Supercharacter theories for symmetric group:S3
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.
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|
|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.