Lecture notes for Introduction to Representation Theory

(a) Z 2
(b) Z 3
(c) Z 5
(d) A 4
(e) Z 2 × Z 2
4.12 Representations of S n
In this subsection we give a description of the representations of the symmetric group S n for any
n.
Definition 4.35. A partition ∂ of n is a representation of n in the form n = ∂ 1 + ∂ 2 + ... + ∂ p ,
where ∂ i are positive integers, and ∂ i ⊂ ∂ i+1 .
To such ∂ we will attach a Young diagram Y , which is the union of rectangles −i ∗ y ∗ −i+1,
0 ∗ x ∗ ∂ i in the coordinate plane, for i = 1, ..., p. Clearly, Y is a collection of n unit squares. A
Young tableau corresponding to Y is the result of filling the numbers 1, ..., n into the squares of
Y in some way (without repetitions). For example, we will consider the Young tableau T obtained
by filling in the numbers in the increasing order, left to right, top to bottom.
We can define two subgroups of S n corresponding to T :
1. The row subgroup P : the subgroup which maps every element of {1, ..., n} into an element
standing in the same row in T .
2. The column subgroup Q : the subgroup which maps every element of {1, ..., n} into an
element standing in the same column in T .
Clearly, P ∈ Q = {1}.
Define the Young projectors:
a :=
1
g,
|P | gP
b :=
1
(−1) g g,
|Q | gQ
where (−1) g denotes the sign of the permutation g. Set c = a b . Since P ∈ Q = {1}, this
element is nonzero.
The irreducible representations of S n are described by the following theorem.
Theorem 4.36. The subspace V := C[S n ]c of C[S n ] is an irreducible representation of S n under
left multiplication. Every irreducible representation of S n is isomorphic to V for a unique ∂.
The modules V are called the Specht modules.
The proof of this theorem is given in the next subsection.
Example 4.37.
For the partition ∂ = (n), P = S n , Q = {1}, so c is the symmetrizer, and hence V is the trivial
representation.
58