# Linear representation theory of unitriangular matrix group:UT(3,3)

From Groupprops

This article gives specific information, namely, linear representation theory, about a particular group, namely: unitriangular matrix group:UT(3,3).

View linear representation theory of particular groups | View other specific information about unitriangular matrix group:UT(3,3)

This article describes the linear representation theory of unitriangular matrix group:UT(3,3), which is the unitriangular matrix group of degree three over field:F3. It is the unique non-abelian group of order and exponent three.

## Summary

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

degrees of irreducible representations over a splitting field | 1,1,1,1,1,1,1,1,1,3,3 (1 occurs 9 times, 3 occurs 2 times) maximum: 3, lcm: 3, number: 11, sum of squares: 27 |

Schur index values of irreducible representations | 1,1,1,1,1,1,1,1,1,1,1 |

smallest field of realization (characteristic zero) | or |

condition for a field to be a splitting field | characteristic not 3, contains a primitive cube root of unity, i.e., the polynomial splits. For a finite field of size , equivalent to 3 dividing |

smallest size splitting field | field:F4, i.e., the field with 4 elements |

orbit structure of irreducible representations under automorphism group | ? |

## Family contexts

Family | Parameter values | General discussion of linear representation theory of family |
---|---|---|

unitriangular matrix group of degree three, more specifically unitriangular matrix group:UT(3,p) | field:F3, i.e., | specific: linear representation theory of unitriangular matrix group:UT(3,p) more generic: linear representation theory of unitriangular matrix group of degree three over a finite field |

extraspecial group | extraspecial group of order of "+" type | linear representation theory of extraspecial groups |

Burnside group | linear representation theory of Burnside groups |

This page is currently under development, but you can probably get the information you're interested in from the following alternative page: linear representation theory of unitriangular matrix group:UT(3,p)

## GAP implementation

### Degrees of irreducible representations

The degrees of irreducible representations can be found using GAP's CharacterDegrees function:

gap> CharacterDegrees(SmallGroup(27,3)); [ [ 1, 9 ], [ 3, 2 ] ]

### Character table

The full character table can be obtained as follows:

gap> Irr(CharacterTable(SmallGroup(27,3))); [ Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 1, E(3), 1, 1, E(3)^2, E(3), 1, 1, E(3)^2, E(3), E(3)^2 ] ), Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 1, E(3)^2, 1, 1, E(3), E(3)^2, 1, 1, E(3), E(3)^2, E(3) ] ), Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 1, 1, E(3), 1, 1, E(3), E(3)^2, 1, E(3), E(3)^2, E(3)^2 ] ), Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 1, E(3), E(3), 1, E(3)^2, E(3)^2, E(3)^2, 1, 1, 1, E(3) ] ), Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 1, E(3)^2, E(3), 1, E(3), 1, E(3)^2, 1, E(3)^2, E(3), 1 ] ), Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 1, 1, E(3)^2, 1, 1, E(3)^2, E(3), 1, E(3)^2, E(3), E(3) ] ), Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 1, E(3), E(3)^2, 1, E(3)^2, 1, E(3), 1, E(3), E(3)^2, 1 ] ), Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 1, E(3)^2, E(3)^2, 1, E(3), E(3), E(3), 1, 1, 1, E(3)^2 ] ), Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 3, 0, 0, 3*E(3), 0, 0, 0, 3*E(3)^2, 0, 0, 0 ] ), Character( CharacterTable( <pc group of size 27 with 3 generators> ), [ 3, 0, 0, 3*E(3)^2, 0, 0, 0, 3*E(3), 0, 0, 0 ] ) ]

A more display-friendly form:

gap> Display(CharacterTable(SmallGroup(27,3))); CT1 3 3 2 2 3 2 2 2 3 2 2 2 1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A 1 1 /A A 1 1 /A A /A X.3 1 /A 1 1 A /A 1 1 A /A A X.4 1 1 A 1 1 A /A 1 A /A /A X.5 1 A A 1 /A /A /A 1 1 1 A X.6 1 /A A 1 A 1 /A 1 /A A 1 X.7 1 1 /A 1 1 /A A 1 /A A A X.8 1 A /A 1 /A 1 A 1 A /A 1 X.9 1 /A /A 1 A A A 1 1 1 /A X.10 3 . . B . . . /B . . . X.11 3 . . /B . . . B . . . A = E(3) = (-1+ER(-3))/2 = b3 B = 3*E(3) = (-3+3*ER(-3))/2 = 3b3

### Irreducible representations

The complete list of irreducible representations can be output using the IrreducibleRepresentations function.

gap> IrreducibleRepresentations(SmallGroup(27,3)); [ Pcgs([ f1, f2, f3 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ E(3) ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ E(3)^2 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ 1 ] ], [ [ E(3) ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ E(3) ] ], [ [ E(3) ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ E(3)^2 ] ], [ [ E(3) ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ 1 ] ], [ [ E(3)^2 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ E(3) ] ], [ [ E(3)^2 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ E(3)^2 ] ], [ [ E(3)^2 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], [ [ 1, 0, 0 ], [ 0, E(3), 0 ], [ 0, 0, E(3)^2 ] ], [ [ E(3), 0, 0 ], [ 0, E(3), 0 ], [ 0, 0, E(3) ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], [ [ 1, 0, 0 ], [ 0, E(3)^2, 0 ], [ 0, 0, E(3) ] ], [ [ E(3)^2, 0, 0 ], [ 0, E(3)^2, 0 ], [ 0, 0, E(3)^2 ] ] ] ]