Lectures in Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 27 August 2013

Lecture 16

2.3Induction and Restriction

1. Basics
2. Induced characters for semisimple algebras

Recall Schur’s Lemma: given a finite dimensional algebra A, with representations W1 and W2, one has that homA(W2,W1)= { 0 ifW1W2 ifW1W2

Recall that for V a completely reducible A-module, we had that V λAˆ (Wλ)mλ, where Wλ are irreducible modules, and the mλ are nonnegative integers (the multiplicities of the Wλ). Note also that in the above homA(Wλ,V)Cmλ; that is that dim(homA(Wλ,V))=mλ.

Definition 2.53 Let AB, with both A and B semisimple algebras. A is a subalgebra of B if A has the same multiplication and same identity as B. (A is usually not an ideal of B.)

Definition 2.54 Given AB, with A a subalgebra of B, and V, a representation of B. We can define the restriction of V to A, denoted VAB, by VAB:AMd():aV(a).

Note: if V is a B-module, then A acts on V in a natural way, and V is an A-module under this restriction of the action of B.

Property 2.55

1. If V1 and V2 are representations of B, then (V1V2) AB!=V1 ABV2 AB.
2. If ABC are subalgebras, and V is a representation of C, then VAC (VBC) AB.

Let AB be semisimple algebras, with A a subalgebra of B. Let Wλ, for λAˆ, be the irreducible representations of A, and Vμ, for μBˆ, be the irreducible representations of B. For each (irreducible) representation Vμ of B, we can obtain a (usually not irreducible) representation of A by restriction. This representation of A can be expressed as a direct sum of irreducible A-representations; such a family of decompositions for all of the irreducible B-representations is called a branching rule from B to A. By a slight abuse of notation, one has VμAB λAˆ (Wλ)gλμ =λAˆgλμ Wλ 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 Aˆ, while those of the bottom row are labeled by elements of Bˆ. The vertex labeled by μ in Bˆ is connected to the vertex labeled by λ in Aˆ by gλμ many edges (i.e. the number of edges is the multiplicity of Wλ in the decomposition of the A-module VAB). For example, suppose B has three irreducible representations, and A five. Further, suppose we have that Vμ(1) AB (Wλ(1))2 Wλ(3) Wλ(5) Vμ(2) AB Wλ(2) (Wλ(3))3 Vμ(3) AB Wλ(4) This information can be represented by the following Bratteli diagram. Aˆ Bˆ λ(1) λ(2) λ(3) λ(4) λ(5) μ(1) μ(2) μ(3)

Example 2.56 We can consider 𝒮m as a subgroup of 𝒮m+1, one such way is as the stabilizer of m+1. Specifically, when m=3, the embedding one gets is 𝒮3 𝒮4 This embedding extends to an embedding of group rings 𝒮3𝒮4. The irreducible representations of 𝒮m are indexed by partitions of m, so the vertices of the Bratteli diagram for the above group ring containment will be 𝒮3 𝒮4 To determine the edges, recall that if μ=(μ1,,μk-1,1)m and μˆ=(μ1,,μk-1)m-1, then χλ(μ)= νχν (μˆ) where the sum is over those ν such that μ/ν is a border strip of length one. (We can take μk=1 by virtue of the means of embedding 𝒮3 into 𝒮4.) In terms of modules, if V is a 𝒮m-representation, then Vλ𝒮m-1𝒮m νm-1λ/ν= Vν. Thus the Bratteli diagram one obtains is 𝒮3 𝒮4 We recognize this Bratteli diagram as a part of the Young Lattice.

Definition 2.57 Let AB be semisimple algebras, with A a subalgebra of B. Let W be a (left) A-module. Then W induces a B-module WAB=B AW, where BAW is the C-span of all elements of the form bw, for bB and wW, subject to the relations (b1+b2)w= b1w+b2w b(w1+w2) =bw1+bw2 c(bw)= (cb)w= b(cw) (ba)w=b (a·w) where b,biB, w,wiW, c, and aA. BAW is a B-module via the action b·(b1w) =(bb1)w.

Problem 2.58

1. If W1 and W2 are A-modules, and AB, then (W1W2) ABW1 ABW2 AB.
2. If ABC are (sub)algebras, and W a (left) A-module, then (WAB) BCW AC.

Lemma 2.59 (Frobenius Reciprocity) Let AB be (sub)algebras, let W be a representation of A, and V a representation of B. Then homA(W,VAB) homB (WAB,V).

Sketch of proof.

We need to produce homomorphisms homA (W,VAB) homB (WAB,V) 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 ϕ:WVAB, one wants to produce a ψ:BWV. Such a ψ must be a B-module morphism, so it must send ψ(bw) to b·ψ(1w). A reasonable choice of ψ(1w) may well be ϕ(w), yielding the definition ϕψ,where ψ(bw)=b· ϕ(w). Going from right to left, given a map ψ:BWV we want to produce a map ϕ:WV. A reasonable choice for ϕ is ϕ(w)=ψ(1w).

Notes and References

This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.

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