Inequalities( tbl )
The condition pi(g) geq 0 for every permutation character candidate pi places restrictions on the multiplicities a_i of the irreducible constituents chi_i of pi = sum_{i=1}^r a_i chi_i. For every group element g holds sum_{i=1}^r a_i chi_i(g) geq 0. The power map provides even stronger conditions.
This system of inequalities is kind of diagonalized, resulting in a
system of inequalities restricting a_i in terms of a_j, j < i.
These inequalities are used to construct characters with nonnegative
values (see PermChars). PermChars either calls Inequalities or
takes this information from the record field ineq of its argument
record.
The number of inequalities arising in the process of diagonalization may grow very strong.
There are two strategies to perform this diagonalization. The default is to simply eliminate one unknown a_i after the other with decreasing i. In some cases it turns out to be better first to look which choice for the next unknown will yield the fewest new inequalities.
GAP 3.4.4