# Linear representation theory of M16

This article gives specific information, namely, linear representation theory, about a particular group, namely: M16.

View linear representation theory of particular groups | View other specific information about M16

This article discusses the linear representation theory of the group M16, a group of order 16 given by the presentation:

## Summary

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

degrees of irreducible representations over a splitting field (such as or ) | 1,1,1,1,1,1,1,1,2,2 (1 occurs 8 times, 2 occurs 2 times) maximum: 2, lcm: 2, number: 10, sum of squares: 16 |

Schur index values of irreducible representations | 1 (all of them) |

smallest ring of realization (characteristic zero) | -- ring of Gaussian integers |

ring generated by character values (characteristic zero) | |

minimal splitting field, i.e., smallest field of realization (characteristic zero) | Same as field generated by character values, because all Schur index values are 1. |

condition for a field to be a splitting field | The characteristic should not be equal to 2, and the polynomial should split. For a finite field of size , this is equivalent to saying that |

minimal splitting field in characteristic | Case : prime field Case : Field , quadratic extension of prime field |

smallest size splitting field | Field:F5, i.e., the field with five elements. |

degrees of irreducible representations over the rational numbers | 1,1,1,1,2,2,4 (1 occurs 4 times, 2 occurs 2 times, 4 occurs 1 time) number: 7 |

orbits of irreducible representations over a splitting field under action of automorphism group | 2 orbits of size 1 of degree 1 representations, 3 orbits of size 2 of degree 1 representations, 1 orbit of size 2 of degree 2 representations. number: 6 |

## Representations

### Summary information

Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2, except in the last column, where we consider what happens in characteristic 2.

Name of representation type | Number of representations of this type | Degree | Schur index | Criterion for field | Kernel (a normal subgroup of M16 comprising the elements that map to identity matrices; see subgroup structure of M16) | Quotient by kernel (on which it descends to a faithful representation) | Characteristic 2? |
---|---|---|---|---|---|---|---|

trivial | 1 | 1 | 1 | any | whole group | trivial group | works |

sign, kernel a non-cyclic maximal subgroup | 1 | 1 | 1 | any | direct product of Z4 and Z2 in M16 -- | cyclic group:Z2 | works, same as trivial |

sign, kernel a cyclic maximal subgroup | 2 | 1 | 1 | any | Z8 in M16 -- either or | cyclic group:Z2 | works, same as trivial |

representation with kernel | 2 | 1 | 1 | must contain a primitive fourth root of unity, or equivalently, must split | non-central Z4 in M16 | cyclic group:Z4 | works, same as trivial |

representation with kernel | 2 | 1 | 1 | must contain a primitive fourth root of unity, or equivalently, must split | V4 in M16 | cyclic group:Z4 | works, same as trivial |

faithful irreducible representation of M16 | 2 | 2 | 1 | must contain a primitive fourth root of unity, or equivalently, must split | trivial subgroup | M16 | indecomposable but not irreducible |

Below are representations that are irreducible over a non-splitting field, but split over a splitting field.

Name of representation type | Number of representations of this type | Degree | Criterion for field | What happens over a splitting field? | Kernel | Quotient by kernel (on which it descends to a faithful representation) |
---|---|---|---|---|---|---|

two-dimensional representation with kernel | 1 | 2 | must not contain a primitive fourth root of unity, or equivalently, does not split |
splits into the one-dimensional representations with kernel | non-central Z4 in M16 | cyclic group:Z4 |

two-dimensional representation with kernel | 1 | 2 | must not contain a primitive fourth root of unity, or equivalently, does not split |
splits into the one-dimensional representations with kernel | Klein four-group | cyclic group:Z4 |

four-dimensional representation | 1 | 4 | must not contain a primitive fourth root of unity, or equivalently, does not split |
splits into the two faithful irreducible representations of degree two | trivial subgroup | M16 |

## Character table

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):Orthogonality relations: Character orthogonality theorem | Column orthogonality theoremSeparation results(basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zeroNumerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integersCharacter value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

Below is the character table over a splitting field. Here denotes a square root of in the field.

Representation/conjugacy class and size | (size 1) | (size 1) | (size 1) | (size 1) | (size 2) | (size 2) | (size 2) | (size 2) | (size 2) | (size 2) |
---|---|---|---|---|---|---|---|---|---|---|

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

-kernel | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |

-kernel | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |

-kernel | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 |

-kernel (first) | 1 | 1 | -1 | -1 | -1 | 1 | ||||

-kernel (second) | 1 | 1 | -1 | -1 | -1 | 1 | ||||

-kernel (first) | 1 | 1 | -1 | -1 | 1 | -1 | ||||

-kernel (second) | 1 | 1 | -1 | -1 | 1 | -1 | ||||

faithful irreducible representation of M16 (first) | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ||

faithful irreducible representation of M16 (second) | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |

Below are the size-degree-weighted characters, obtained by multiplying each character value by the size of the conjugacy class and dividing by the degree of the irreducible representation:

Representation/conjugacy class and size | (size 1) | (size 1) | (size 1) | (size 1) | (size 2) | (size 2) | (size 2) | (size 2) | (size 2) | (size 2) |
---|---|---|---|---|---|---|---|---|---|---|

trivial | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |

-kernel | 1 | 1 | 1 | 1 | 2 | 2 | -2 | -2 | -2 | -2 |

-kernel | 1 | 1 | 1 | 1 | -2 | -2 | 2 | 2 | -2 | -2 |

-kernel | 1 | 1 | 1 | 1 | -2 | -2 | -2 | -2 | 2 | 2 |

-kernel (first) | 1 | 1 | -1 | -1 | -2 | 2 | ||||

-kernel (second) | 1 | 1 | -1 | -1 | -2 | 2 | ||||

-kernel (first) | 1 | 1 | -1 | -1 | 2 | -2 | ||||

-kernel (second) | 1 | 1 | -1 | -1 | 2 | -2 | ||||

faithful irreducible representation of M16 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ||

faithful irreducible representation of M16 (second) | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |