Representing Product Groups
An important construction for groups is their direct product. Given two groups and
we take the cartesian product of their underlying sets
and put a group structure on it by multiplying component-by-component. That is, the product of
and
is
. Representations of product groups aren’t really any different from those of any other group, but we have certain ways of viewing them that will come in handy.
The thing to notice is that we have copies of both and
inside
. Indeed, we can embed
into
by sending
to
, which clearly preserves the multiplication. Similarly, the map
embeds
into
. The essential thing here is that the transformations coming from
and those coming from
commute with each other. Indeed, we calculate
Also, every transformation in is the product of one from
and one from
.
The upshot is that a representation of on a space
provides us with a representation of
, and also one of
on the space
. Further, transformation in the representation of
must commute with every transformation in the representation of
. Conversely, if we have a representation of each factor group on the same space
, then we have a representation of the product group, but only if all the transformations in each representation commute with all the transformations in the other.
So what can we do with this? Well, it turns out that it’s pretty common to have two separate group actions on the same module, and to have these two actions commute with each other like this. Whenever this happens we can think of it as a representation of the product group, or as two commuting representations.
In fact, there’s another way of looking at it: remember that a representation of a group on a space
can be regarded as a module for the group algebra
. If we then add a commuting representation of a group
, we can actually regard it as a representation on the
-module instead of just the underlying vector space. That is, instead of just having a homomorphism
that sends each element of
to a linear endomorphism of
, we actually get a homomorphism
that sends each element of
to a
-module endomorphism of
.
Indeed, let’s write our action of with the group homomorphism
and our action of
with the group homomorphism
. Now, I’m asserting that each
is an intertwinor for the action of
. This means that for each
, it satisfies the equation
. But this is exactly what it means for the two representations to commute!
Some notation will be helpful here. If the vector space carries a representation of a group
, we can hang a little tag on it to remind ourselves of this, writing
. That is,
is a
-module, and not just a vector space. If we now add a new representation of a group
that commutes with the original representation, we just add a new tag:
. Of course, the order of the tags doesn’t really matter, so we could just as well write
. Either way, this means that we have a representation of
on
.
[…] can extend the notation from last time. If the space carries a right representation of a group , then we hang a tag on the right: . If we […]
Pingback by Right Representations « The Unapologetic Mathematician | November 2, 2010 |
[…] . It turns out that the tensor product naturally carries a representation of the product group . Equivalently, it carries a representation of each of and , and these representations commute with each other. […]
Pingback by Outer Tensor Products « The Unapologetic Mathematician | November 4, 2010 |
[…] actions of the same group , and these two actions commute with each other. That is, carries a representation of the product group . This representation is a homomorphism […]
Pingback by Inner Tensor Products « The Unapologetic Mathematician | November 5, 2010 |