==============================================================================
(A) irreducible dimensions of S_N^+ and decompositions of an N-dim corepresentation
==============================================================================
N= 4: dims r_0..r_7 = [1, 3, 5, 7, 9, 11, 13, 15];  irreps of dim<=N: [0, 1];  N-dim decompositions: ['r0 + r0 + r0 + r0', 'r0 + r1']
N= 5: dims r_0..r_7 = [1, 4, 11, 29, 76, 199, 521, 1364];  irreps of dim<=N: [0, 1];  N-dim decompositions: ['r0 + r0 + r0 + r0 + r0', 'r0 + r1']
N= 6: dims r_0..r_7 = [1, 5, 19, 71, 265, 989, 3691, 13775];  irreps of dim<=N: [0, 1];  N-dim decompositions: ['6*r0', 'r0 + r1']
N= 7: dims r_0..r_7 = [1, 6, 29, 139, 666, 3191, 15289, 73254];  irreps of dim<=N: [0, 1];  N-dim decompositions: ['7*r0', 'r0 + r1']
N= 8: dims r_0..r_7 = [1, 7, 41, 239, 1393, 8119, 47321, 275807];  irreps of dim<=N: [0, 1];  N-dim decompositions: ['8*r0', 'r0 + r1']
N= 9: dims r_0..r_7 = [1, 8, 55, 377, 2584, 17711, 121393, 832040];  irreps of dim<=N: [0, 1];  N-dim decompositions: ['9*r0', 'r0 + r1']
N=10: dims r_0..r_7 = [1, 9, 71, 559, 4401, 34649, 272791, 2147679];  irreps of dim<=N: [0, 1];  N-dim decompositions: ['10*r0', 'r0 + r1']
N=11: dims r_0..r_7 = [1, 10, 89, 791, 7030, 62479, 555281, 4935050];  irreps of dim<=N: [0, 1];  N-dim decompositions: ['11*r0', 'r0 + r1']
N=12: dims r_0..r_7 = [1, 11, 109, 1079, 10681, 105731, 1046629, 10360559];  irreps of dim<=N: [0, 1];  N-dim decompositions: ['12*r0', 'r0 + r1']
OK: for every N in 4..12 an N-dimensional unitary corepresentation of S_N^+ is either
    N copies of the trivial one or r_0 + r_1 (= the magic fundamental representation).
    (For all N >= 4 this follows from d_{k+1}-d_k = (N-3)d_k - d_{k-1} > 0 by induction.)

==============================================================================
(B) transitive actions of S_N on k points (homomorphisms S_N -> S_k, transitive image)
==============================================================================
N=4: index 2: 1 subgroups; index 3: 3 subgroups; index 4: 4 subgroups;    [k=N: #fixed points of phi(transposition) over transitive homs: {(2,): 24}]
N=5: index 2: 1 subgroups; index 3: 0 subgroups; index 4: 0 subgroups; index 5: 5 subgroups;    [k=N: #fixed points of phi(transposition) over transitive homs: {(3,): 120}]
N=6: index 2: 1 subgroups; index 3: 0 subgroups; index 4: 0 subgroups; index 5: 0 subgroups; index 6: 12 subgroups;    [k=N: #fixed points of phi(transposition) over transitive homs: {(4,): 720, (0,): 720}]
N=7: index 2: 1 subgroups; index 3: 0 subgroups; index 4: 0 subgroups; index 5: 0 subgroups; index 6: 0 subgroups; index 7: 7 subgroups;    [k=N: #fixed points of phi(transposition) over transitive homs: {(5,): 5040}]
Reading: for N>=5 there is no subgroup of index k with 3<=k<=N-1; for N=6 the index-6
subgroups split into point stabilisers (transposition fixes N-2=4 points) and exotic ones
(transposition acts fixed-point-freely), which the permutation character excludes;
for N=4 the index-3 subgroups (the three D_4's) exist, so the sketch needs an extra case.

==============================================================================
(C) N=4: standard rep of S_4 in the Hadamard basis; characters of D_4
==============================================================================
H P_sigma H^T/4 = diag(1, signed 3x3 permutation matrix) for all 24 sigma: True
number of linear characters of D_4: 4
  chi on generators {'(01)': 1, '(23)': 1, '(02)(13)': 1}:  Ind(chi) == standard character: False
  chi on generators {'(01)': 1, '(23)': 1, '(02)(13)': -1}:  Ind(chi) == standard character: True
  chi on generators {'(01)': -1, '(23)': -1, '(02)(13)': 1}:  Ind(chi) == standard character: False
  chi on generators {'(01)': -1, '(23)': -1, '(02)(13)': -1}:  Ind(chi) == standard character: False
characters of D_4 inducing the standard representation: 1
character realised on the Hadamard line f_{01|23}: {'(01)': 1, '(23)': 1, '(02)(13)': -1}
DONE s1
