# Projective representation theory of symmetric group:S3

From Groupprops

This article gives specific information, namely, projective representation theory, about a particular group, namely: symmetric group:S3.

View projective representation theory of particular groups | View other specific information about symmetric group:S3

This article describes the projective representation theory of symmetric group:S3, which we call , in characteristic zero.

First, note that the Schur multiplier is isomorphic to trivial group. For more, see group cohomology of symmetric group:S3.

Therefore, the only kinds of projective representations are those arising from linear representations, i.e., those for the trivial cohomology class. Thus, the irreducible projective representations are the same thing as the equivalence classes of irreducible linear representations under the multiplicative action of one-dimensional representations.

Here is a list of the irreducible projective representations and linear representations that give rise to them:

Representation | Degree | Number of linear representations that give rise to it | List of linear representations |
---|---|---|---|

trivial | 1 | 2 | trivial representation and sign representation |

standard | 2 | 1 | standard representation of symmetric group:S3 |