# Size-degree-weighted characters are algebraic integers

## Statement

Suppose $k$ is an algebraically closed field of characteristic zero, and $G$ is a finite group. Let $\rho$ be an irreducible linear representation of $G$ over $k$, and $\chi$ be the character corresponding to $\rho$. Let $c$ be a conjugacy class in $G$ and $g \in c$ be an element. Then:

$\frac{|c|\chi(g)}{\chi(1)}$

is an algebraic integer.

## Related facts

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

Further information: cyclic group:Z3

Let $G$ be the cyclic group of order three and $\R$ be the field. $G$ has an irreducible two-dimensional linear representation over $\R$ given by rotation by multiples of $2\pi/3$. For a non-identity element $g$ of $G$, $\chi(g) = -1$ for the corresponding character, while $\chi(1) = 2$. Thus, the expression works out to $-1/2$, which is not an algebraic integer.

## Proof

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

### Description of the convolution algebra on conjugacy classes

Let $C(G,\mathbb{Z})$ be a $\mathbb{Z}$-subalgebra of the group ring $\mathbb{Z}(G)$ 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 $C(G,\mathbb{Z})$ is defined as follows: given conjugacy classes $c_1, c_2, c_3$, the coefficient of the $c_3$-indicator function in the product of the $c_1$-indicator function and the $c_2$-indicator function is the number of ways of writing $g_1g_2 = g_3$ where $g_i \in c_i$.

Note that all the structure constants are integers.

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

The representation $\rho$ gives rise to a homomorphism from $C(G,\mathbb{Z})$ to the matrix ring $M_n(k)$. The indicator function for a conjugacy class $c$ goes to the matrix given by:

$\sum_{g \in c} \rho(g)$.

This sum commutes with $\rho(h)$ for all $h$, and thus, by Schur's lemma, the sum is a scalar matrix. The trace of the sum is $|c|\chi(g)$, so the sum must be a scalar matrix with scalar entry:

$\frac{|c|\chi(g)}{\chi(1)}$.

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.