# Linear representation theory of symmetric group:S7

## Contents

View linear representation theory of particular groups | View other specific information about symmetric group:S7

## Summary

Item Value
degrees of irreducible representations over a splitting field 1,1,6,6,14,14,14,14,15,15,20,21,21,35,35
maximum: 35, lcm: 420, number: 15, sum of squares: 5040
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 (all 1s)
smallest ring of realization (characteristic zero) $\mathbb{Z}$ -- ring of integers
smallest field of realization (characteristic zero) $\mathbb{Q}$ -- field of rational numbers
condition for a field to be a splitting field Any field of characteristic not 2, 3, 5, or 7.
smallest size splitting field field:F11, i.e., the field with 11 elements

## Family contexts

Family name Parameter values General discussion of linear representation theory of family
symmetric group 7 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(7));
[ [ 1, 2 ], [ 6, 2 ], [ 14, 4 ], [ 15, 2 ], [ 20, 1 ], [ 21, 2 ], [ 35, 2 ] ]```

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

```gap> Irr(CharacterTable(SymmetricGroup(7)));
[ Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1 ] ), Character( CharacterTable( Sym(
[ 1 .. 7 ] ) ), [ 6, -4, 2, 0, 3, -1, -1, 0, -2, 0, 1, 1, 1, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ),
[ 14, -6, 2, -2, 2, 0, 2, -1, 0, 0, 0, -1, -1, 1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 14, -4, 2, 0, -1, -1, -1, 2, 2, 0, -1, -1,
1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 15, -5, -1, 3, 3, 1, -1, 0, -1, -1, -1, 0, 0, 0, 1 ] ),
Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 35, -5, -1, -1, -1, 1, -1, -1, 1, 1, 1, 0, 0, -1, 0 ] ), Character( CharacterTable( Sym(
[ 1 .. 7 ] ) ), [ 21, -1, 1, 3, -3, -1, 1, 0, 1, -1, 1, 1, -1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ),
[ 21, 1, 1, -3, -3, 1, 1, 0, -1, -1, -1, 1, 1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 20, 0, -4, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0,
-1 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 35, 5, -1, 1, -1, -1, -1, -1, -1, 1, -1, 0, 0, 1, 0 ] ), Character( CharacterTable( Sym(
[ 1 .. 7 ] ) ), [ 14, 4, 2, 0, -1, 1, -1, 2, -2, 0, 1, -1, -1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ),
[ 15, 5, -1, -3, 3, -1, -1, 0, 1, -1, 1, 0, 0, 0, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 14, 6, 2, 2, 2, 0, 2, -1, 0, 0, 0, -1, 1,
-1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 6, 4, 2, 0, 3, 1, -1, 0, 2, 0, -1, 1, -1, 0, -1 ] ), Character( CharacterTable( Sym(
[ 1 .. 7 ] ) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ]```