Lectures in Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 27 August 2013

Lecture 16

2.3$$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 $${\text{hom}}_A(W_2,W_1)=\{\begin{array}{cc}0& \text{if}\hspace{0.17em}{W}_1\not\simeq W_2\\ ℂ& \text{if}\hspace{0.17em}{W}_1\simeq W_2\end{array}$$

Recall that for $V$ a completely reducible $A\text{-module,}$ we had that $$V\simeq {\oplus }_{\lambda \in \hat{A}}{\left(W^{\lambda }\right)}^{\oplus m_{\lambda }},$$ 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 ${\text{hom}}_A(W^{\lambda },V)\simeq C^{\oplus m_{\lambda }}\text{;}$ that is that ${\text{dim}}_{ℂ}\hspace{0.17em}\left({\text{hom}}_A(W^{\lambda },V)\right)=m_{\lambda }\text{.}$

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 $V{\downarrow }_A^B,$ by $V{\downarrow }_A^B:A\to M_d\left(ℂ\right):a\mapsto V\left(a\right)\text{.}$

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 $$(V_1\oplus V_2){\downarrow }_A^B!=V_1{\downarrow }_A^B\oplus V_2{\downarrow }_A^B\text{.}$$
2.	If $A\subseteq B\subseteq C$ are subalgebras, and $V$ is a representation of $C,$ then $$V{\downarrow }_A^C\simeq \left(V{\downarrow }_B^C\right){\downarrow }_A^B\text{.}$$

Let $A\subseteq B$ be semisimple algebras, with $A$ a subalgebra of $B\text{.}$ Let $W^{\lambda },$ for $\lambda \in \hat{A},$ be the irreducible representations of $A,$ and $V^{\mu },$ for 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 $$V^{\mu }{\downarrow }_A^B\simeq \underset{\lambda \in \hat{A}}{⨁}{\left(W^{\lambda }\right)}^{\oplus g_{\lambda \mu }}=\sum _{\lambda \in \hat{A}}{g}_{\lambda \mu }{W}^{\lambda }$$ 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 $\hat{A},$ while those of the bottom row are labeled by elements of $\hat{B}\text{.}$ The vertex labeled by $\mu$ in $\hat{B}$ is connected to the vertex labeled by $\lambda$ in $\hat{A}$ by $g_{\lambda \mu }$ many edges (i.e. the number of edges is the multiplicity of $W^{\lambda }$ in the decomposition of the $A\text{-module}$ $V{\downarrow }_A^B\text{).}$ For example, suppose $B$ has three irreducible representations, and $A$ five. Further, suppose we have that $$\begin{array}{ccc}{V}^{{\mu }^{\left(1\right)}}{\downarrow }_A^B& \simeq & {\left(W^{{\lambda }^{\left(1\right)}}\right)}^{\oplus 2}\oplus W^{{\lambda }^{\left(3\right)}}\oplus W^{{\lambda }^{\left(5\right)}}\\ V^{{\mu }^{\left(2\right)}}{\downarrow }_A^B& \simeq & W^{{\lambda }^{\left(2\right)}}\oplus {\left(W^{{\lambda }^{\left(3\right)}}\right)}^{\oplus 3}\\ V^{{\mu }^{\left(3\right)}}{\downarrow }_A^B& \simeq & W^{{\lambda }^{\left(4\right)}}\end{array}$$ 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 $$\begin{array}{ccccc} & \in {𝒮}_3& \mapsto & & \in {𝒮}_4\end{array}$$ 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 $\mu =({\mu }_1,\dots ,{\mu }_{k-1},1)⊢m$ and $\hat{\mu }=({\mu }_1,\dots ,{\mu }_{k-1})⊢m-1,$ then $${\chi }^{\lambda }\left(\mu \right)=\sum _{\nu }{\chi }^{\nu }\left(\hat{\mu }\right)$$ 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 $$V^{\lambda }{\downarrow }_{{𝒮}_{m-1}}^{{𝒮}_m}\simeq \underset{\underset{\lambda /\nu =\square }{\nu ⊢m-1}}{⨁}{V}^{\nu }\text{.}$$ 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 $B\text{-module}$ $$W{\uparrow }_A^B=B{\otimes }_{A}W,$$ 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 $$\begin{array}{c}(b_1+b_2)\otimes w=b_1\otimes w+b_2\otimes w\\ b\otimes (w_1+w_2)=b\otimes w_1+b\otimes w_2\\ c(b\otimes w)=\left(cb\right)\otimes w=b\otimes \left(cw\right)\\ \left(ba\right)\otimes w=b\otimes (a·w)\end{array}$$ 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 $$b·(b_1\otimes w)=\left(b{b}_1\right)\otimes w\text{.}$$

Problem 2.58

1.	If $W_1$ and $W_2$ are $A\text{-modules,}$ and $A\subseteq B,$ then $$(W_1\oplus W_2){\uparrow }_A^B\simeq W_1{\uparrow }_A^B\oplus W_2{\uparrow }_A^B\text{.}$$
2.	If $A\subseteq B\subseteq C$ are (sub)algebras, and $W$ a (left) $A\text{-module,}$ then $$\left(W{\uparrow }_A^B\right){\uparrow }_B^C\simeq W{\uparrow }_A^C\text{.}$$

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 $${\text{hom}}_A(W,V{\downarrow }_A^B)\simeq {\text{hom}}_B(W{\uparrow }_A^B,V)\text{.}$$

		Sketch of proof.
	We need to produce homomorphisms $${\text{hom}}_A(W,V{\downarrow }_A^B)\leftrightarrows {\text{hom}}_B(W{\uparrow }_A^B,V)$$ 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 $\varphi :W\to V{\downarrow }_A^B,$ 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 $\psi (b\otimes w)$ to $b·\psi (1\otimes w)\text{.}$ A reasonable choice of $\psi (1\otimes w)$ may well be $\varphi \left(w\right),$ yielding the definition $$\varphi \mapsto \psi ,\hspace{0.17em}\text{where}\hspace{0.17em}\psi (b\otimes w)=b·\varphi \left(w\right)\text{.}$$ 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 $\varphi \left(w\right)=\psi (1\otimes w)\text{.}$ $\text{“}\square \text{”}$

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.

page history