Spinorial representations of symmetric groups

Abstract

A real representation ( )pi of a finite group may be regarded as a homomorphism to an orthogonal group O(V). For symmetric groups S-n, alternating groups A(n), and products S-n x S-n' of symmetric groups, we give criteria for whether it lifts to the double cover Pin(V) of O(V), in terms of character values. From these criteria we compute the second Stiefel-Whitney classes of these representations.