The representation tensor product of two group representations 
and 
of a group 
over the same field is the group
representation on their vector space
tensor product 
.
An element 
of 
acts on an element 
, with 
and 
,
by
 |
If 
is a finite group and 
is a finite-dimensional faithful group representation over the complex
numbers, then every irreducible representation
is contained in 
for some nonnegative
integer 
. If 
is a group representation
of 
and 
is a group representation of 
over the same field, then 
is a group representation of the group
direct product 
,
called the external tensor product. For
, the representation tensor product is obtained
by restricting this group
representation along the diagonal embedding 
of 
in 
.
