Smith Equivalence of Representations for Finite Perfect Groups

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group G, we compute a certain subgroup IO′(G) of the representation ring RO(G). This allows us to prove that a finite perfect group G has a smooth 2–proper action on a sphere with isolated fixed points at which the tangent representations of G are mutually nonisomorphic if and only if G contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer n ≥ 1 and primes p, q, we prove similar results for the group G = An, SL2(Fp), or PSL2(Fq). In particular, G has Smith equivalent representations that are not isomorphic if and only if n ≥ 8, p ≥ 5, q ≥ 19.