Lectures in Representation Theory

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

Last update: 12 August 2013

Lecture 1

$1\text{Basics}$

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

$$\left(c{a}_1\right)a_2=c\left(a_1{a}_2\right)=a_1\left(c{a}_2\right)\forall c\in ℂ,\hspace{0.17em}{a}_1,a_2\in A$$

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 $d\times d$ matrices over the complex numbers with the usual matrix multiplication. We will write

$$I_d=\left(\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& \cdots & 0\\ & ⋮& \ddots & \\ 0& 0& \cdots & 1\end{array}\right)$$

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

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

$$a_i{a}_j=\sum _{k=1}^d{c}_{i,j}^k{a}_k$$

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

$$\left\{E_{i,j}\hspace{0.17em}\right|\hspace{0.17em}1\le i,j\le d\}$$

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

$$E_{i,j}{E}_{r,s}={\delta }_{j,r}{E}_{i,s}$$

and so the structure constants are given by

$$c_{\{i,j\}\{r,s\}}^{\{k,\ell \}}=\{\begin{array}{cc}1& j=r,\hspace{0.17em}k=i,\hspace{0.17em}\ell =s\\ 0& \text{otherwise}\end{array}$$

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.	$f(a_1+a_2)=f\left(a_1\right)+f\left(a_2\right)\text{(}f$ 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 $d\in \mathbb{N}$ 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 $\varphi :G⟶\text{GL}(d,ℂ)\text{.}$ 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 $G=\{g_1,g_2,\dots ,g_5\}$ then

$$ℂG=\{c_1{g}_1+c_2{g}_2+\dots c_5{g}_5\hspace{0.17em}|\hspace{0.17em}{c}_i\in ℂ\}$$

with multiplication defined as in the following example:

$$(5g_1+3g_2)(2g_3+4g_4)=10\left(g_1{g}_3\right)+20\left(g_1{g}_4\right)+6\left(g_2{g}_3\right)+12\left(g_2{g}_4\right)$$

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 $A\times V⟶V$ denoted $⟨a,v⟩\mapsto a·v$ such that

1.	$(c_1{a}_1+c_2{a}_2)·v=c_1\left(a_{1}v\right)+c_2\left(a_{2}v\right)$
2.	$a_1·(c_1{v}_1+c_2{v}_2)=c_1\left(a_{1}v\right)+c_2\left(a_{2}v\right)$
3.	$a_1·(a_2·v_1)=\left(a_1{a}_2\right)·v_1$
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 $\{v_1,v_2,\dots ,v_d\}$ 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 $T\mapsto \left(t_{j,i}\right)$ 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

$$ab·v_i=a·\left(\sum _{j=1}^d{b}_{j,i}{v}_j\right)=\sum _{j,k=1}^d{a}_{k,j}{b}_{j,i}{v}_k=\sum _{k=1}^d\left(\sum _{j=1}^d{a}_{k,j}{b}_{j,i}\right)v_k=\left(\varphi \left(a\right)\varphi \left(b\right)\right)v_i\text{.}$$

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 $\{w_1,w_2,\dots ,w_d\}$ 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 $P\in \text{GL}(d,ℂ)$ such that

$$\psi \left(a\right)=P^{-1}\varphi \left(a\right)P$$

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

$$P^{-1}\varphi \left(a\right)P{w}_i=P^{-1}\varphi \left(a\right)v_i=P^{-1}\sum _j{a}_{j,i}{v}_j=\sum _j{a}_{j,i}{w}_j=\psi \left(a\right)w_i,$$

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 $d\times 1$ 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

$$a\mapsto \left(\begin{array}{cc}\psi \left(a\right)& *\\ 0& *\end{array}\right)$$

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.

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