# Linear representation theory of symmetric group:S8

## Contents

View linear representation theory of particular groups | View other specific information about symmetric group:S8

## Summary

Item Value
degrees of irreducible representations over a splitting field 1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90
maximum: 90, lcm: 20160, number: 22, sum of squares: 40320
Schur index values of irreducible representations over a splitting field 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
smallest ring of realization (characteristic zero) $\mathbb{Z}$ -- ring of integers
smallest field of realization (characteristic zero), i.e., smallest splitting field in characteristic zero $\mathbb{Q}$ -- hence it is a rational-representation group
condition for a field to be a splitting field any field of characteristic not 2,3,5,7
smallest size finite splitting field field:F11

## Family contexts

Family name Parameter values General discussion of linear representation theory of family
symmetric group 8 linear representation theory of symmetric groups

## GAP implementation

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

```gap> CharacterDegrees(SymmetricGroup(8));
[ [ 1, 2 ], [ 7, 2 ], [ 14, 2 ], [ 20, 2 ], [ 21, 2 ], [ 28, 2 ], [ 35, 2 ], [ 42, 1 ], [ 56, 2 ], [ 64, 2 ], [ 70, 2 ], [ 90, 1 ] ]```

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

```gap> Irr(CharacterTable(SymmetricGroup(8)));
[ Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, -1 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 7, -5, 3, -1, -1, 4, -2, 0, 1, 1, -3, 1, 1, 0, -1, 2, 0, -1, -1, -1, 0, 1 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 20, -10, 4, -2, 4, 5, -1, 1, -1, -1, -2, 0, -2, 1, 0, 0, 0, 0, 1, 1, -1, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 28, -10, 4, -2, -4, 1, -1, 1, 1, -1, 2, 0, 2, -1, 0, -2, 0, 1, 1, -1, 0, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 14, -4, 2, 0, 6, -1, -1, -1, 2, 2, 2, 0, -2, -1, 2, -1, 1, -1, 0, 0, 0, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 21, -9, 1, 3, -3, 6, 0, -2, 0, 0, -3, -1, 1, 0, 1, 1, 1, 1, 0, 0, 0, -1 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 64, -16, 0, 0, 0, 4, 2, 0, -2, 2, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 1, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 70, -10, 2, 2, -2, -5, -1, -1, 1, -1, 4, 0, 0, 1, -2, 0, 0, 0, -1, 1, 0, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 56, -4, 0, -4, 8, -4, 2, 0, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, -1, -1, 0, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 42, 0, 2, 0, -6, -6, 0, 2, 0, 0, 0, -2, 0, 0, 2, 2, 0, -1, 0, 0, 0, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 35, -5, -5, 3, 3, 5, 1, 1, 2, -2, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 90, 0, -6, 0, -6, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 56, 4, 0, 4, 8, -4, -2, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 1, 1, -1, 0, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 70, 10, 2, -2, -2, -5, 1, -1, 1, 1, -4, 0, 0, -1, -2, 0, 0, 0, 1, 1, 0, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 14, 4, 2, 0, 6, -1, 1, -1, 2, -2, -2, 0, 2, 1, 2, -1, -1, -1, 0, 0, 0, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 35, 5, -5, -3, 3, 5, -1, 1, 2, 2, 1, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, -1 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 64, 16, 0, 0, 0, 4, -2, 0, -2, -2, 0, 0, 0, 0, 0, -1, 1, -1, 0, 0, 1, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 28, 10, 4, 2, -4, 1, 1, 1, 1, 1, -2, 0, -2, 1, 0, -2, 0, 1, -1, -1, 0, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 21, 9, 1, -3, -3, 6, 0, -2, 0, 0, 3, -1, -1, 0, 1, 1, -1, 1, 0, 0, 0, 1 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 20, 10, 4, 2, 4, 5, 1, 1, -1, 1, 2, 0, 2, -1, 0, 0, 0, 0, -1, 1, -1, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 7, 5, 3, 1, -1, 4, 2, 0, 1, -1, 3, 1, -1, 0, -1, 2, 0, -1, 1, -1, 0, -1 ] ),
Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ]```