Let be an abelian group and be a collection of endomorphisms of . The commutant of is the set of all endomorphisms of commuting with every element of ; symbolically,
The commutant of is equal to the commutant of the subring of generated by the , so we may assume without loss of generality that is already such a subring. In that case, is just the ring of endomorphisms of as a left -module. The use of the term commutant instead can be thought of as emphasizing the role of and de-emphasizing the role of .
The assignment is a contravariant Galois connection on the lattice of subsets of , so the double commutant may be thought of as a closure operator. Today we will prove a basic but important theorem about this operator.
If is a finite group and a finite-dimensional complex representation of , then breaks up into a direct sum
of irreducible representations with some multiplicities . However, this direct sum decomposition is not canonical if the multiplicities are greater than . In the worst case, may act trivially on , and then is a direct sum of copies of the trivial representation. Actually choosing such a direct sum decomposition is equivalent to choosing a basis of .
However, there is an alternate and completely canonical way of describing a representation in terms of its irreducible subrepresentations without choosing a direct sum decomposition as above. As a first hint, note that
This suggests that it might be useful to replace with the vector space . And, in fact, this turns out to be a great idea: there is a canonical evaluation map
whose image is precisely the -isotypic component of , and this gives an alternate canonical decomposition of as
which does not require making any choices. One can think of as the multiplicity space associated to , the correct canonical replacement for the multiplicity .
The idea of the double commutant theorem is to think about what kind of structure multiplicity spaces have. So far we have been using them only as vector spaces, but in fact they are -modules. Note that is precisely the commutant of the image of in .
Basic properties of commutants
Now that our warmup is done, we list some basic properties of the commutant operation
- is a subring of .
- implies .
- if and only if .
- (by 3).
- (by 2 and 4).
- (by 4 and 5).
The second and third properties assert that the commutant establishes a special type of Galois connection. In the language of category theory, the second and third properties assert that the commutant is a contravariant functor from the poset of subsets of to itself which happens to be its own adjoint. The remaining properties verify something slightly stronger than the statement that the double commutant is a closure operator: they also verify that the subsets of which are their own double commutant are precisely the commutants of other subsets of .
The double commutant theorem
Theorem (double commutant): Let be an abelian group and let be a subring of such that
- is a semisimple ring, and
- is a finite direct sum of simple -modules.
Then is its own double commutant. Moreover, is also semisimple, and as a -module, decomposes as a direct sum
where is a complete list of the simple -modules, is a complete list of the simple -modules, and . In particular, there is a canonical bijection between simple -modules and simple -modules.
Proof. Choose a finite direct sum decomposition
where the are the simple -modules. Since acts faithfully on , it follows (for example by Artin-Wedderburn) that the multiplicities are all positive. By Schur’s lemma,
where are division rings; in particular, is semisimple. Now, acts on the multiplicity spaces , and by inspection of the two decompositions above these are precisely the simple -modules. More precisely, is the unique simple
-module on which the factor acts nontrivially, and it is in particular an -dimensional -vector space (since acts on on the left, it acts on on the right). Hence, as in the finite group case above, the natural map
is an isomorphism. Writing , we may now think of the as the multiplicity spaces of the decomposition of as a -module, we conclude that is also the finite direct sum of simple -modules (with multiplicities given by the dimensions of the as -vector spaces), and it follows from here that by Artin-Wedderburn.
If you don’t like division rings, feel free to assume that is a finite-dimensional vector space over an algebraically closed field , which case everything above is a -vector space.
Example. Let be a finite group and a subgroup, and consider the representation of . The double commutant theorem tells us that decomposes into a direct sum as
where the are irreducible representations of and the are a complete list of the simple -modules. Understanding thus gives us a information about the decomposition of as a -module.
is one definition of the Hecke algebra . It may be described explicitly as spanned by double cosets , which have a well-defined product by a left coset on the left as a left coset. This construction is morally responsible for many of the Hecke algebras appearing in mathematics by making particular choices of and (usually and are not finite groups and so one passes from to a suitable space of functions on , but the idea is the same).