On some applications of group representation theory to algebraic problems related to the congruence principle for equivariant maps

Given a finite group G and two unitary G-representations V and W, possible restrictions on Brouwer degrees of equivariant maps between the representation spheres S(V) and S(W) are usually expressed in terms of congruences modulo the greatest common divisor of lengths of orbits in S(V) (denoted α(V)).  Effective application of these congruences is limited by answers to the following questions:

(i) Under which conditions is α(V)>1?

(ii) Does there exist an equivariant map whose degree is easy to calculate?

In my talk I'll address mainly the first question. It will be shown that α(V)>1 for every irreducible non-trivial C[G]-module if and only if G is solvable. So this result provides a new solvability criterion for finite groups.

This is a joint work with Z. Balanov and Haopin Wu.