{
  "schema": "groups.qft.org/conventions/1",
  "versions": {
    "schema": "2.0.0",
    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
  "profiles": [
    {
      "id": "finite-ordinary-complex-v1",
      "version": "1.0.0",
      "title": "Ordinary complex finite-group characters",
      "field": "C",
      "characteristic": 0,
      "category": "ordinary-complex",
      "description": "Rows are irreducible complex characters; columns are named conjugacy classes. Real/complex/quaternionic is the Frobenius–Schur type. D8 denotes order eight. Exact character fields and power maps are recorded with each table."
    },
    {
      "id": "lie-bourbaki-long-root-2-v1",
      "version": "1.0.0",
      "title": "Finite highest weights and root normalization",
      "field": "C",
      "characteristic": 0,
      "category": "finite-dimensional-continuous",
      "description": "Dynkin labels pair with numbered simple coroots. The displayed Cartan matrix is the definitive ordering. Long roots have squared length 2 and C2(lambda)=(lambda,lambda+2rho)/2. The integration default is the simply connected compact group."
    },
    {
      "id": "su3-dynkin-pq-v1",
      "version": "1.0.0",
      "title": "SU(3) Dynkin labels",
      "field": "C",
      "characteristic": 0,
      "category": "finite-dimensional-continuous",
      "description": "Fundamental 3=(1,0), conjugate 3-bar=(0,1); complex conjugation exchanges p and q. Triality is p+2q mod 3. Multiplicities are independent of a basis; no Clebsch–Gordan coefficients are supplied."
    },
    {
      "id": "su3-regular-su2-v1",
      "version": "1.0.0",
      "title": "Regular SU(2) restriction of SU(3)",
      "field": "C",
      "characteristic": 0,
      "category": "finite-dimensional-continuous",
      "description": "The injective homomorphism is U ↦ diag(U,1). The SU(2) torus diag(z,z^-1) maps to diag(z,z^-1,1). Highest weight n=2j has dimension n+1. Long-root-length-two trace embedding index is one; no commuting U(1) charges are supplied."
    },
    {
      "id": "su3-principal-su2-v1",
      "version": "1.0.0",
      "title": "Principal SU(2) restriction of SU(3)",
      "field": "C",
      "characteristic": 0,
      "category": "finite-dimensional-continuous",
      "description": "The spin-one homomorphism has kernel Z2 and image SO(3). Its torus sends diag(z,z^-1) to diag(z^2,1,z^-2). Every output n=2j is even. Long-root-length-two trace embedding index is four."
    },
    {
      "id": "su-n-dynkin-center-v1",
      "version": "1.0.0",
      "title": "SU(N) central character and matter descent",
      "field": "C",
      "characteristic": 0,
      "category": "finite-dimensional-continuous",
      "description": "The fundamental is (1,0,...,0); z=exp(2 pi i/N)I has central charge sum(i*a_i) mod N. Z_d is the subgroup generated by z^(N/d). Descent requires d to divide every matter charge. Matter does not select line operators or discrete theta data."
    },
    {
      "id": "su-n-young-lr-v1",
      "version": "1.0.0",
      "title": "SU(N) Young diagrams and Littlewood–Richardson products",
      "field": "C",
      "characteristic": 0,
      "category": "finite-dimensional-continuous",
      "description": "A=(a1,...,aN−1) corresponds to the canonical partition lambda_i=sum(j≥i)a_j and lambda_N=0. Schur products use ordinary Littlewood–Richardson coefficients. Full columns of height N are determinant factors and are removed only on restriction to SU(N). Irrep identity remains the Dynkin labels, not dimension alone."
    },
    {
      "id": "su-n-block-u1-v1",
      "version": "1.0.0",
      "title": "Block restriction with a primitive U(1)",
      "field": "C",
      "characteristic": 0,
      "category": "finite-dimensional-continuous",
      "description": "For p+q=N and g=gcd(p,q), (A,B,z) maps to diag(z^(q/g)A,z^(-p/g)B). The U(1) parameter has period 2pi. The kernel is cyclic of order pq/g. A pair of GL partitions (mu,nu) has charge X=(q|mu|−p|nu|)/g. For SU(5) with p=3, physical hypercharge is Y=−X/6, so the integer Standard Model charge Q=6Y is −X."
    },
    {
      "id": "product-su-u1-center-v1",
      "version": "1.0.0",
      "title": "Product SU centers and an integer U(1) charge",
      "field": "C",
      "characteristic": 0,
      "category": "finite-dimensional-continuous",
      "description": "The covering product is SU(N1) [× SU(N2)] × U(1), with z_i=exp(2pi i/N_i) and a U(1) phase exp(2pi i theta). A multiplet has phase sum(t_i*k_i/N_i)+Q*theta modulo one, where t_i=sum(j*a_ij) mod N_i and Q is integral. Standard Model examples use Q=6Y and gamma=(omega3,−1,exp(2pi i/6)). Quotients are specified by embedded generators, not only their abstract isomorphism type."
    },
    {
      "id": "four-d-local-polynomial-v1",
      "version": "1.0.0",
      "title": "Four-dimensional local polynomial invariants",
      "field": "C",
      "characteristic": 0,
      "category": "finite-dimensional-algebraic",
      "description": "Spacetime is four-dimensional and Lorentzian. A scalar is (0,0), a left Weyl field is (1/2,0), and a right Weyl field is (0,1/2) under the complexified proper Lorentz spin algebra. Compact SU(2) × SU(2) character projection computes these finite-dimensional algebraic invariants; it does not replace the physical spacetime group. For each species, symmetric (bosonic) or exterior (fermionic) powers act on Lorentz × color × weak × flavor components together. Flavor is ungauged multiplicity, and complex conjugate fields are separate polynomial generators. The result is the complex dimension at one fixed multidegree, with algebraic index identities included. Derivatives, EOM, IBP, Hermiticity, flavor-basis equivalences, gauge-field strengths, and anomaly tests are outside this convention. Gauge charges are integers for a 2pi circle; Standard Model examples use Q=6Y and the embedded diagonal quotient generated by (omega3,−1,exp(2pi i/6))^(6/d)."
    }
  ],
  "scope": "Ten convention profiles for the ten Research workbench I–IV additions. The five inherited design-edition tools retain their declared conventions and mathematical contracts."
}