---
tag: 1VQC05EEYW
kind: concept
name: Representation theory of the symmetric group
msc: [20C30]
sources:
  - url: https://en.wikipedia.org/wiki/Representation_theory_of_the_symmetric_group
same_as:
  wikidata: Q7314231
  wikipedia: Representation theory of the symmetric group
---

## Summary

The representation theory of the [[symmetric group|0515A6D2Y1]] $S_n$ studies its [[representations|WYGRXCG2RN]] and characters.
Over the complex numbers, the irreducible representations of $S_n$ correspond to the [[partitions|0JV30XN73Z]] of $n$, and are realised by the [[Specht modules|NTFM7X5555]].
The theory is fully combinatorial: [[Young diagrams|096AHMB3P4]] and [[Young tableaux|Q6TH9PH2R7]] give dimensions and characters.
It connects with symmetric functions, the representations of general linear groups and many parts of combinatorics and physics.

## Explanations

### L1 human

The symmetric group consists of all shufflings of $n$ objects.
Its irreducible representations are labelled by the ways of writing $n$ as a sum of positive integers, drawn as left-justified rows of boxes.
For $n = 3$ there are three partitions, $3$, $2 + 1$ and $1 + 1 + 1$, matching the trivial, the two-dimensional and the sign representations of $S_3$.
The dimension of each one is the number of ways to fill its diagram with $1, \ldots, n$ increasing along rows and down columns.