# Lcm of degrees of irreducible representations

## Definition

### For a group over a field

Suppose $G$ is a group and $K$ is a field. The lcm of degrees of irreducible representations of $G$ is defined as the least common multiple of all the degrees of irreducible representations of $G$ over $K$.

### Typical context: finite group and splitting field

The typical context is where $G$ is a finite group and $K$ is a splitting field for $G$. In particular, the characteristic of $K$ is either zero or is a prime not dividing the order of $G$, and every irreducible representation of $G$ over any extension field of $K$ can be realized over $K$.

Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field $K$. This is because the degrees of irreducible representations over a splitting field depend only on the characteristic of the field.

### Default case: characteristic zero

By default, when referring to the lcm of degrees of irreducible representations, we refer to the case of characteristic zero, and we can in particular take $K = \mathbb{C}$.

## Related facts

### What it divides

Any divisibility fact stating that the degree of every irreducible representation over a splitting field must divide some fixed number implies that the lcm also divides that fixed number. Some of these are listed below: