50.32 Faithful Permutation Characters

PermChars( tbl, arec )

PermChars may as well determine faithful candidates for permutation characters. In that case arec requires the fields normalsubgrp, nonfaithful, chars, lower, upper, and torso.

Let tbl be the character table of the group G, arec.normalsubgrp a list of classes forming a normal subgroup N of G. arec.nonfaithful is a permutation character candidate (see Permutation Character Candidates) of G with kernel N. arec.chars is a list of (in most cases all) rational irreducible characters of tbl.

PermChars computes all those permutation character candidates pi having following properties:

:

arec.chars contains every rational irreducible constituent of pi.

:

pi[i] geq <arec>.'lower'[i] for all integer values of the list arec.lower.

:

pi[i] leq <arec>.'upper'[i] for all integer values of the list arec.upper.

:

pi[i] = <arec>.'torso'[i] for all integer values of the list arec.torso.

:

No irreducible constituent of pi-<arec>.'nonfaithful' has N in its kernel.

If there exists a subgroup V of G, V geq N, with <nonfaithful> = (1_V)^G, the last condition means that the candidates for those possible subgroups U with V = UN are constructed.

Note: At least the degree <torso>[1] must be an integer. If chars does not contain all rational irreducible characters of G, it may happen that any scalar product of pi with an omitted character is negative; there should be nontrivial reasons for excluding a character that is known to be not a constituent of pi.

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