Last update: 27 August 2013
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 $${\text{hom}}_{A}({W}_{2},{W}_{1})=\{\begin{array}{cc}0& \text{if}\hspace{0.17em}{W}_{1}\not\simeq {W}_{2}\\ \u2102& \text{if}\hspace{0.17em}{W}_{1}\simeq {W}_{2}\end{array}$$
Recall that for $V$ a completely reducible $A\text{-module,}$ we had that $$V\simeq {\oplus}_{\lambda \in \stackrel{\u02c6}{A}}{\left({W}^{\lambda}\right)}^{\oplus {m}_{\lambda}},$$ 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 ${\text{dim}}_{\u2102}\hspace{0.17em}\left({\text{hom}}_{A}({W}^{\lambda},V)\right)={m}_{\lambda}\text{.}$
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 $V{\downarrow}_{A}^{B}:A\to {M}_{d}\left(\u2102\right):a\mapsto V\left(a\right)\text{.}$
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 $\lambda \in \stackrel{\u02c6}{A},$ be the irreducible representations of $A,$ and ${V}^{\mu},$ for $\mu \in B\u02c6,$ 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 $${V}^{\mu}{\downarrow}_{A}^{B}\simeq \underset{\lambda \in \stackrel{\u02c6}{A}}{\u2a01}{\left({W}^{\lambda}\right)}^{\oplus {g}_{\lambda \mu}}=\sum _{\lambda \in \stackrel{\u02c6}{A}}{g}_{\lambda \mu}{W}^{\lambda}$$ 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 $\stackrel{\u02c6}{A},$ while those of the bottom row are labeled by elements of $\stackrel{\u02c6}{B}\text{.}$ The vertex labeled by $\mu $ in $\stackrel{\u02c6}{B}$ is connected to the vertex labeled by $\lambda $ in $\stackrel{\u02c6}{A}$ 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. $$\nA\u02c6\nB\u02c6\n\lambda (1)\n\lambda (2)\n\lambda (3)\n\lambda (4)\n\lambda (5)\n\mu (1)\n\mu (2)\n\mu (3)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n$$
Example 2.56 We can consider ${\mathcal{S}}_{m}$ as a subgroup of ${\mathcal{S}}_{m+1},$ one such way is as the stabilizer of $m+1\text{.}$ Specifically, when $m=3,$ the embedding one gets is $$\begin{array}{ccccc}\n\n\n\n\n\n\n\n\n\n\n\n\n& \in {\mathcal{S}}_{3}& \mapsto & \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n& \in {\mathcal{S}}_{4}\end{array}$$ This embedding extends to an embedding of group rings $\u2102{\mathcal{S}}_{3}\subseteq \u2102{\mathcal{S}}_{4}\text{.}$ The irreducible representations of ${\mathcal{S}}_{m}$ are indexed by partitions of $m,$ so the vertices of the Bratteli diagram for the above group ring containment will be $$\n\u2102\mathcal{S}3\n\u2102\mathcal{S}4\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n$$ To determine the edges, recall that if $\mu =({\mu}_{1},\dots ,{\mu}_{k-1},1)\u22a2m$ and $\stackrel{\u02c6}{\mu}=({\mu}_{1},\dots ,{\mu}_{k-1})\u22a2m-1,$ then $${\chi}^{\lambda}\left(\mu \right)=\sum _{\nu}{\chi}^{\nu}\left(\stackrel{\u02c6}{\mu}\right)$$ 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 ${\mathcal{S}}_{3}$ into ${\mathcal{S}}_{4}\text{.)}$ In terms of modules, if $V$ is a $\u2102{\mathcal{S}}_{m}\text{-representation,}$ then $${V}^{\lambda}{\downarrow}_{{\mathcal{S}}_{m-1}}^{{\mathcal{S}}_{m}}\simeq \underset{\underset{\lambda /\nu =\square}{\nu \u22a2m-1}}{\u2a01}{V}^{\nu}\text{.}$$ Thus the Bratteli diagram one obtains is $$\n\u2102\mathcal{S}3\n\u2102\mathcal{S}4\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n$$ 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 $$\begin{array}{c}({b}_{1}+{b}_{2})\otimes w={b}_{1}\otimes w+{b}_{2}\otimes w\\ b\otimes ({w}_{1}+{w}_{2})=b\otimes {w}_{1}+b\otimes {w}_{2}\\ c(b\otimes w)=\left(cb\right)\otimes w=b\otimes \left(cw\right)\\ \left(ba\right)\otimes w=b\otimes (a\xb7w)\end{array}$$ where $b,{b}_{i}\in B,$ $w,{w}_{i}\in W,$ $c\in \u2102,$ and $a\in A\text{.}$ $B{\otimes}_{A}W$ is a $B\text{-module}$ via the action $$b\xb7({b}_{1}\otimes w)=\left(b{b}_{1}\right)\otimes w\text{.}$$
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 $b\xb7\psi (1\otimes w)\text{.}$ A reasonable choice of $\psi (1\otimes w)$ may well be $\varphi \left(w\right),$ yielding the definition $$\varphi \mapsto \psi ,\hspace{0.17em}\text{where}\hspace{0.17em}\psi (b\otimes w)=b\xb7\varphi \left(w\right)\text{.}$$ 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 $\varphi \left(w\right)=\psi (1\otimes w)\text{.}$ $\text{\u201c}\square \text{\u201d}$ |
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.