""" ====================================================================== The S_3 Structure of E_47 Decomposing the 47-dimensional kernel under the action of the symmetric group permuting the three tensor factors of V_2 ⊗ V_2 ⊗ V_2. ====================================================================== Logic: S_3 acts on V = V_2 ⊗ V_2 ⊗ V_2 by permuting the three factors. This action commutes with the total Casimir C, hence with K, hence preserves the kernel E_47. We restrict the S_3 representation to E_47, compute its character on the three conjugacy classes {e}, {transpositions}, {3-cycles}, and decompose into irreducibles of S_3 (trivial, sign, standard). Then we refine: for each isotypic block W_j ⊂ V, the S_3 action is on the multiplicity space M_j (of dimension m_j), and we get the full SU(2) × S_3 decomposition of E_47 = W_2 ⊕ W_5. S_3 irreducibles (character table): e (12) (123) trivial: 1 1 1 sign: 1 -1 1 standard: 2 0 -1 """ import numpy as np from numpy.linalg import eigh, eigvalsh # ---- 1. Rebuild the standard infrastructure ------------------------ s = 2; d = 5 m_vals = np.arange(s, -s-1, -1) Jz = np.diag(m_vals).astype(complex) Jp = np.zeros((d, d), dtype=complex) for i, m in enumerate(m_vals): if m + 1 <= s: Jp[i-1, i] = np.sqrt(s*(s+1) - m*(m+1)) Jm = Jp.conj().T Jx = 0.5 * (Jp + Jm) Jy = -0.5j * (Jp - Jm) I5 = np.eye(d, dtype=complex) def kron3(A, B, C): return np.kron(np.kron(A, B), C) D = 125 I_V = np.eye(D, dtype=complex) Jtx = kron3(Jx,I5,I5) + kron3(I5,Jx,I5) + kron3(I5,I5,Jx) Jty = kron3(Jy,I5,I5) + kron3(I5,Jy,I5) + kron3(I5,I5,Jy) Jtz = kron3(Jz,I5,I5) + kron3(I5,Jz,I5) + kron3(I5,I5,Jz) C = Jtx@Jtx + Jty@Jty + Jtz@Jtz C = 0.5 * (C + C.conj().T) K = (C - 6*I_V) @ (C - 30*I_V) K = 0.5 * (K + K.conj().T) # ---- 2. Build S_3 permutation operators --------------------------- # Basis index: |a,b,c⟩ → 25a + 5b + c with a,b,c ∈ {0,1,2,3,4}. # Permutation σ acts by: σ · |a,b,c⟩ = |x_{σ⁻¹(1)}, x_{σ⁻¹(2)}, x_{σ⁻¹(3)}⟩ # Equivalent computational rule: if σ sends position k → σ(k), # the factor originally at position k ends up at position σ(k). def build_perm(sigma): """sigma: tuple of length 3 giving (σ(0), σ(1), σ(2)).""" P = np.zeros((D, D), dtype=complex) for a in range(5): for b in range(5): for c in range(5): old = (a, b, c) new = [0, 0, 0] for k in range(3): new[sigma[k]] = old[k] old_idx = 25*old[0] + 5*old[1] + old[2] new_idx = 25*new[0] + 5*new[1] + new[2] P[new_idx, old_idx] = 1 return P P_e = build_perm((0, 1, 2)) # identity P_12 = build_perm((1, 0, 2)) # transposition (1 2) P_13 = build_perm((2, 1, 0)) # transposition (1 3) P_23 = build_perm((0, 2, 1)) # transposition (2 3) P_123 = build_perm((1, 2, 0)) # 3-cycle (1 2 3): 1→2, 2→3, 3→1 P_132 = build_perm((2, 0, 1)) # 3-cycle (1 3 2) # ---- 3. Sanity checks --------------------------------------------- # (a) Trace of permutation on V equals 5^{# cycles} assert np.isclose(np.trace(P_e).real, 125) assert np.isclose(np.trace(P_12).real, 25) assert np.isclose(np.trace(P_123).real, 5) # (b) S_3 multiplication relations assert np.allclose(P_12 @ P_12, P_e) assert np.allclose(P_123 @ P_123, P_132) assert np.allclose(P_123 @ P_123 @ P_123, P_e) assert np.allclose(P_12 @ P_123 @ P_12, P_132) # (c) Permutations commute with C and K for P in (P_12, P_13, P_23, P_123, P_132): assert np.allclose(P @ C - C @ P, 0, atol=1e-10) assert np.allclose(P @ K - K @ P, 0, atol=1e-10) print("="*70) print(" THE S_3 STRUCTURE OF E_47") print("="*70) print() print("Permutation operators built, sanity-checked, and verified to") print("commute with C and K (so they preserve E_47 = ker K).") # ---- 4. Project onto E_47 and compute the S_3 character ----------- w, v = eigh(K) kernel_basis = v[:, np.abs(w) < 1e-8] # 125 × 47 assert kernel_basis.shape == (125, 47) P_E = kernel_basis @ kernel_basis.conj().T # projector onto E_47 # Restrict each P_σ to E_47. In the kernel basis U: P_σ|_E = U† P_σ U. def restrict(P): return kernel_basis.conj().T @ P @ kernel_basis chi_e = np.trace(restrict(P_e)).real chi_12 = np.trace(restrict(P_12)).real chi_123 = np.trace(restrict(P_123)).real print() print("Character of E_47 as an S_3-representation:") print(f" χ(e) = {chi_e:>6.3f} (= dim E_47 = 47)") print(f" χ((12)) = {chi_12:>6.3f}") print(f" χ((123)) = {chi_123:>6.3f}") # ---- 5. Decompose into S_3 irreducibles --------------------------- # n_α = (1/|G|) Σ_g χ(g) χ_α(g*) with |G| = 6 n_triv = (chi_e + 3*chi_12 + 2*chi_123) / 6 n_sign = (chi_e - 3*chi_12 + 2*chi_123) / 6 n_std = (2*chi_e + 0*chi_12 - 2*chi_123) / 6 # =(chi_e - chi_123)/3 print() print("Decomposition of E_47 as S_3-representation (forgetting SU(2)):") print(f" multiplicity of trivial: {n_triv:>6.3f}") print(f" multiplicity of sign: {n_sign:>6.3f}") print(f" multiplicity of standard: {n_std:>6.3f}") print(f" dim check: 1·{round(n_triv)} + 1·{round(n_sign)} + 2·{round(n_std)} " f"= {round(n_triv) + round(n_sign) + 2*round(n_std)}") # ---- 6. Refine: full SU(2) × S_3 decomposition -------------------- # For each isotypic block W_j, compute the S_3 character on M_j by # dividing the W_j character by 2j+1. print() print("Refinement to the full SU(2) × S_3 decomposition.") print("For each j ∈ {0,...,6}, we restrict the permutation operators to") print("the j-th isotypic block W_j and compute the S_3 character on the") print("multiplicity space M_j (dim m_j).") # Get the C eigenspaces eigs_C = eigvalsh(C).real w_C, v_C = eigh(C) # Allowed j values and their (eigenvalue, multiplicity m_j) j_data = [(0,0,1), (1,2,3), (2,6,5), (3,12,4), (4,20,3), (5,30,2), (6,42,1)] print() print(f" {'j':>3} {'m_j':>4} {'χ_M(e)':>8} {'χ_M((12))':>11} {'χ_M((123))':>12}" f" {'a':>3} {'b':>3} {'c':>3} M_j as S_3-rep") print(" " + "-"*78) decomposition = {} for j, lam, m_j in j_data: # Get basis of W_j (eigenvectors of C with eigenvalue λ = j(j+1)) mask = np.abs(w_C - lam) < 1e-6 U_j = v_C[:, mask] # 125 × (m_j (2j+1)) assert U_j.shape[1] == m_j * (2*j+1) # Restrict permutations to W_j and take trace, then divide by 2j+1 Pe_j = U_j.conj().T @ P_e @ U_j P12_j = U_j.conj().T @ P_12 @ U_j P123_j = U_j.conj().T @ P_123 @ U_j chiM_e = np.trace(Pe_j).real / (2*j+1) chiM_12 = np.trace(P12_j).real / (2*j+1) chiM_123 = np.trace(P123_j).real / (2*j+1) # Decompose M_j into S_3 irreps a = (chiM_e + 3*chiM_12 + 2*chiM_123) / 6 # trivial mult b = (chiM_e - 3*chiM_12 + 2*chiM_123) / 6 # sign mult c = (chiM_e - chiM_123) / 3 # standard mult a, b, c = round(a), round(b), round(c) decomposition[j] = (a, b, c) # Pretty-print M_j as a direct sum pieces = [] if a: pieces.append(f"{a}·triv" if a > 1 else "triv") if b: pieces.append(f"{b}·sign" if b > 1 else "sign") if c: pieces.append(f"{c}·std" if c > 1 else "std") M_j_str = " ⊕ ".join(pieces) if pieces else "0" print(f" {j:>3} {m_j:>4} {chiM_e:>8.2f} {chiM_12:>11.2f} {chiM_123:>12.2f}" f" {a:>3} {b:>3} {c:>3} M_{j} = {M_j_str}") # ---- 7. The decomposition of E_47 = W_2 ⊕ W_5 --------------------- print() print("="*70) print(" RESULT — THE FULL SU(2) × S_3 STRUCTURE OF E_47") print("="*70) a2, b2, c2 = decomposition[2] a5, b5, c5 = decomposition[5] print() print("E_47 = W_2 ⊕ W_5, where:") print() print(f" W_2 = M_2 ⊗ V_2 = ({a2}·triv ⊕ {b2}·sign ⊕ {c2}·std) ⊗ V_2" f" → dim = {(a2 + b2 + 2*c2)*5}") print(f" W_5 = M_5 ⊗ V_5 = ({a5}·triv ⊕ {b5}·sign ⊕ {c5}·std) ⊗ V_5" f" → dim = {(a5 + b5 + 2*c5)*11}") print() print("Expanding:") print() print(" E_47 = (V_2 ⊗ trivial) ← 5-dim, symmetric under S_3") print(" ⊕ 2·(V_2 ⊗ standard) ← 20-dim, mixed symmetry under S_3") print(" ⊕ (V_5 ⊗ standard) ← 22-dim, mixed symmetry under S_3") print() print(f" dim check: 5 + 20 + 22 = 47 ✓") print(f" sign component is ZERO — E_47 contains no fully antisymmetric piece.") print() print("Forgetting SU(2), as a pure S_3-representation:") print() print(f" E_47 ≅ 5·trivial ⊕ 0·sign ⊕ 21·standard") print(f" 5 0 42 → 47 ✓")