On the irreducible characters of the groups Sn and An

Two characters phi and psi of a finite group G are called semiproportional if they are not proportional and there exists a set M in G such that the restrictions of phi and psi to M and G \ M are proportional. We obtain a description for all pairs of proportional irreducible characters of symmetric groups. Namely, in Theorem 1 we prove equivalence of the following conditions for a pair ( phi, psi) of different irreducible characters of S-n (n is an element of N): (1) phi and psi are semiproportional; (2) phi and psi have the same roots; and (3) phi and psi are associated (i.e., psi = phixi where xi is a linear character of S-n with kernel A(n)) Note that (1) and (2) are in general not equivalent for arbitrary finite groups. For the symmetric groups, the equivalence of (1) and (3) validates the following conjecture proven earlier by the author for a number of group classes: serniproportional irreducible characters of a finite group have the same degree. The alternating groups seem to have no serniproportional irreducible characters. Theorem 2 of this article is a step towards proving this conjecture.