---
tag: NTFM7X5555
kind: concept
name: Specht module
msc: [20C30]
specializes: [WYGRXCG2RN]
sources:
  - url: https://en.wikipedia.org/wiki/Specht_module
same_as:
  wikidata: Q7574439
  wikipedia: Specht module
---

## Summary

For each [[partition|0JV30XN73Z]] $\lambda$ of $n$, the Specht module $S^\lambda$ is a representation of the [[symmetric group|0515A6D2Y1]] $S_n$ built from [[Young tableaux|Q6TH9PH2R7]] of shape $\lambda$.
Over a field of characteristic $0$, the Specht modules are irreducible and give every irreducible representation of $S_n$ exactly once.
They are defined over any field, even over the integers.
The dimension of $S^\lambda$ is the number of standard Young tableaux of shape $\lambda$, given by the [[hook length formula|VCMS1CP91N]].

## Explanations

### L1 human

A Specht module turns a picture, a [[Young diagram|096AHMB3P4]], into a representation of the symmetric group.
One fills the diagram with the numbers $1$ to $n$ and forms a signed sum that is symmetric along rows and antisymmetric down columns.
The span of all such sums is the module.
A single row gives the trivial representation, and a single column gives the sign representation.