The external tensor product of a group representation 
of 
and a group representation 
of 
over the same field is the group
representation of the group direct product 
on the vector
space tensor product 
. An element 
of 
acts on an element 
, with 
and 
, by
 |
To distinguish it from the representation tensor product, the external tensor product is denoted ![V□AdjustmentBox[x, BoxMargins -> {{-0.65, 0.13913}, {-0.5, 0.5}}, BoxBaselineShift -> -0.1]W](/images/equations/ExternalTensorProduct/Inline12.svg)
.
For finite groups 
and 
and irreducible
representations 
and 
over the complex numbers,
the external tensor product is irreducible.
In fact, all irreducible representations
of 
arise as external tensor products of irreducible
representations.
