# Size-degree-weighted characters are algebraic integers

## Contents

## Statement

Suppose is an algebraically closed field of characteristic zero, and is a finite group. Let be an Irreducible linear representation (?) of over , and be the character corresponding to . Let be a conjugacy class in and be an element. Then:

is an algebraic integer.

## Related facts

- Characters are cyclotomic integers: This statement holds in much greater generality. In particular, it holds over any field and it does not require the linear representation to be irreducible.
- Characters are algebraic integers

### Applications and further results

- Degree of irreducible representation divides group order
- Zero-or-scalar lemma
- Conjugacy class of prime power order implies not simple
- Order has only two prime factors implies solvable

### Breakdown for a field that is not algebraically closed

`Further information: cyclic group:Z3`

Let be the cyclic group of order three and be the field. has an irreducible two-dimensional linear representation over given by rotation by multiples of . For a non-identity element of , for the corresponding character, while . Thus, the expression works out to , which is not an algebraic integer.

### Breakdown for a representation that is not irreducible

The same example as the above (the one for breakdown over a field that is not algebraically closed) works. Specifically, the irreducible representation over can be viewed as a reducible representation over .

## Proof

The proof is based on the idea of the convolution algebra on conjugacy classes.

### Description of the convolution algebra on conjugacy classes

Let be a -subalgebra of the group ring defined as follows: as a group, it is the free Abelian group on all *indicator* class functions for conjugacy classes. In other words, for each conjugacy class, we have a free generator that corresponds to the *sum* of elements of that conjugacy class.

The structure constant for multiplication of elements of is defined as follows: given conjugacy classes , the coefficient of the -indicator function in the product of the -indicator function and the -indicator function is the number of ways of writing where , and a fixed element of .

Note that all the structure constants are integers.

### A homomorphism from this convolution algebra to the matrix ring

The representation gives rise to a homomorphism from to the matrix ring . The indicator function for a conjugacy class goes to the matrix given by:

.

This sum commutes with for all , and thus, by Schur's lemma, the sum is a scalar matrix. The trace of the sum is , so the sum must be a scalar matrix with scalar entry:

.

Thus, the set of scalar matrices with entries described as above additively generate a group that is a ring under multiplication. The structure constants for this ring are the same as the structure constants for the convolution algebra. A result from algebraic number theory now tells us that this forces the entire ring to be a ring of algebraic integers, and in particular, the generating elements are algebraic integers.