Linear representation theory of projective special linear group:PSL(3,2)

From Groupprops
Jump to: navigation, search
This article gives specific information, namely, linear representation theory, about a particular group, namely: projective special linear group:PSL(3,2).
View linear representation theory of particular groups | View other specific information about projective special linear group:PSL(3,2)

Summary

Item Value
degrees of irreducible representations over a splitting field 1,3,3,6,7,8
maximum: 8, lcm: 168, number: 6, sum of squares: 168

Family contexts

Family name Parameter values General discussion of linear representation theory of family
projective special linear group of degree two field:F7 linear representation theory of projective special linear group of degree two over a finite field
general linear group of degree three field:F2 linear representation theory of general linear group of degree three over a finite field

GAP implementation

The degrees of irreducible representations can be computed using GAP's CharacterDegrees and PSL functions:

gap> CharacterDegrees(PSL(3,2));
[ [ 1, 1 ], [ 3, 2 ], [ 6, 1 ], [ 7, 1 ], [ 8, 1 ] ]

The characters of irreducible representations can be computed using GAP's CharacterTable function:

gap> Irr(CharacterTable(PSL(3,2)));
[ Character( CharacterTable( Group([ (4,6)(5,7), (1,2,4)(3,6,5) ]) ), [ 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Group(
    [ (4,6)(5,7), (1,2,4)(3,6,5) ]) ), [ 3, -1, 1, 0, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4 ] ), Character( CharacterTable( Group(
    [ (4,6)(5,7), (1,2,4)(3,6,5) ]) ), [ 3, -1, 1, 0, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6 ] ), Character( CharacterTable( Group(
    [ (4,6)(5,7), (1,2,4)(3,6,5) ]) ), [ 6, 2, 0, 0, -1, -1 ] ), Character( CharacterTable( Group([ (4,6)(5,7), (1,2,4)(3,6,5) ]) ), [ 7, -1, -1, 1, 0, 0
     ] ), Character( CharacterTable( Group([ (4,6)(5,7), (1,2,4)(3,6,5) ]) ), [ 8, 0, 0, -1, 1, 1 ] ) ]