# Modular representation theory of symmetric group:S3 at 3

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

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

This article describes the modular representation theory for symmetric group:S3 at the prime three, i.e., for fields of characteristic three, specifically field:F3 and its extensions.

For more information on the (modular) linear representation theory in characteristic two, see modular representation theory of symmetric group:S3 at 2.

For more information on the linear representation theory in other characteristics, see linear representation theory of symmetric group:S3.

## Contents

## Irreducible representations

There are two irreducible representations, both of them arising from irreducible representations in characteristic zero, namely:

Name of representation type | Number of representations of this type | Field of realization | Kernel | Degree | Liftable to ordinary representation (characteristic zero)? |
---|---|---|---|---|---|

trivial | 1 | field:F3 | whole group | 1 | Yes (lift is also called trivial representation) |

sign | 1 | field:F3 | A3 in S3 | 1 | Yes (lift is also called sign representation) |

### Trivial representation

The trivial or principal representation is a one-dimensional representation sending every element of the symmetric group to the identity matrix of order one.

### Sign representation

The sign representation or alternating representation sends all the even permutations (the identity element and the two 3-cycles) to the matrix and the three 2-cycles to the matrix . It arises by reducing modulo 3 the sign representation in characteristic zero.

## Character table

The character table (taking values in field:F3) is as follows:

Rep/Conj class | (-regular) | (-regular) | (not -regular) |
---|---|---|---|

trivial | 1 | 1 | 1 |

sign | 1 | -1 | 1 |

## Brauer characters

Symmetric group:S3 is a rational representation group, in the sense that all irreducible representations are realized with integer entries. Further, all irreducible representations in characteristic three are realized over field:F3.

Moreover, the relevant eigenvalues do not require adjoining roots of unity, because the fields already come equipped with square roots of unity.

### Brauer character table

There are two 3-regular conjugacy classes: the identity element and the 2-cycles. There are two Brauer characters: arising from the trivial and the sign representation respectively.

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 identity element | Value of Brauer character on conjugacy class of , , and |
---|---|---|---|

trivial representation | trivial representation | 1 | 1 |

sign representation | sign representation | 1 | -1 |

## GAP implementation

The character degrees can be computed using GAP's CharacterDegrees function as follows:

gap> CharacterDegrees(SymmetricGroup(3),3); [ [ 1, 2 ] ]

The character table can be computed using GAP's CharacterTable function as follows:

gap> Irr(CharacterTable(SymmetricGroup(3),3)); [ Character( BrauerTable( Sym( [ 1 .. 3 ] ), 3 ), [ 1, -1 ] ), Character( BrauerTable( Sym( [ 1 .. 3 ] ), 3 ), [ 1, 1 ] ) ]