# Supercharacter theories for symmetric group:S3

## Contents

View supercharacter theories for particular groups | View other specific information about symmetric group:S3

This page discusses the various possible supercharacter theories for symmetric group:S3. Thus, it builds on a thorough understanding of the element structure of symmetric group:S3, subgroup structure of symmetric group:S3, and linear representation theory of symmetric group:S3.

We describe the group $S_3$ as the symmetric group on $\{ 1,2,3\}$, and elements of the group are described by means of their cycle decompositions.

## Character table

Below, the character table of $S_4$ is given. This table is crucial for understanding the possible supercharacter theories.

Representation/Conjugacy class representative $()$ (identity element) -- size 1 $(1,2,3)$ (3-cycle) -- size 2 $(1,2)$ (2-transposition) -- size 3
Trivial representation 1 1 1
Sign representation 1 1 -1
Standard representation 2 -1 0

## Supercharacter theories

### Summary

There are only two supercharacter theories possible for symmetric group:S3, namely the two extreme cases:

Quick description of supercharacter theory Number of blocks of conjugacy classes = number of blocks of irreducible representations Block sizes for conjugacy classses (in number of conjugacy class terms) (should add up to 3, the total number of conjugacy classes) Block sizes for conjugacy classes (in number of elements terms) (should add up to 6, the order of the group) Block sizes for irreducible representations (in number of representations terms) (should add up to 3, the total number of conjugacy classes) Block sizes for irreducible representations (in sum of squares of degrees terms) (should add up to 6, the order of the group)
ordinary character theory 3 1,1,1 1,2,3 1,1,1 1,1,4
all non-identity elements form one block 2 1,2 1,5 1,2 1,5

There are no other possibilities to consider because the group has only three conjugacy classes.