# Linear representation theory of symmetric group:S5

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This article describes the linear representation theory of symmetric group:S5, a group of order $120$. We take this to be the group of permutations on the set $\{1,2,3,4,5 \}$.

## Summary

Item Value
Degrees of irreducible representations over a splitting field 1,1,4,4,5,5,6
maximum: 6, lcm: 60, number: 7
Smallest ring of realization for all irreducible representations (characteristic zero) $\mathbb{Z}$ -- ring of integers
Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero) $\mathbb{Q}$ -- 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.

## 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 $1,1,4,4,5,5,6$.

### Explanation of degrees from the perspective of symmetric group of degree five

Common name of representation Degree Partition corresponding to representation Hook length formula for degree Conjugate partition Representation for conjugate partition
trivial representation 1 5 $\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ 1 + 1 + 1 + 1 + 1 sign representation
sign representation 1 1 + 1 + 1 + 1 + 1 $\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ 5 trivial representation
standard representation 4 4 + 1 (or is it 2 + 1 + 1 + 1?) $\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ 2 + 1 + 1 + 1 (or is it 4 + 1?) product of standard and sign representation
product of standard and sign representation 4 2 + 1 + 1 + 1 (or is it 4 + 1?) $\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ 4 + 1 (or is it 2 + 1 + 1 + 1?) standard representation
irreducible five-dimensional representation 5 3 + 2 $\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ 2 + 2 + 1 other irreducible five-dimensional representation
irreducible five-dimensional representation 5 2 + 2 + 1 $\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ 3 + 2 other irreducible five-dimensional representation
exterior square of standard representation 6 3 + 1 + 1 $\frac{5!}{5 \cdot 2 \cdot 2 \cdot 1 \cdot 1}$ 3 + 1 + 1 the same representation, because the partition is self-conjugate.