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

From Groupprops

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| [[degrees of irreducible representations]] over a [[splitting field]] || 1,3,3,4,5<br>[[maximum degree of irreducible representation|maximum]]: 5, [[lcm of degrees of irreducible representations|lcm]]: 60, [[number of irreducible representations equals number of conjugacy classes|number]]: 5 | | [[degrees of irreducible representations]] over a [[splitting field]] || 1,3,3,4,5<br>[[maximum degree of irreducible representation|maximum]]: 5, [[lcm of degrees of irreducible representations|lcm]]: 60, [[number of irreducible representations equals number of conjugacy classes|number]]: 5 | ||

+ | |} | ||

+ | |||

+ | ==Family contexts== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! 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]] | ||

|} | |} | ||

## Revision as of 01:42, 20 May 2011

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 |

## 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 ] ) ]