## Lectures in Representation Theory

Last update: 12 August 2013

## $1\phantom{\rule{1em}{0ex}}\text{Basics}$

Definition 1.1 An algebra is a vector space $A$ over the complex numbers $ℂ$ with multiplication such that $A$ is an associative ring with identity and

$(ca1)a2= c(a1a2)= a1(ca2) ∀c∈ℂ, a1, a2∈A$

The last condition merely says that the subalgebra $ℂ1\subseteq A$ is central, i.e. the scalars are scalars.

Homework Problem 1.2

 1 Develop the following theory for algebras over other fields. Identify places where prime characteristic or lack of algebraic closure causes problems. 2 Develop this theory for Lie algebras, Jordan algebras, or alternative algebras, etc. 3 Develop a specific base theoretic cohomology for semisimple algebras.

Example 1.3 An important example of an associative algebra with identity is the set ${M}_{d}\left(ℂ\right)$ of all $d×d$ matrices over the complex numbers with the usual matrix multiplication. We will write

$Id= ( 10⋯0 01⋯0 ⋮⋱ 00⋯1 )$

for the identity element. Then the scalars $ℂ$ are isomorphic to the subalgebra of scalar matrices $ℂ{I}_{d}\text{.}$

We will often work with particular bases of an algebra.

Definition 1.4 Let $A$ be a finite dimensional algebra and choose a basis ${a}_{1},{a}_{2},\dots ,{a}_{d}\text{.}$ The structure constants of $A$ are the scalars ${c}_{i,j}^{k}\in ℂ$ defined by the multiplication rules

$aiaj= ∑k=1d ci,jkak$

Alternatively, one may define an algebra structure on a vector space by specifying a set of structure constants. Of course, the structure constants depend on the particular choice of basis.

For example, in ${M}_{d}\left(ℂ\right),$ one might choose the basis of matrix units

${Ei,j | 1≤i,j≤d}$

where ${E}_{i,j}$ is the matrix with a 1 in the $i\text{th}$ row and $j\text{th}$ column, and zeros elsewhere. If $T=\left({t}_{i,j}\right)$ is a matrix, then we express $T=\sum _{i,j}{t}_{i,j}{E}_{i,j}$ in this basis. Matrix multiplication is merely the linear extension of the multiplication rule

$Ei,jEr,s =δj,rEi,s$

and so the structure constants are given by

$c {i,j}{r,s} {k,ℓ} = { 1 j=r, k=i, ℓ=s 0 otherwise$

We turn next to defining our notions of representation and module.

Definition 1.5 An algebra homomorphism is a map $f:A⟶B$ such that

 1 $f\left({a}_{1}+{a}_{2}\right)=f\left({a}_{1}\right)+f\left({a}_{2}\right)\phantom{\rule{1em}{0ex}}\text{(}f$ is an additive group homomorphism) 2 $f\left(c{a}_{1}\right)=cf\left({a}_{1}\right)$ 3 $f\left({a}_{1}{a}_{2}\right)=f\left({a}_{1}\right)f\left({a}_{2}\right)$ 4 $f\left({1}_{A}\right)={1}_{B}$
for all ${a}_{1},{a}_{2}\in A$ and all $c\in ℂ\text{.}$

Definition 1.6 An algebra homomorphism $\varphi :A⟶{M}_{d}\left(ℂ\right)$ is called a representation. The number $d\in ℕ$ is called the dimension of the representation.

Group representations fit into this scheme as follows. One notion is that a representation of a group $G$ is a group homomorphism $\varphi :G⟶\text{GL}\left(d,ℂ\right)\text{.}$ Equivalently, we may consider group representations as a special case of algebra representations as follows:

Definition 1.7 The group algebra $ℂG$ (often written $ℂ\left[G\right]\text{)}$ is the vector space of finite $ℂ\text{–linear}$ combinations of elements of $G$ with multiplication given as the linear extension of the group multiplication.

Example 1.8 If $G=\left\{{g}_{1},{g}_{2},\dots ,{g}_{5}\right\}$ then

$ℂG= { c1g1+c2g2+ …c5g5 | ci∈ℂ }$

with multiplication defined as in the following example:

$(5g1+3g2) (2g3+4g4)= 10(g1g3)+20 (g1g4)+6 (g2g3)+12 (g2g4)$

where the products ${g}_{i}{g}_{j}$ are taken in the group $G\text{.}$

By definition, the elements of $G$ form a basis for $ℂG\text{.}$

Definition 1.9 A group representation of a group $G$ is an algebra representation of the group algebra $ℂG\text{.}$

Definition 1.10 A module for an algebra $A$ is a vector space $V$ over $ℂ$ together with an $A\text{-action,}$ that is a map $A×V⟶V$ denoted $⟨a,v⟩↦a·v$ such that

 1 $\left({c}_{1}{a}_{1}+{c}_{2}{a}_{2}\right)·v={c}_{1}\left({a}_{1}v\right)+{c}_{2}\left({a}_{2}v\right)$ 2 ${a}_{1}·\left({c}_{1}{v}_{1}+{c}_{2}{v}_{2}\right)={c}_{1}\left({a}_{1}v\right)+{c}_{2}\left({a}_{2}v\right)$ 3 ${a}_{1}·\left({a}_{2}·{v}_{1}\right)=\left({a}_{1}{a}_{2}\right)·{v}_{1}$ 4 $1·{v}_{1}={v}_{1}$
for all ${v}_{i}\in V,$ ${c}_{i}\in ℂ$ and ${a}_{i}\in A\text{.}$

We next explore the relationship between modules and representations. Let $V$ be a vector space of dimension $d$ and let $\left\{{v}_{1},{v}_{2},\dots ,{v}_{d}\right\}$ be a basis of $V\text{.}$ If $T\in \text{End}\left(V\right)$ is a linear transformation then in terms of this basis we may express $T{v}_{i}=\sum _{j}{t}_{j,i}{v}_{j}$ for some scalars ${t}_{j,i}\in ℂ\text{.}$ The map $\text{End}\left(V\right)⟶{M}_{d}\left(ℂ\right)$ given by $T↦\left({t}_{j,i}\right)$ is an algebra isomorphism.

Now suppose that $V$ is an $A\text{-module.}$ If $a\in A$ then $a{v}_{i}=\sum _{j}{a}_{j,i}{v}_{j}$ for some scalars ${a}_{j,i}\text{.}$ We claim that $\varphi :A⟶{M}_{d}\left(ℂ\right)$ defined by $\varphi \left(a\right)=\left({a}_{j,i}\right)$ is a representation of $A\text{.}$ The bilinearity of the $A\text{-module}$ action implies $\varphi$ is a linear transformation. Clearly $\varphi \left({1}_{A}\right)={I}_{d}$ and it remains to check that $\varphi$ is multiplicative. Let $a,b\in A$ and write $\varphi \left(a\right)=\left({a}_{j,i}\right)$ and $\varphi \left(b\right)=\left({b}_{j,i}\right)\text{.}$ Then

$ab·vi=a· (∑j=1dbj,ivj)= ∑j,k=1d ak,jbj,i vk=∑k=1d (∑j=1dak,jbj,i) vk= (ϕ(a)ϕ(b)) vi.$

By definition, $\varphi \left(ab\right)=\varphi \left(a\right)\varphi \left(b\right),$ hence $\varphi$ is a representation of $A\text{.}$

Thus, given an $A\text{-module}$ and a choice of basis, we can produce a representation of $A\text{.}$ What happens if we choose a different basis of $V\text{?}$

Let $\left\{{w}_{1},{w}_{2},\dots ,{w}_{d}\right\}$ be another basis of $V,$ and let $\psi :A⟶{M}_{d}\left(ℂ\right)$ be the representation defined as above using this new basis.

Definition 1.11 Two representations $\varphi :A⟶{M}_{d}\left(ℂ\right)$ and $\psi :A⟶{M}_{d}\left(ℂ\right)$ are equivalent provided there exists a matrix $P\in \text{GL}\left(d,ℂ\right)$ such that

$ψ(a)= P-1ϕ(a)P$

for all $a\in A\text{.}$

If we take $P$ to be the change of basis matrix such that $P{w}_{i}={v}_{i}$ then

$P-1ϕ(a)P wi=P-1ϕ(a) vi=P-1∑j aj,ivj=∑j aj,iwj=ψ (a)wi,$

hence ${P}^{-1}\varphi \left(a\right)P=\psi \left(a\right),$ i.e. the representations are equivalent. We have shown that up to equivalence, a module corresponds to a unique representation.

On the other hand, if we have a representation $\varphi :A⟶{M}_{d}\left(ℂ\right),$ then letting $W={ℂ}^{d}$ be the space of $d×1$ column vectors produces an $A\text{-module}$ by $a·w=\varphi \left(a\right)w\text{.}$

Definition 1.12 Given modules $V$ and $W,$ a linear transformation $f:V⟶W$ is a module homomorphism provided $f\left(av\right)=af\left(v\right)$ for all $a\in A$ and $v\in V\text{.}$ We say $V$ and $W$ are isomorphic (and write $V\cong W\text{)}$ provided $f$ is invertible.

Homework Problem 1.13

 1 Show that two $A\text{-modules}$ $V$ and $W$ are isomorphic if and only if their associated representations are equivalent. 2 Given two equivalent representations $V:A⟶{M}_{d}\left(ℂ\right)$ and $W:A⟶{M}_{d}\left(ℂ\right)\text{.}$ Show that $V$ and $W$ induce isomorphic $A\text{-module}$ structures on the space of column vectors ${ℂ}^{d}\text{.}$

Thus module and representation are equivalent notions. In the future we will not separate the two halves of the brain, but instead use these concepts simultaneously. Often, we will denote a module and its corresponding representation by a single letter, i.e. the module $V$ affords the representation $V:A⟶{M}_{d}\left(ℂ\right)$ defined by $a·v=V\left(a\right)v\text{.}$

Definition 1.14 A submodule $V\prime \subseteq V$ is a vector subspace of $V$ that is invariant under the action of $A$ that is $a·v\prime \in V\prime$ for all $A\in A$ and $v\prime \in V\prime \text{.}$

A representation $\psi :A⟶{M}_{n}\left(ℂ\right)$ is a subrepresentation of $\varphi :A⟶{M}_{d}\left(ℂ\right)$ provided $d\ge n$ and $\varphi$ is equivalent to a representation of the form

$a↦ ( ψ(a)* 0* )$

Definition 1.15 A module $V$ is simple provided $V$ has no nonzero proper submodules.

A representation $\varphi :A⟶{M}_{d}\left(ℂ\right)$ irreducible provided $\varphi$ has no nonzero proper subrepresentations.

## 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.