# Supercharacter theories for symmetric group:S3

From Groupprops

This article gives specific information, namely, supercharacter theories, about a particular group, namely: symmetric group:S3.

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 as the symmetric group on , and elements of the group are described by means of their cycle decompositions.

## Character table

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

Representation/Conjugacy class representative | (identity element) -- size 1 | (3-cycle) -- size 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.