- How many non-isomorphic irreducible complex representations does the symmetric group \(S_n\) have?
- What classical function in mathematics does this number equal, and how are the irreducible representations explicitly indexed?
Solution
For a finite group over \(\mathbb C\), the number of isomorphism classes of irreducible representations equals the number of conjugacy classes. In \(S_n\), two permutations are conjugate exactly when they have the same cycle type.
A cycle type is precisely a partition \[ \lambda=(\lambda_1\ge\cdots\ge\lambda_r)\vdash n. \] Hence the conjugacy classes of \(S_n\) are indexed by partitions of \(n\), so \[ |\operatorname{Irr}(S_n)|=p(n), \] where \(p(n)\) is the integer partition function.
More precisely, the irreducible complex representations are the Specht modules \[ S^\lambda,\qquad \lambda\vdash n. \] Their dimensions are given by the hook-length formula \[ \dim S^\lambda=\frac{n!}{\prod_{u\in\lambda} h(u)}. \] Thus the partitions of \(n\) simultaneously index the conjugacy classes and the irreducible complex representations of \(S_n\).