Degrees of irreducible representations of direct product are pairwise products of degrees of irreducible representations of direct factors
From Groupprops
Contents
Statement for two groups
Over any field
Suppose and
are finite groups. Suppose
is a field. Suppose
are the degrees of irreducible representations (possibly with repetitions) of
over
and
are the degrees of irreducible representations of
.
Then, the degrees of irreducible representations of over
are:
Over a splitting field
If we take to be a splitting field for both
and
, then the above relates the degrees of irreducible representations for
over a splitting field. For instance, we could take
to be
or
. 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
.
Statement for multiple groups
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]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