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

This article gives specific information, namely, linear representation theory, about a particular group, namely: prime-cube order group:U(3,3).
View linear representation theory of particular groups | View other specific information about prime-cube order group:U(3,3)

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)	$\mathbb{Q}(e^{2\pi i/3})$ or $\mathbb{Q}[x]/(x^2+x+1)$
condition for a field to be a splitting field	characteristic not 3, contains a primitive cube root of unity, i.e., the polynomial $x^2+x+1$ splits. For a finite field of size $q$, equivalent to 3 dividing $q-1$
smallest size splitting field	field:F4, i.e., the field with 4 elements
orbit structure of irreducible representations under automorphism group	?

GAP implementation

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

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

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 ] ) ]