Projective representation theory of alternating groups

This article describes the projective representation theory of alternating groups.

For alternating groups of degree 1,2,3, the group is a Schur-trivial group (i.e., the Schur multiplier is trivial) and hence all the irreducible projective representations arise from irreducible linear representations.

For the alternating group of degree 4, the Schur covering group is a double cover, namely, special linear group:SL(2,3), and all the irreducible projective representations arise from irreducible linear representations of the double cover.

For the alternating groups of degree 6 and 7, the Schur covering group is a 6-fold cover, whereas for alternating group:A5 and all alternating groups of degree 8 or higher, the Schur covering group is double cover of alternating group. In all these cases, the Schur covering group, which is also the universal central extension, is a perfect group, so the irreducible linear representations of the Schur covering group correspond precisely to irreducible projective representations of the alternating group.

Note that among the irreducible representations of the Schur covering group, those that contain the center in their kernel are precisely the ones that descend to linear representations of the alternating group itself.