# Projective representation theory of alternating groups

## Contents

This article gives specific information, namely, projective representation theory, about a family of groups, namely: alternating group.
View projective representation theory of group families | View other specific information about alternating group

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.

## Particular cases

$n$ alternating group $A_n$ order of $A_n$ Schur covering group of $A_n$ order of Schur covering group of $A_n$ Degrees of irreducible linear representations of $A_n$ Degrees of irreducible linear representations of the Schur covering group (includes those of $A_n$, and more) Degrees of irreducible projective representations of $A_n$ (for $n \le 4$, different linear representations may be projectively equivalent if they differ multiplicatively by a one-dimensional representation) Information on projective representation theory
1 trivial group 1 trivial group 1 1 1 1 projective representation theory of trivial group
2 trivial group 1 trivial group 1 1 1 1 projective representation theory of trivial group
3 cyclic group:Z3 3 cyclic group:Z3 3 1,1,1 1,1,1 1 projective representation theory of cyclic group:Z3
4 alternating group:A4 12 special linear group:SL(2,3) 24 1,1,1,3 1,1,1,2,2,2,3 1,2,3 projective representation theory of alternating group:A4
5 alternating group:A5 60 special linear group:SL(2,5) 120 1,3,3,4,5 1,2,2,3,3,4,4,5,6 1,2,2,3,3,4,4,5,6 projective representation theory of alternating group:A5
6 alternating group:A6 360 Schur cover of alternating group:A6 2160 1,5,5,8,8,9,10  ?  ? projective representation theory of alternating group:A6
7 alternating group:A7 2520 Schur cover of alternating group:A7 15120 1,6,10,10,14,14,15,21,35  ?  ? projective representation theory of alternating group:A7
8 alternating group:A8 20160 double cover of alternating group:A8 40320 1,7,14,20,21,21,21,28,35,45,45,56,64,70  ?  ? projective representation theory of alternating group:A8