## Lectures in Representation Theory

Last update: 12 August 2013

## Lecture 2

Homework Problem 1.16 Study ${L}^{2}\left(G\right)$ for large groups (the algebra of ${L}^{2}$ functions on $G\text{).}$ Go find out about compact groups and representation theory.

If $V$ is a (finite dimensional) $A\text{-module}$ of dimension $d,$ then there is an associated representation, which we shall also denote by $V,$ such that $V:A⟶{M}_{d}\left(ℂ\right)\text{.}$

Definition 1.17 Let $A$ be an algebra, and ${V}_{1}$ and ${V}_{2}$ be $A\text{-modules.}$ The direct sum ${V}_{1}\oplus {V}_{2}$ is the $A\text{-module}$ of all pairs $\left({v}_{1},{v}_{2}\right),$ where ${v}_{i}\in {V}_{i}\text{.}$ Addition is defined componentwise; the action of $A$ is $a·\left({v}_{1},{v}_{2}\right)=\left(a{v}_{1},a{v}_{2}\right)\text{.}$

We can make a parallel definition for representations.

Definition 1.18 Let $A$ be an algebra, and ${V}_{1}:A\to {M}_{{d}_{1}}\left(ℂ\right)$ and ${V}_{2}:A\to {M}_{{d}_{2}}\left(ℂ\right)$ representations of $A\text{.}$ Then the direct sum $\left({V}_{1}\oplus {V}_{2}\right):A\to {M}_{{d}_{1}+{d}_{2}}\left(ℂ\right)$ is the representation given by

$(V1⊕V2)(a)= ( V1(a)0 0V2(a) ) .$

Definition 1.19 An $A\text{-module}$ $V$ is completely decomposable if it is isomorphic (as an $A\text{-module)}$ to a direct sum of simple $A\text{-modules.}$

Problem 1.20 Let $A$ be an algebra.

 1 What are the irreducible representations of $A\text{?}$ 2 Are the other representations of $A$ completely decomposable?

Definitions 1.21 Given an algebra $A\text{.}$ A trace on $A$ is a linear functional $\stackrel{\to }{t}:A\to ℂ$ such that $\stackrel{\to }{t}\left(ab\right)=\stackrel{\to }{t}\left(ba\right),$ for all $a,b\in A\text{.}$ The center of $A$ is the subalgebra $Z\left(A\right)=\left\{t\in A | ta=at \forall a\in A\right\}\text{.}$ An idempotent is an element $p\in A$ such that $p\ne 0$ and ${p}^{2}=p\text{.}$ A central idempotent is an idempotent $z$ such that $z\in Z\left(A\right)\text{.}$ An ideal of $A$ is an ideal in the ring theory sense.

We will use the words “trace” and “character” interchangeably; some would have us call our traces “virtual characters”.

Definition 1.22 Let $V:A\to {M}_{d}\left(ℂ\right)$ be a representation. Then the character of the representation $V$ is given by ${\chi }_{V}:A\to ℂ:a↦\text{tr}\left(V\left(a\right)\right),$ where $\text{tr}$ denotes the vanilla trace of high school matrix theory. Any such character is a trace in the sense defined above.

The following propositions can be nicely summarized as follows:

Theorem 1.23 There exists a unique nonzero

${ trace element of the center central idempotent ideal irreducible representation } in Md(ℂ) { up to scalar multiples up to scalar multiples up to equivalence } .$

Proposition 1.24 There is a unique (up to scalar multiples) nonzero trace on ${M}_{d}\left(ℂ\right)\text{.}$

 Proof. Let $\stackrel{\to }{t}$ be a trace, and let $a\in {M}_{d}\left(ℂ\right)\text{.}$ $t→(a) = t→ ( ∑ij aijEij ) = ∑ijaij t→(Eij)= ∑ijaij t→(Ei1E1j) = ∑ijaijt→ (E1jEi1)= ∑ijaij δijt→(E11) =t→(E11)tr(a)$ Thus any such $\stackrel{\to }{t}$ is a scalar multiple of the ordinary trace functional. $\square$

Proposition 1.25 There is a unique nonzero element in $Z\left({M}_{d}\left(ℂ\right)\right)$ (up to scalar multiples).

 Proof. Let $z\in Z\left({M}_{d}\left(ℂ\right)\right),$ $z\ne 0\text{.}$ Now $z=\sum _{ij}{z}_{ij}{E}_{ij},$ and $z\ne 0$ implies that for some $m$ and $n$ the coefficient ${z}_{mn}\ne 0,$ Fix these $m$ and $n\text{.}$ Let $k\ne s\text{.}$ Since $z$ is central, ${E}_{mk}z{E}_{sn}={E}_{mk}{E}_{sn}z=0\text{.}$ On the other hand, $EmkzEsn = Emk ( ∑ijzij Eij ) Esn = Emkzks EksEsn = zksEmn$ Thus if $k\ne s,$ then ${z}_{ks}=0\text{.}$ Thus $z$ central implies that $z$ is a diagonal matrix. To show that $z$ central implies $z$ is scalar, fix $1\le i\le d,$ and consider ${E}_{ii}z\text{.}$ For any $j,$ ${E}_{ii}z={E}_{ij}{E}_{ji}z={E}_{ij}z{E}_{ji}={z}_{jj}{E}_{ii}\text{.}$ Since this holds for all $j,$ it follows that ${z}_{jj}={z}_{11}$ for all $j\text{.}$ $\square$

Proposition 1.26 There is a unique nonzero ideal in ${M}_{d}\left(ℂ\right)\text{.}$

 Proof. Exercise. $\square$

Proposition 1.27 There is a unique central idempotent in ${M}_{d}\left(ℂ\right)\text{.}$

 Proof. Let $z\in Z\left({M}_{d}\left(ℂ\right)\right)$ with ${z}^{2}=z\text{.}$ $z$ central implies that $z={z}_{11}{I}_{d}\text{.}$ Now ${z}^{2}={\left({z}_{11}{I}_{d}\right)}^{2}={{z}_{11}}^{2}{I}_{d}\text{;}$ thus ${z}_{11}$ is a complex number satisfying ${{z}_{11}}^{2}={z}_{11},$ so either ${z}_{11}=0$ or ${z}_{11}=1\text{.}$ This is one place where these developments will not go through so nicely in other characteristics. $\square$

Proposition 1.28 There is a unique (nonzero) irreducible representation of ${M}_{d}\left(ℂ\right)$ (up to equivalence).

 Proof. First we shall prove that such a representation exists. Let $V$ be the space of all column vectors of length $d,$ ${ℂ}^{d}\text{.}$ ${M}_{d}\left(ℂ\right)$ acts on $V$ by matrix multiplication. Claim: $V$ is irreducible. Let ${e}_{1},\dots ,{e}_{d}$ be the standard basis for $V\text{.}$ Let $W$ be a subrepresentation, with $W\ne 0\text{.}$ Choose $v\in W$ with $v\ne 0,$ $v=\sum {v}_{i}{e}_{i},$ and fix an $i$ such that ${v}_{i}\ne 0\text{.}$ Claim: ${M}_{d}\left(ℂ\right)v=V$ (whence $W=V,$ proving the previous claim.) $1viEjiv = 1viEji ∑kvkek = 1vi∑kvk Ejiek = 1vi∑kvk δikej = 1viviej=ej$ Thus for every $j,$ ${e}_{j}\in {M}_{d}\left(ℂ\right)v\text{.}$ Thus ${M}_{d}\left(ℂ\right)v=V\text{.}$ Continued in next lecture.

## 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.