Last update: 6 November 2012
Let be a subalgebra of an algebra
Let be a representation of The restriction of to to be the representation of given by the action of on Let be a representation of Define to be all formal linear combinations of elements where with the relations
for all and The induced representation is the representation of on given by the action
for all and
(5.3) Proposition. Let be such that is a subalgebra of and is a subalgebra of Let be representations of and let be representations of
1) and 2) are trivial consequences of the definition. he fact that the map
is a isomorphism gives 3). The map
and the map
are both isomorphisms. So
Note: Proving that these maps are isomorphisms is not a complete triviality. One must show that they are well defined (by showing that they preserve the bilinearity relations (5.1)) and that the inverse maps are also well defined. It is helpful to use the fact that the tensor product is a universal object as given in Ex. 1.
(5.4) Theorem. (Frobenius reciprocity) Let be algebras and and be irreducible representations of and respectively. Then
is an isomorphism. The inverse map is given by where is given by
so that is a homomorphism.
Now suppose that is a subalgebra of and that both and are semisimple. Let and be index sets for the irreducible representations of and respectively. Let and be the irreducible representations of and labelled by and respectively. Let be such that
for each pair Frobenius reciprocity implies that
for each An equation of the form (5.5) or (5.5') is called a branching rule between and
One can produce a visual representation of branching rules in the form of a graph. Construct a graph with two rows of vertices, the vertices in the first row labelled by the elements of and the vertices of the second row labelled by the elements of such that the vertex labelled by and the vertex labelled by are connected by edges. This graph is the Bratteli diagram of
As an example, the following diagram is the Bratteli diagram of where denotes the symmetric group. Recall that the irreducible representations of and are indexed by partitions of 2 and of 3 respectively.
Note that in this example each is either 0 or 1; there are no multiple edges.
Let and consider the representation of given by left multiplication on the space Then
To see this, informally, one notes that since we can move across the tensor product to give,
since More formally we should show that the map
is well defined and has well defined inverse given by
Now let be a minimal idempotent of such that the action of by left multiplication on is a representation of isomorphic to the irreducible representation of (3.6). Suppose that
is a decomposition (§1 Ex. 7) of the minimal idempotent of into minimal orthogonal idempotents of Then gives a decomposition of into irreducible representations. So, by (5.6) and the branching rule (5.5), for exactly of the we will have that is isomorphic to the irreducible representation of We can write the decomposition of as
where each is such that is isomorphic to the irreducible representation of
Let be a representation of where is a subalgebra of an algebra and both and are semisimple. Let be the character of and let be the character of For each let denote the element of the dual basis to with respect to the trace, tr, of the regular representation of such that
Let be a basis of and let be a nondegenerate trace on For each let denote the element of the dual basis to with respect to the trace such that For any element we set (as in §3 EX.7)
In keeping with the notations of earlier sections, let and be index sets for the irreducible representations of and respectively and let and denote the irreducible characters of and respectively. Let and denote the minimal central idempotents of and respectively. Let so that is the dimension of the irreducible representation of corresponding to
We have the following facts:
Let be a subalgebra of an algebra and let be a representation of Let and be the centralizers of and respectively. Then is a subalgebra of and
(5.9) Theorem. Suppose that
are the branching rules for and respectively. Then for all
We know, Theorem (4.11), that, as representations,
and as representations,
where and are irreducible representations of and respectively.
is a subalgebra of both and We have that as representations
On the other hand as representations
The tensor product is given by a vector space and a map such that for every bilinear map there exists a linear map such that the following diagram commutes:
One constructs the tensor product as the vector space of elements with relations for all and The map is given by Using the above universal mapping property one gets easily that the tensor product is unique in the sense that any two tensor products of and are isomorphic.
If is an algebra and is a right -module (a vector space that affords an antirepresentation of ) and is a left -module them one forms the vector space as above except that we require a bilinear map to satisfy the additional condition for all Then the tensor product once again is constructed by using the vector space of elements with the relations above and the additional relation for all
[Bra1972] There exists a complete set of matrix units of that is a refinement of the in the sense that for each and each , for some set of .
Suppose that . Let be the minimal central idempotent of such that is the minimal ideal corresponding to the block of matrices in
For each and each decompose into minimal orthogonal idempotents of (Section 1, Ex 7), Label each appearing in this sum by the element which indexes the minimal ideal of . Then Now If then the space for all Since and , we know that is not zero for any Furthermore, since the dimension of is each of the spaces is one dimensional.
For each define For each and each let be some element of Then choose such that This defines a complete set of matrix units of
Let be a representation of Let be a basis of Then the elements where span The fourth relation in 5.1 gives that the set forms a basis of
Let and suppose that where and Then Then
Since characters are constant on conjugacy classes we have that where denotes the conjugacy class of This is an alternate proof of Theorem 5.8 for the special case of inducing from a subgroup of a group to the group
Let and be representations of Then the restriction of the representation to the algebra is the Kronecker product (Section 4, Ex 1) of and Since we can view as a representation of
Let and be irreducible representations of such that appears as an irreducible component of the representation . The decomposition of the Kronecker product into irreducible representations of is given by the branching rule for Let and be the centralisers of the representations and respectively. Let be the centraliser of the representation Applying Theorem 5.9 to where and shows that the are also given by the branching rule for
The main result, Theorem (5.8), of this section is a generalization of the formula for the induced character for finite groups, see [Ser1977] §7.2. I have been unable to find any similar result in previous literature.
This is an excerpt from the unpublished first chapter of Arun Ram's dissertation entitled Representation Theory, written July 4, 1990.