# Modular representation theory of alternating group:A4 at 2

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

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

This article discusses the modular representation theory of alternating group:A4 in characteristic two, i.e., the linear representation theory in chraacteristic two, specifically, for field:F2 and its extensions.

For information on the linear representation theory in characteristic three (the other modular case) see modular representation theory of alternating group:A4 at 3.

For information on linear representation theory in other characteristics (including characteristic zero, the typical case), see linear representation theory of alternating group:A4.

## Summary

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

degrees of irreducible representations over a field realizing all irreducible representations (or equivalently, degrees of irreducible Brauer characters) | 1,1,1 maximum: 1, lcm: 1, number: 3 |

unique smallest field of realization of irreducible representations in characteristic 2 |
field:F4, the field of four elements |

degrees of irreducible representations over the prime field field:F2 | 1,2 |

## Irreducible representations

### Summary information

Below are the absolutely irreducible representations, i.e., representations that are irreducible in any field of characteristic two realizing them:

Name of representation type | Number of representations of this type | Degree | Criterion for field | Kernel | Liftable to ordinary representation (characteristic zero)? | How does the behavior differ from the non-modular case? |
---|---|---|---|---|---|---|

trivial | 1 | 1 | field:F2 | whole group | Yes | No difference |

one-dimensional mapping to primitive cube roots of unity | 2 | 1 | field:F4 | Klein four-subgroup of alternating group:A4 -- comprising the identity and the three double transpositions | Yes, over where is a primitive cube root of unity | The map surjects to |

Below are representations that are irreducible over some fields but are not absolutely irreducible, i.e., they split in a bigger field:

Name of representation type | Number of representations of this type | Degree | Criterion for field | What happens over a field where it reduces completely? | Kernel | Liftable to ordinary representation (characteristic zero)? | How does the behavior differ from the non-modular case? |
---|---|---|---|---|---|---|---|

two-dimensional representation over a non-splitting field | 1 | field:F2 | splits into the two nontrivial one-dimensionals | Klein four-subgroup of alternating group:A4 | 2 | Yes | No difference |

## Character table

Denote by one of the elements in field:F4 not in field:F2, hence is a primitive cube root of unity. Note that is the other primitive cube root of unity. The character table is as follows:

Rep/conj class representative | -- 2-regular | -- 2-regular | -- 2-regular | (not 2-regular) |
---|---|---|---|---|

trivial | 1 | 1 | 1 | 1 |

one of the one-dimensional nontrivial ones | 1 | 1 | ||

the other one-dimensional nontrivial one | 1 | 1 |

## Brauer characters

The Brauer characters can all be realized over where is a primitive cube root of unity. This field is isomorphic to and can be viewed as a subfield of the complex number where .

To establish Brauer characters, we need to choose a bijection between the primitive cube roots of unity in field:F4 and those in characteristic zero. One such bijection is . The other bijection would be . The choice of bijection simply affects the correspondence between modular characters and ordinary characters, and does not affect the set of Brauer characters.

Since all characters are one-dimensional, the Brauer characters actually *are* ordinary characters in the most straightforward way imaginable.

### Brauer character table

Irreducible representation in characteristic two 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 | Value of Brauer character on conjugacy class of | Value of Brauer character on conjugacy class of |
---|---|---|---|---|

trivial | trivial | 1 | 1 | 1 |

one of the one-dimensional nontrivial ones | one of the one-dimensional nontrivial ones | 1 | ||

the other one-dimensional nontrivial one | the other one-dimensional nontrivial one | 1 |