# 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 |

## GAP implementation

### Degrees of irreducible representations

The degrees of irreducible representation can be computed using the CharacterDegrees function:

gap> CharacterDegrees(SmallGroup(16,6)); [ [ 1, 8 ], [ 2, 2 ] ]

### Character table

The character table can be computed using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(SmallGroup(16,6))); [ Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, -1, 1, 1, 1, -1, -1, 1, 1, -1 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, 1, -1, 1, 1, -1, 1, -1, 1, -1 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, -1, -1, 1, 1, 1, -1, -1, 1, 1 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, E(4), 1, -1, 1, E(4), -E(4), -1, -1, -E(4) ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, -E(4), 1, -1, 1, -E(4), E(4), -1, -1, E(4) ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, E(4), -1, -1, 1, -E(4), -E(4), 1, -1, E(4) ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, -E(4), -1, -1, 1, E(4), E(4), 1, -1, -E(4) ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 2, 0, 0, 2*E(4), -2, 0, 0, 0, -2*E(4), 0 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 2, 0, 0, -2*E(4), -2, 0, 0, 0, 2*E(4), 0 ] ) ]

A nicer display can be achieved using the Display function:

gap> Display(CharacterTable(SmallGroup(16,6))); CT3 2 4 3 3 4 4 3 3 3 4 3 1a 8a 2a 4a 2b 8b 8c 4b 4c 8d X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 -1 -1 1 1 -1 X.3 1 1 -1 1 1 -1 1 -1 1 -1 X.4 1 -1 -1 1 1 1 -1 -1 1 1 X.5 1 A 1 -1 1 A -A -1 -1 -A X.6 1 -A 1 -1 1 -A A -1 -1 A X.7 1 A -1 -1 1 -A -A 1 -1 A X.8 1 -A -1 -1 1 A A 1 -1 -A X.9 2 . . B -2 . . . -B . X.10 2 . . -B -2 . . . B . A = E(4) = ER(-1) = i B = 2*E(4) = 2*ER(-1) = 2i

### Irreducible representations

The irreducible representations can be computed explicitly using the IrreducibleRepresentations function:

gap> IrreducibleRepresentations(SmallGroup(16,6)); [ Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ E(4) ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -E(4) ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ E(4) ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -E(4) ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, E(4) ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], [ [ E(4), 0 ], [ 0, E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ] , Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, -E(4) ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], [ [ -E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ] ]