RING ATLAS / CONSTRUCTIONS CATALOGUE
Specific constructions, their known ingredients, and the proofs that identify their census rings.
36 examples · ordered by rank, multiplicity and identifier
No constructions match these filters. Clear the filters to see all examples.
Condense the diagonal central boson in the product of two identical half-E₆ categories, then equivariantize the factor-swap action with a sign on the condensed algebra.
Ingredients and context. The half-E₆ category; condensation of the diagonal central boson and equivariantization.
Take the category of modules over the diagonal central boson in the product of half-E₆ and PSU(2)₆.
Ingredients and context. The known half-E₆ and PSU(2)₆ categories and their central fermions.
Form the relative center Z_Fib(TT₃), using the Fibonacci subcategory of the dual even part of the 2D2 subfactor.
Ingredients and context. The known category TT₃ and the standard relative-center construction.
Construct a near-group orbifold, descend a coherent inversion action, and take the Morita dual associated to the reflection algebra. The resulting invertible group is C₂ × C₂.
Ingredients and context. The Yu–Zhang near-group category NG(C₄ × C₄, 16) and near-group orbifold machinery.
Apply the orbifold, inversion-descent and reflection-algebra Morita construction to the cyclic near-group input. Here the invertible group is C₄, and the two large noninvertible simples are mutually dual.
Ingredients and context. Budinski’s NG(C₁₆, 16) category and Izumi’s near-group orbifold machinery.
Transport an odd automorphism through a crossed product to obtain an extension in which the odd invertible a satisfies a² = g, with g the nontrivial invertible in the neutral 2D2 category.
Ingredients and context. The known unitary 2D2 category and the Grossman–Izumi–Snyder construction framework.
Use an explicit Davydov–Runkel extension of Kashina’s self-dual 16-dimensional Hopf algebra. The resulting simple dimensions are 1, 1, 1, 1, 2, 2, 2, 4.
Ingredients and context. Kashina’s Hopf algebra and the general Davydov–Runkel extension construction.
Equivariantize the coherent order-two phase-rotation action on Izumi’s unitary near-group category. Its sign on the unit-output channels distinguishes this construction from the reflection variant.
Ingredients and context. Izumi’s NG(C₃, 6) solution and its explicit tensor automorphisms.
Equivariantize the coherent order-two reflection action on the same near-group category. The reflection and phase-rotation actions yield different complete fusion tensors.
Ingredients and context. Izumi’s NG(C₃, 6) solution and its explicit tensor automorphisms.
Equivariantize Fib ⊠ Fib ⊠ Fib by the canonical C₃ action that cyclically permutes its factors. Character averaging on the fusion spaces determines the complete tensor.
Ingredients and context. The Fibonacci category and the standard permutation action on a tensor power.
Take bimodules over the inherited Q-system 1 + r in the nonsplit C₂-extension represented by R8_2_37. The neutral component is TT₃; the odd component and its products determine this rank-eight ring.
Ingredients and context. The nonsplit extension above, the known Q-system 1 + r, and the category TT₃.
Construct C(A₆, ω, A₅, ψ) from a specified relative μ₂-cocycle class. Stable-element and projective-stabilizer calculations determine the twisting and the full fusion tensor.
Ingredients and context. The alternating groups A₆ and A₅, the stable-elements theorem, and the standard group-theoretical category construction.
An explicit central C₃ condensation produces a generalized Tambara–Yamagami category with invertible group C₉. Equivariantize its inversion action with the identity square coherence.
Ingredients and context. TY(C₃), a specified pointed C₉ category, and central condensation. R9_2_1209 is the other parity-coherence variant.
Use the same central C₃ condensation and coherent inversion action, with the negative square coherence on the odd component. This yields the second cyclic generalized Tambara–Yamagami orbifold.
Ingredients and context. The same TY(C₃) and pointed C₉ inputs as R9_2_746; the parity natural automorphism changes the coherence.
A mixed order-four tensor functor on a covering category descends, after central boson condensation, to a coherent C₂ action on a generalized Moore–Read category. Equivariantization gives the recorded rank-nine model.
Ingredients and context. An Ising category, a specified pointed C₁₆ category, and the generalized Moore–Read construction.
Recover a coherent S₃ action from the known LPR-24 category by Tannakian descent, pull it back along Dih(C₉) → S₃, and equivariantize.
Ingredients and context. The known LPR-24 category and coherent equivariantization. The project derives the central-lift argument and this specific pullback model.
Pull back the same coherent S₃ action along Dih(C₃ × C₃) → S₃ and equivariantize. The different covering group gives a second rank-nine fusion tensor.
Ingredients and context. The known LPR-24 category and the same Tannakian descent as R9_2_1396, with an elementary-abelian covering group.
Group-theoretical category on A5 x C2 with graph D10 and relative binary cocycle. This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
C2 Morita dual of Liptrap feudal category 'group': 'F20', 'sigma': [0, 3, 2, 1], 'p': 0. This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
C2-algebra Morita dual of the matched-grading subcategory of Ising x Vec(D16,omega). This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
C2 Morita dual of D8-twisted F20-Liptrap category [cocycle 2]. This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
S3-algebra dual of a (C3^2 semidirect C4)-extension of Vec(V4). This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
S3 triality gauging with bosonic C2 zest (bosonic zest). This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
S3 triality gauging with bosonic C2 zest (bosonic zest). This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
Group-theoretical category on A5 x C2 with graph D10 and relative binary cocycle. This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
C2-algebra Morita dual of the matched-grading subcategory of Ising x Vec(SD16,omega). This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
Relative-cocycle subgroup-bimodule category: S4 x C2 with V4. This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
Nonsplit generalized-Ising extension, rotation-cocycle twist, and C2 algebra dual. This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
C(A4 x C2, relative invariant cocycle, diagonal C2). This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
Group-theoretical category on A5 x C2 with graph D10 and relative binary cocycle. This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
TY(C5, chi)^C4, inversion with odd parity coherence. This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
C(A4 x C2, relative invariant cocycle, diagonal C2). This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
Nonsplit generalized-Ising extension, rotation-cocycle twist, and C2 algebra dual. This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
Relative-cocycle subgroup-bimodule category: S4 x_C2 C4 with C4. This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
C(A4 x C2, relative invariant cocycle, diagonal C2). This supplies a unitary model with the full rank-twelve tensor identified exactly.
Ingredients and context. Known tensor-category constructions, with the specified coherence, cocycle or separable algebra checked in the linked proof.
Condense the central regular C6 in the algebraically reconstructed C36+36 near-group category. The quotient has inversion mixed action and the nontrivial pointed square term g³. This supplies a complex and pseudo-unitary model; unitarity remains unproved. The same construction works for k congruent to 2 modulo 4.
Ingredients and context. The cyclic near-group existence theorem, a Lagrangian subgroup H = 6Z/36Z, explicit central half-braidings, and character idempotents. No unitary classification premise is used.