# Projective representation theory of alternating group:A4

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This article gives specific information, namely, projective representation theory, about a particular group, namely: alternating group:A4.

View projective representation theory of particular groups | View other specific information about alternating group:A4

This article describes the projective linear representations of alternating group:A4.

The Schur multiplier of alternating group:A4 is cyclic group:Z2, and the corresponding Schur covering group is special linear group:SL(2,3). Thus, the projective representations of alternating group:A4 all arise from (ordinary) linear representations of special linear group:SL(2,3) (see linear representation theory of special linear group:SL(2,3)).

Representation | Degree | Corresponding element of Schur multiplier, which is cyclic group:Z2 | Number of ordinary representations of special linear group:SL(2,3) that give rise to it | List of these (see linear representation theory of special linear group:SL(2,3)) |
---|---|---|---|---|

trivial | 1 | trivial (identity element) | 3 | all the one-dimensional representations |

three-dimensional | 3 | trivial (identity element) | 1 | the unique three-dimensional representation |

two-dimensional | 2 | nontrivial (non-identity element) | 3 | all the two-dimensional representations |

Note that a representation corresponding to the trivial element of the Schur multiplier is precisely a representation that descends to a linear representation of alternating group:A4.