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