Loewy length at exactly 3. All statements are machine-checked in Lean 4.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
22d0c59c8fe57319361a66bb87cafb7b64a70c4f47b325c499a2e76b96a99827
Claim ledger
Stated results
gq22-01known2026-08-23
The doily is GQ(2,2): 15 points, 15 lines, order (2,2), the generalized-quadrangle axiom, and both graphs srg(15,6,1,3)
gq22-02routine2026-08-23
The source's ten sigmaᵢ span a 10-dimensional algebra with explicit structure constants, and the A₁ / A₂ convention is immaterial
gq22-03known2026-08-23
Gate: dim_(F₂) Rad(F₂ Y) = 1 for the point scheme, with B = J_P - I_P - A₁ and B² = 0
gq22-04known2026-08-23
Gate: u = sigma₂ sigma₇ is nonzero, square-zero, in Rad(F₂ X), and outside F₂ Y
gq22-05known2026-08-23
Ring.jacobson(F₂ X) = span_(F₂)sigma₃, sigma₆, sigma₈, sigma₁0, so dim_(F₂) Rad(F₂ X) = 4
gq22-06known2026-08-23
Rad² = span_(F₂)sigma₃, sigma₆ has dimension 2, Rad² is nonzero, Rad³ = 0, so the Loewy length is exactly 3
gq22-07candidate2026-08-23
Problem (a): F₂ X / Rad(F₂ X) is isomorphic to M₂(F₂) x F₂ x F₂ – i.e. F₂ x F₂ and not F₄, the one bit the source's own constraints leave open
gq22-08routine2026-08-23
The Problem's dichotomy is already forced by the source's own data: sum nᵢ² fᵢ = 6 with sum fᵢ <= 4 leaves exactly M₂(F₂) x F₄ and M₂(F₂) x F₂ x F₂, both containing a matrix algebra
gq22-09candidate2026-08-23
Problem (c): the Gabriel quiver of F₂ X has exactly two arrows, 1 <-> 3, and no loops; the M₂(F₂) vertex is isolated
gq22-10candidate2026-08-23
Four orthogonal primitive idempotents summing to 1, and the Cartan-TYPE (corner-dimension) matrix [[2,0,1,0],[0,1,0,1],[1,0,2,0],[0,1,0,1]]
gq22-n1routine2026-08-23
Negative controls on the radical: dimension is not 3 and not 5, sigma₂ is not radical, and Rad is neither the whole ring nor zero
gq22-n2routine2026-08-23
Negative controls on the quotient and the point fibre: not M₃(F₂), not F₂³, the point fibre is commutative while F₂ X is not, and dim Rad(F₂ Y) is not 2
gq22-11routine2026-08-23
The Gabriel quiver of F₂𝔛 has exactly three vertices e₁, e₂,e₄, e₃, with simple modules of dimensions 1, 2, 1 and the M₂(F₂) vertex isolated
gq22-12routine2026-08-23
The Cartan matrix of F₂𝔛 is the 3×3 [[2,0,1],[0,1,0],[1,0,2]], with three projective indecomposables of dimensions 3, 2, 3 — the landed 4×4 cornerCard is the corner-dimension matrix, not the Cartan matrix
gq22-n3routine2026-08-23
Controls for the vertex count: the equivalence test accepts the true pair (e₂,e₄), four vertices and one vertex are both refuted, and the corner e₁𝔛e₃ is nonzero
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source: arXiv:2512.06541, Osamu Shimabukuro, *Frame Numbers and Jacobson Radicals for Partial Geometries and Related Coherent Configurations*, December 2025.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7