Supercharacter theories for symmetric group:S4

This page discusses the various possible supercharacter theories for symmetric group:S4. Thus, it builds on a thorough understanding of the element structure of symmetric group:S4, subgroup structure of symmetric group:S4, and linear representation theory of symmetric group:S4.

We describe the group $$S_4$$ as the symmetric group on $$\{ 1,2,3,4 \}$$, 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.

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.

Summary
There is a total of five possible supercharacter theories:

Ordinary character theory
One extreme case of a supercharacter theory is the one where we simply use blocks of size one both on the conjugacy class side and the linear representation side. The "supercharacter table" in this case is just the usual character table:

Supercharacter theory with all non-identity elements in one block
This is the other extreme supercharacter theory: all the non-identity elements form a single block, and all the nontrivial representations form a single block. The supercharacter table is as follows:

Supercharacter theory with a block for the Klein four-subgroup as kernel
This supercharacter theory corresponds to the normal series:

trivial subgroup $$\le$$ $$\{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$$ (normal V4 in S4) $$\le$$ whole group

The superconjugacy classes are the set differences between adjacent members of the normal series, and the blocks on the representation side are the collections of representations whose kernel contains one member but not the preceding member of the normal series.

Explicitly, the superconjugacy classes are as follows:

The blocks for representations, and corresponding supercharacters, are as follows:

The supercharacter table is:

Note that the multiplication by $$1/3$$ in the last row is to maintain our convention of choosing the smallest possible positive combination of the characters that works.

Supercharacter theory with a block for the alternating group as kernel
This supercharacter theory corresponds to the normal series:

trivial subgroup $$\le$$ A4 in S4 $$\le$$ whole group

The superconjugacy classes are the set differences between adjacent members of the normal series, and the blocks on the representation side are the collections of representations whose kernel contains one member but not the preceding member of the normal series.

Explicitly, the superconjugacy classes are:

The blocks for representations, and corresponding supercharacters, are as follows:

The supercharacter table is:

Supercharacter theory with a block for the full chief series
This supercharacter theory corresponds to the normal series:

trivial subgroup $$\le$$ $$\{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$$ (normal V4 in S4) $$\le$$ A4 in S4 $$\le$$ whole group

The superconjugacy classes are the set differences between adjacent members of the normal series, and the blocks on the representation side are the collections of representations whose kernel contains one member but not the preceding member of the normal series.

Explicitly, the superconjugacy classes are:

The blocks for representations, and corresponding supercharacters, are as follows:

The supercharacter table is: