Last update: 27 August 2013
|2.||Induced characters for semisimple algebras|
Recall Schur’s Lemma: given a finite dimensional algebra with representations and one has that
Recall that for a completely reducible we had that where are irreducible modules, and the are nonnegative integers (the multiplicities of the Note also that in the above that is that
Definition 2.53 Let with both and semisimple algebras. is a subalgebra of if has the same multiplication and same identity as is usually not an ideal of
Definition 2.54 Given with a subalgebra of and a representation of We can define the restriction of to denoted by
Note: if is a then acts on in a natural way, and is an under this restriction of the action of
|1.||If and are representations of then|
|2.||If are subalgebras, and is a representation of then|
Let be semisimple algebras, with a subalgebra of Let for be the irreducible representations of and for be the irreducible representations of For each (irreducible) representation of we can obtain a (usually not irreducible) representation of by restriction. This representation of can be expressed as a direct sum of irreducible such a family of decompositions for all of the irreducible is called a branching rule from to By a slight abuse of notation, one has This branching rule can be presented geometrically in a Bratteli diagram, consisting of two rows of vertices connected by vertical line segments. The vertices of the top row are labeled by elements of while those of the bottom row are labeled by elements of The vertex labeled by in is connected to the vertex labeled by in by many edges (i.e. the number of edges is the multiplicity of in the decomposition of the For example, suppose has three irreducible representations, and five. Further, suppose we have that This information can be represented by the following Bratteli diagram.
Example 2.56 We can consider as a subgroup of one such way is as the stabilizer of Specifically, when the embedding one gets is This embedding extends to an embedding of group rings The irreducible representations of are indexed by partitions of so the vertices of the Bratteli diagram for the above group ring containment will be To determine the edges, recall that if and then where the sum is over those such that is a border strip of length one. (We can take by virtue of the means of embedding into In terms of modules, if is a then Thus the Bratteli diagram one obtains is We recognize this Bratteli diagram as a part of the Young Lattice.
Definition 2.57 Let be semisimple algebras, with a subalgebra of Let be a (left) Then induces a where is the of all elements of the form for and subject to the relations where and is a via the action
|1.||If and are and then|
|2.||If are (sub)algebras, and a (left) then|
Lemma 2.59 (Frobenius Reciprocity) Let be (sub)algebras, let be a representation of and a representation of Then
|Sketch of proof.|
We need to produce homomorphisms which are inverses of one another. We will identify the appropriate maps, and leave the details (checking that the maps are well-defined, morphisms, and mutual inverses) to the interested reader.
Going from left to right, given a map one wants to produce a Such a must be a morphism, so it must send to A reasonable choice of may well be yielding the definition Going from right to left, given a map we want to produce a map A reasonable choice for is
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.