# Supercharacter theory corresponding to a normal series

## Definition

Suppose is a finite group. Consider a normal series for , i.e., groups:

such that each is a normal subgroup of . The **supercharacter theory corresponding to this normal series** is a supercharacter theory defined as follows:

### Blocks of conjugacy classes

There are blocks of conjugacy classes, i.e., there are superconjugacy classes. As sets of elements, these are the set differences:

As blocks of conjugacy classes, the block corresponding to each is just the set of conjugacy classes in it.

### Blocks of representations and corresponding supercharacters

The blocks of linear representations are as follows. There are blocks. The block corresponding to the containment is the set of those irreducible linear representations whose kernel contains but not . In addition to these blocks for , there is the trivial representation, which forms its own block.

We can choose the supercharacters as follow. For the trivial representation, the supercharacter is the trivial character itself. For a containment , the supercharacter is:

(Character of the regular representation of composed with the quotient map) - (Character of the regular representation of composed with the quotient map)

Alternatively, this supercharacter is a weighted sum of the characters in the corresponding block with the weight on a character equal to its degree.