table A7.2partmod3.pdf  sha256 524faffa36f864877d54f2a3cb8aacbdb26e438b061cf8f49f9a7ca2d14c96e0
S_7 mod 3; 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 3)'
partitions of 7: 15; rows found: 15; columns: 9; distinct label strings among the partitions: 15
parse problems: 0
r-regular partitions: 9; columns = r-regular partitions with unitriangular block (James): True
r-regular classes: 9 (must equal the number of columns: True)
M B = X on all 15 rows x 9 classes: inconsistent pairs = 0; degrees positive: True
block check (nonzero decomposition numbers only between partitions with the same 3-core): violations = 0; blocks: 3, of defect zero (3-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) (13): 1; D^(5, 1, 1) (15): 3; D^(4, 3) (1): 1; D^(4, 2, 1) (20): 2; D^(3, 3, 1) (6): 0; D^(3, 2, 2) (15): 3; D^(3, 2, 1, 1) (13): 1
7-cycle: fixed-space dimensions integral and >= 0: True; fixed-point-free on 2 simple module(s): ϕ3 = D^(6, 1) dim 6 [N]; ϕ4 = D^(3, 3, 1) dim 6 [N(x)sgn]; max |beta(x)| = 1
  list is exactly {N, N (x) sgn}: True
  dim V^x for the 5-cycle on every simple module: D^(7,) (1): 1; D^(6, 1) (6): 2; D^(5, 2) (13): 1; D^(5, 1, 1) (15): 3; D^(4, 3) (1): 1; D^(4, 2, 1) (20): 4; D^(3, 3, 1) (6): 2; D^(3, 2, 2) (15): 3; D^(3, 2, 1, 1) (13): 1
5-cycle: fixed-space dimensions integral and >= 0: True; fixed-point-free on 0 simple module(s): none; max |beta(x)| = 2
negative controls: entry (1341, ϕ1) + 1 -> 9 inconsistent pairs; entry (3141, ϕ1) + 1 -> 27 inconsistent pairs
