# nLab Specht module

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

Specht modules are linear representations of the symmetric group $Sym(n)$ (for any $\in \mathbb{N}$), hence modules over the group ring $k(Sym(n))$, which are indexed by the partitions of $n$. In characteristic 0, they are irreducible and exhaust the isomorphism classes of irreps (e.g. Sagan 01, Thm. 2.4.6).

Over a field of positive characteristic $p$, where $p \mid n!$, the Specht modules are not irreducible, but every irreducible module does appear as the cosocle of a Specht module.

## References

Textbook accounts: