# Modular representation theory of special linear group:SL(2,3) at 3

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This article gives specific information, namely, modular representation theory, about a particular group, namely: special linear group:SL(2,3).

View modular representation theory of particular groups | View other specific information about special linear group:SL(2,3)

This article describes the modular representation theory of special linear group:SL(2,3), i.e. ,the linear representation theory in characteristic three -- over field:F3 and its extensions.

## Summary

Item | Value |
---|---|

degrees of irreducible representations (or degrees of irreducible Brauer characters) | 1,2,3 number: 3 |

smallest field of realization of all irreducible representations | field:F3 |

## Family contexts

Family name | Parameter value | General discussion of modular representation theory of family |
---|---|---|

special linear group of degree two | field:F3, i.e., the field with three elements | modular representation theory of special linear group of degree two over a finite field in its defining characteristic |

## Character table

Below is the character table in characteristic 3. The entries are given modulo 3, and are written as 0,1,2.

Representation/conjugacy class representative and size | -- 3-regular, size 1 | -- 3-regular, size 1 | -- 3-regular, size 6 | -- not 3-regular, 4 elements | -- not 3-regular, 4 elements | -- not 3-regular, 4 elements | -- not 3-regular, 4 elements |
---|---|---|---|---|---|---|---|

trivial | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

two-dimensional, given by identity mapping | 2 | 1 | 0 | 2 | 2 | 1 | 1 |

three-dimensional, given by symmetric square of two-dimensional | 0 | 0 | 2 | 0 | 0 | 0 | 0 |

## Brauer characters

### Brauer character table

Irreducible representation in characteristic three whose Brauer character we are computing | Irreducible representation in characteristic zero whose character equals the Brauer character | Value of Brauer character on conjugacy class of -- size 1 | Value of Brauer character on conjugacy class of -- size 1 | Value of Brauer character on conjugacy class of -- size 6 |
---|---|---|---|---|

trivial | trivial | 1 | 1 | 1 |

two-dimensional, given by identity mapping | quaternionic representation of special linear group:SL(2,3) | 2 | -2 | 0 |

three-dimensional, given by symmetric square of two-dimensional | kernel is center, reduces to standard representation of alternating group:A4 on quotient | 3 | 3 | -1 |