## Lectures in Representation Theory

Last update: 27 August 2013

## Lecture 16

### 2.3$\phantom{\rule{1em}{0ex}}$Induction and Restriction

 1 Basics 2 Induced characters for semisimple algebras

Recall Schur’s Lemma: given a finite dimensional algebra $A,$ with representations ${W}_{1}$ and ${W}_{2},$ one has that $homA(W2,W1)= { 0 if W1≄W2 ℂ if W1≃W2$

Recall that for $V$ a completely reducible $A\text{-module,}$ we had that $V≃ ⊕λ∈Aˆ (Wλ)⊕mλ,$ where ${W}^{\lambda }$ are irreducible modules, and the ${m}_{\lambda }$ are nonnegative integers (the multiplicities of the ${W}^{\lambda }\text{).}$ Note also that in the above ${\text{hom}}_{A}\left({W}^{\lambda },V\right)\simeq {C}^{\oplus {m}_{\lambda }}\text{;}$ that is that ${\text{dim}}_{ℂ} \left({\text{hom}}_{A}\left({W}^{\lambda },V\right)\right)={m}_{\lambda }\text{.}$

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

Definition 2.54 Given $A\subseteq B,$ with $A$ a subalgebra of $B,$ and $V,$ a representation of $B\text{.}$ We can define the restriction of $V$ to $A,$ denoted $V{↓}_{A}^{B},$ by $V{↓}_{A}^{B}:A\to {M}_{d}\left(ℂ\right):a↦V\left(a\right)\text{.}$

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

Property 2.55

 1 If ${V}_{1}$ and ${V}_{2}$ are representations of $B,$ then $(V1⊕V2) ↓AB!=V1 ↓AB⊕V2 ↓AB.$ 2 If $A\subseteq B\subseteq C$ are subalgebras, and $V$ is a representation of $C,$ then $V↓AC≃ (V↓BC) ↓AB.$

Let $A\subseteq B$ be semisimple algebras, with $A$ a subalgebra of $B\text{.}$ Let ${W}^{\lambda },$ for $\lambda \in \stackrel{ˆ}{A},$ be the irreducible representations of $A,$ and ${V}^{\mu },$ for $\mu \in Bˆ,$ be the irreducible representations of $B\text{.}$ For each (irreducible) representation ${V}^{\mu }$ 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 $A\text{-representations;}$ such a family of decompositions for all of the irreducible $B\text{-representations}$ is called a branching rule from $B$ to $A\text{.}$ By a slight abuse of notation, one has $Vμ↓AB≃ ⨁λ∈Aˆ (Wλ)⊕gλμ =∑λ∈Aˆgλμ Wλ$ 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 $\stackrel{ˆ}{A},$ while those of the bottom row are labeled by elements of $\stackrel{ˆ}{B}\text{.}$ The vertex labeled by $\mu$ in $\stackrel{ˆ}{B}$ is connected to the vertex labeled by $\lambda$ in $\stackrel{ˆ}{A}$ by ${g}_{\lambda \mu }$ many edges (i.e. the number of edges is the multiplicity of ${W}^{\lambda }$ in the decomposition of the $A\text{-module}$ $V{↓}_{A}^{B}\text{).}$ 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 ${𝒮}_{m}$ as a subgroup of ${𝒮}_{m+1},$ one such way is as the stabilizer of $m+1\text{.}$ Specifically, when $m=3,$ the embedding one gets is $∈𝒮3 ↦ ∈𝒮4$ This embedding extends to an embedding of group rings $ℂ{𝒮}_{3}\subseteq ℂ{𝒮}_{4}\text{.}$ The irreducible representations of ${𝒮}_{m}$ are indexed by partitions of $m,$ so the vertices of the Bratteli diagram for the above group ring containment will be $ℂ𝒮3 ℂ𝒮4$ To determine the edges, recall that if $\mu =\left({\mu }_{1},\dots ,{\mu }_{k-1},1\right)⊢m$ and $\stackrel{ˆ}{\mu }=\left({\mu }_{1},\dots ,{\mu }_{k-1}\right)⊢m-1,$ then $χλ(μ)= ∑νχν (μˆ)$ where the sum is over those $\nu$ such that $\mu /\nu$ is a border strip of length one. (We can take ${\mu }_{k}=1$ by virtue of the means of embedding ${𝒮}_{3}$ into ${𝒮}_{4}\text{.)}$ In terms of modules, if $V$ is a $ℂ{𝒮}_{m}\text{-representation,}$ then $Vλ↓𝒮m-1𝒮m ≃⨁ν⊢m-1λ/ν=□ Vν.$ Thus the Bratteli diagram one obtains is $ℂ𝒮3 ℂ𝒮4$ We recognize this Bratteli diagram as a part of the Young Lattice.

Definition 2.57 Let $A\subseteq B$ be semisimple algebras, with $A$ a subalgebra of $B\text{.}$ Let $W$ be a (left) $A\text{-module.}$ Then $W$ induces a $B\text{-module}$ $W↑AB=B ⊗AW,$ where $B{\otimes }_{A}W$ is the $C\text{-span}$ of all elements of the form $b\otimes w,$ for $b\in B$ and $w\in W,$ subject to the relations $(b1+b2)⊗w= b1⊗w+b2⊗w b⊗(w1+w2) =b⊗w1+b⊗w2 c(b⊗w)= (cb)⊗w= b⊗(cw) (ba)⊗w=b⊗ (a·w)$ where $b,{b}_{i}\in B,$ $w,{w}_{i}\in W,$ $c\in ℂ,$ and $a\in A\text{.}$ $B{\otimes }_{A}W$ is a $B\text{-module}$ via the action $b·(b1⊗w) =(bb1)⊗w.$

Problem 2.58

 1 If ${W}_{1}$ and ${W}_{2}$ are $A\text{-modules,}$ and $A\subseteq B,$ then $(W1⊕W2) ↑AB≃W1 ↑AB⊕W2 ↑AB.$ 2 If $A\subseteq B\subseteq C$ are (sub)algebras, and $W$ a (left) $A\text{-module,}$ then $(W↑AB) ↑BC≃W ↑AC.$

Lemma 2.59 (Frobenius Reciprocity) Let $A\subseteq B$ be (sub)algebras, let $W$ be a representation of $A,$ and $V$ a representation of $B\text{.}$ Then $homA(W,V↓AB) ≃homB (W↑AB,V).$

 Sketch of proof. We need to produce homomorphisms $homA (W,V↓AB) ⇆homB (W↑AB,V)$ 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 $\varphi :W\to V{↓}_{A}^{B},$ one wants to produce a $\psi :B\otimes W\to V\text{.}$ Such a $\psi$ must be a $B\text{-module}$ morphism, so it must send $\psi \left(b\otimes w\right)$ to $b·\psi \left(1\otimes w\right)\text{.}$ A reasonable choice of $\psi \left(1\otimes w\right)$ may well be $\varphi \left(w\right),$ yielding the definition $ϕ↦ψ, where ψ(b⊗w)=b· ϕ(w).$ Going from right to left, given a map $\psi :B\otimes W\to V$ we want to produce a map $\varphi :W\to V\text{.}$ A reasonable choice for $\varphi$ is $\varphi \left(w\right)=\psi \left(1\otimes w\right)\text{.}$ $\text{“}\square \text{”}$

## 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.