Congruence relations characterizing the representation ring of the symmetric group (Q1075425): Difference between revisions

It is proven that an arbitrary mapping, \(\chi\), from the partitions of n to \({\mathbb{Z}}\) is a generalized character of the symmetric group on n letters if and only if this mapping satisfies a specified set of congruences. As a corollary, it is demonstrated that \(\chi\) is an irreducible character if and only if \(\chi (1)>0\), \(\sum K(\pi)\chi^ 2(\pi)=n!\), and \(\chi\) satisfies a specified set of congruences.