Linear representation theory of alternating group:A5

From Groupprops
Revision as of 16:13, 15 June 2011 by Vipul (talk | contribs)
Jump to: navigation, search
This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A5.
View linear representation theory of particular groups | View other specific information about alternating group:A5

Summary

Item Value
degrees of irreducible representations over a splitting field 1,3,3,4,5
maximum: 5, lcm: 60, number: 5

Family contexts

Family name Parameter values General discussion of linear representation theory of family
alternating group 5 linear representation theory of alternating groups
projective general linear group of degree two over a finite field field:F4 linear representation theory of projective general linear group of degree two over a finite field
projective special linear group of degree two over a finite field field:F5 linear representation theory of projective special linear group of degree two over a finite field
COMPARE AND CONTRAST: View linear representation theory of groups of order 60 to compare and contrast the linear representation theory with other groups of order 60.

GAP implementation

The character degrees can be computed using GAP's CharacterDegrees function:

gap> CharacterDegrees(AlternatingGroup(5));
[ [ 1, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5, 1 ] ]

This means that there is 1 degree 1 irreducible representation, 2 degree 3 irreducible representations, 1 degree 4 irreducible representation, and 1 degree 5 irreducible representation.

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

gap> Irr(CharacterTable(AlternatingGroup(5)));
[ Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 5 ] ) ),
    [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), Character( CharacterTable( Alt(
    [ 1 .. 5 ] ) ), [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ),
  Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 4, 0, 1, -1, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 5, 1, -1, 0, 0 ] ) ]