# Difference between revisions of "Linear representation theory of alternating group:A5"

## Contents

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

## 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 ] ) ]```