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

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 $3^{3}=27$ 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)	$\mathbb {Q} (e^{2\pi i/3})$ 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	?

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., $p=3$	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 $3^{1+2(1)}$ of "+" type	linear representation theory of extraspecial groups
Burnside group $B(d,n)$	$d=2,n=3$	linear representation theory of Burnside groups

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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 ] ] ] ]