# nLab representation theory of the symmetric group

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

The representation theory of the symmetric groups.

## Properties

### Irreducible representations

In characteristic zero, the irreducible representations of the symmetric group are, up to isomorphism, given by the Specht modules labeled by partitions $\lambda \in Part(n)$ (e.g. Sagan 01, Thm. 2.4.6).

### Dimension of irreps and hook length

Over the complex numbers:

The dimension of the irrep $S^{(\lambda)}$ (Specht module) equals the number of standard Young tableau of shape $\lambda$:

(1)$dim\big(S^{(\lambda)}\big) \;=\; \left\vert sYTableaux_\lambda \right\vert$

(e.g. Sagan, Thm. 2.6.5)

Moreover, the number of standard Young tableaux of shape $\lambda$ is given by the hook length formula

(2)$\left\vert sYTableaux_\lambda \right\vert \;=\; n! \left( \prod_{ { 1 \leq i \leq rows(\lambda) } \atop { 1 \leq j \leq \lambda_i } } \ell hook_\lambda(i,j) \right)^{-1} \,.$

This is due to Frame, Robinson & Thrall 54. Textbook accounts include Stanley 99, Cor. 7.21.6, Sagan 01 Thm. 3.10.2.

Combining (1) with (2) gives the hook length formula for the dimension of the Specht modules

$dim\big(S^{(\lambda)}\big) \;=\; n! \left( \prod_{ { 1 \leq i \leq rows(\lambda) } \atop { 1 \leq j \leq \lambda_i } } \ell hook_\lambda(i,j) \right)^{-1} \,.$

(e.g. James 78, Thm. 20.1)

## References

Textbook accounts: