Supercharacter theories for alternating group:A5

This page discusses the various possible supercharacter theories for alternating group:A5. Thus, it builds on a thorough understanding of the element structure of alternating group:A5, subgroup structure of alternating group:A5, and linear representation theory of alternating group:A5.

We describe the group $$A_5$$ as the alternating group on $$\{ 1,2,3,4,5 \}$$, and elements of the group are described by means of their cycle decompositions.

Character table
Below, the character table of $$A_5$$ is given. This table is crucial for understanding the possible supercharacter theories.

Supercharacter theories
Note that for each of the supercharacter tables presented, the supercharacter is the smallest positive integer linear combination of the characters in the block that takes constant values on each superconjugacy class.

Ordinary character theory
Here, the blocks for conjugacy classes all have size one and the blocks for irreducible representations all have size one. The supercharacter table is the same as the character table.

All non-identity elements form one block
There are two blocks of conjugacy classes: the identity element is one block, and all non-identity elements form the other block. There are two blocks of irreducible characters: the trivial character is one block, and all other characters form the other block. The supercharacter table is as follows:

Supercharacter theory for automorphism group action
In this case, the two conjugacy classes of 5-cycles get fused into one superconjugacy class. All other conjugacy classes remain intact. On the character side, the two characters of degree three form one block. All other characters form separate blocks.

Below is an explicit description of the superconjugacy classes:

Below is an explicit description of the supercharacters and the blocks of characters:

Below is the supercharacter table: