Suppose that 
is a group representation of 
, and 
is a group representation
of 
.
Then the vector space tensor product 
is a group
representation of the group direct product 
. An element 
of 
acts on a basis element 
by
 |
To distinguish 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/Inline10.svg)
,
although the only possible confusion would occur when 
.
When 
and 
are irreducible representations of
and 
respectively, then so is the external tensor product. In fact, all irreducible
representations of 
arise as external direct products of irreducible
representations.
