Degrees of irreducible representations of direct product are pairwise products of degrees of irreducible representations of direct factors

Statement for two groups

Over any field

Suppose $G_1$ and $G_2$ are finite groups. Suppose $K$ is a field. Suppose $d_{11},d_{12},\dots,d_{1r}$ are the degrees of irreducible representations (possibly with repetitions) of $G_1$ over $K$ and $d_{21},d_{22},\dots,d_{2s}$ are the degrees of irreducible representations of $G_2$.

Then, the degrees of irreducible representations of $G_1 \times G_2$ over $K$ are:

$d_{11}d_{21}, d_{11}d_{22}, \dots,d_{11}d_{2s},d_{12}d_{21},d_{12}d_{22},\dots,d_{12}d_{2s},d_{1r}d_{21},\dots,d_{1r}d_{2s}$

Over a splitting field

If we take $K$ to be a splitting field for both $G_1$ and $G_2$, then the above relates the degrees of irreducible representations for $G_1,G_2,G_1 \times G_2$ over a splitting field. For instance, we could take $K$ to be $\mathbb{C}$ or $\overline{\mathbb{Q}}$. Note that degrees of irreducible representations are the same for all splitting fields (as long as the characteristic does not divide the group order), so this relates universal lists of numbers for $G_1,G_2,G_1 \times G_2$.

Statement for multiple groups

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Related facts

• Conjugacy class sizes of direct product are pairwise products of conjugacy class sizes of direct factors
• Number of conjugacy classes in a direct product is the product of the number of conjugacy classes in each factor

Facts used

1. Tensor product establishes bijection between irreducible representations of direct factors and direct product