Difference between revisions of "Linear representation theory of symmetric group:S5"
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| sign representation || 1 || 1 + 1 + 1 + 1 + 1 || <math>\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}</math> || 5 || trivial representation | | sign representation || 1 || 1 + 1 + 1 + 1 + 1 || <math>\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}</math> || 5 || trivial representation | ||
|- | |- | ||
| − | | standard representation || 4 || 4 + 1 || <math>\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 2 + 1 + 1 + 1 | + | | standard representation || 4 || 4 + 1 || <math>\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 2 + 1 + 1 + 1|| product of standard and sign representation |
|- | |- | ||
| − | | product of standard and sign representation || 4 || 2 + 1 + 1 + 1 || <math>\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 4 + 1 | + | | product of standard and sign representation || 4 || 2 + 1 + 1 + 1 || <math>\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 4 + 1 || standard representation |
|- | |- | ||
| irreducible five-dimensional representation || 5 || 3 + 2 || <math>\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 2 + 2 + 1 || other irreducible five-dimensional representation | | irreducible five-dimensional representation || 5 || 3 + 2 || <math>\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 2 + 2 + 1 || other irreducible five-dimensional representation | ||
Revision as of 21:30, 9 May 2011
This article gives specific information, namely, linear representation theory, about a particular group, namely: symmetric group:S5.
View linear representation theory of particular groups | View other specific information about symmetric group:S5
This article describes the linear representation theory of symmetric group:S5, a group of order 
. We take this to be the group of permutations on the set 
.
Summary
| Item | Value |
|---|---|
| Degrees of irreducible representations over a splitting field | 1,1,4,4,5,5,6 maximum: 6, lcm: 60, number: 7 |
| Schur index values of irreducible representations | 1,1,1,1,1,1,1 maximum: 1, lcm: 1 |
| Smallest ring of realization for all irreducible representations (characteristic zero) |  -- ring of integers
|
| Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero) |  -- hence it is a rational representation group
|
| Criterion for a field to be a splitting field | Any field of characteristic not equal to 2,3, or 5. |
| Smallest size splitting field | field:F7, i.e., the field of 7 elements. |
Family contexts
| Family name | Parameter values | General discussion of linear representation theory of family |
|---|---|---|
| symmetric group | 4 | linear representation theory of symmetric groups |
| projective general linear group of degree two | field:F5 | linear representation theory of projective general linear groups of degree two |
Degrees of irreducible representations
FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization
Note that the linear representation theory of the symmetric group of degree four works over any field of characteristic not equal to two or three, and the list of degrees is 
.
Interpretation as symmetric group
| Common name of representation | Degree | Partition corresponding to representation | Hook length formula for degree | Conjugate partition | Representation for conjugate partition |
|---|---|---|---|---|---|
| trivial representation | 1 | 5 |  |
1 + 1 + 1 + 1 + 1 | sign representation |
| sign representation | 1 | 1 + 1 + 1 + 1 + 1 | |
5 | trivial representation |
| standard representation | 4 | 4 + 1 |  |
2 + 1 + 1 + 1 | product of standard and sign representation |
| product of standard and sign representation | 4 | 2 + 1 + 1 + 1 | |
4 + 1 | standard representation |
| irreducible five-dimensional representation | 5 | 3 + 2 |  |
2 + 2 + 1 | other irreducible five-dimensional representation |
| irreducible five-dimensional representation | 5 | 2 + 2 + 1 | |
3 + 2 | other irreducible five-dimensional representation |
| exterior square of standard representation | 6 | 3 + 1 + 1 |  |
3 + 1 + 1 | the same representation, because the partition is self-conjugate. |
