Linear representation theory of 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.

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(  ),   [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable(  ),   [ 1, E(3), 1, 1, E(3)^2, E(3), 1, 1, E(3)^2, E(3), E(3)^2 ] ), Character( CharacterTable(  ),   [ 1, E(3)^2, 1, 1, E(3), E(3)^2, 1, 1, E(3), E(3)^2, E(3) ] ), Character( CharacterTable(  ),   [ 1, 1, E(3), 1, 1, E(3), E(3)^2, 1, E(3), E(3)^2, E(3)^2 ] ), Character( CharacterTable(  ),   [ 1, E(3), E(3), 1, E(3)^2, E(3)^2, E(3)^2, 1, 1, 1, E(3) ] ), Character( CharacterTable(  ),   [ 1, E(3)^2, E(3), 1, E(3), 1, E(3)^2, 1, E(3)^2, E(3), 1 ] ), Character( CharacterTable(  ),   [ 1, 1, E(3)^2, 1, 1, E(3)^2, E(3), 1, E(3)^2, E(3), E(3) ] ), Character( CharacterTable(  ),   [ 1, E(3), E(3)^2, 1, E(3)^2, 1, E(3), 1, E(3), E(3)^2, 1 ] ), Character( CharacterTable(  ),   [ 1, E(3)^2, E(3)^2, 1, E(3), E(3), E(3), 1, 1, 1, E(3)^2 ] ), Character( CharacterTable(  ),   [ 3, 0, 0, 3*E(3), 0, 0, 0, 3*E(3)^2, 0, 0, 0 ] ), Character( CharacterTable(  ),   [ 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 ] ] ] ]