table A7.2partmod2.pdf  sha256 a50809b8d2ecb9573a327f4558db7d747996ebfab67317488c0cdbbed6426d8f
S_7 mod 2; pypdf 6.16.2
self-test of the Murnaghan-Nakayama code (hook formula, row orthogonality) for S_7: True
pages: 1, sections: 1; row blocks: 1; column blocks: 1
title line: 'S7 (mod 2)'
partitions of 7: 15; rows found: 15; columns: 5; distinct label strings among the partitions: 15
parse problems: 0
r-regular partitions: 5; columns = r-regular partitions with unitriangular block (James): True
r-regular classes: 5 (must equal the number of columns: True)
M B = X on all 15 rows x 5 classes: inconsistent pairs = 0; degrees positive: True
block check (nonzero decomposition numbers only between partitions with the same 2-core): violations = 0; blocks: 2, of defect zero (2-cores of size 7): 0
columns with a single nonzero entry: 0; single +1 alterations invisible to M B = X (and to unitriangularity and positive degrees): 0; all of them violate the block check
  dim V^x for the 7-cycle on every simple module: D^(7,) (1): 1; D^(6, 1) (6): 0; D^(5, 2) (14): 2; D^(4, 3) (8): 2; D^(4, 2, 1) (20): 2
7-cycle: fixed-space dimensions integral and >= 0: True; fixed-point-free on 1 simple module(s): ϕ3 = D^(6, 1) dim 6 [N]; max |beta(x)| = 1
  list is exactly {N} (r = 2, where N (x) sgn = N): True
  dim V^x for the 5-cycle on every simple module: D^(7,) (1): 1; D^(6, 1) (6): 2; D^(5, 2) (14): 2; D^(4, 3) (8): 0; D^(4, 2, 1) (20): 4
5-cycle: fixed-space dimensions integral and >= 0: True; fixed-point-free on 1 simple module(s): ϕ2 = D^(4, 3) dim 8 [OTHER]; max |beta(x)| = 2
  list is exactly {N} (r = 2, where N (x) sgn = N): False
negative controls: entry (1251, ϕ1) + 1 -> 5 inconsistent pairs; entry (2151, ϕ1) + 1 -> 25 inconsistent pairs
