Supercharacter theory corresponding to a normal series

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Definition

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

\{ e \} = H_0 \le H_1 \le \dots \le H_n = G

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

Blocks of conjugacy classes

There are n + 1 blocks of conjugacy classes, i.e., there are n + 1 superconjugacy classes. As sets of elements, these are the set differences:

H_0, H_1 \setminus H_0, H_2 \setminus H_1, \dots, H_n \setminus H_{n-1}

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 n + 1 blocks. The block corresponding to the containment H_i \le H_{i + 1} is the set of those irreducible linear representations whose kernel contains H_i but not H_{i+1}. In addition to these n blocks for i = 0,1,\dots,n-1, 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 H_i \le H_{i+1}, the supercharacter is:

(Character of the regular representation of G/H_i composed with the quotient map) - (Character of the regular representation of G/H_{i+1} 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.