Representations

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

Last update: 6 November 2012

Representations

An algebra is a vector space $A$ over $ℂ$ with a multiplication such that $A$ is a ring with identity and such that for all $a_1,a_2\in A$ and $c\in ℂ,$

$$\begin{array}{cc}\left(c{a}_1\right)a_2=a_1\left(c{a}_2\right)=c\left(a_1{a}_2\right)\text{.}& \text{(1.1)}\end{array}$$

More precisely, an algebra is a vector space over $ℂ$ with a multiplication that is associative, distributive, has an identity, and satisfies (1.1). Suppose that $a_1,a_2,\dots ,a_n$ is a basis of $A$ and that $c_{ij}^k$ are constants in $ℂ$ such that

$$\begin{array}{cc}{a}_i{a}_j=\sum _{k=1}^n{c}_{ij}^k{a}_k\text{.}& \text{(1.2)}\end{array}$$

It follows from (1.1) and the distributive property that the equations (1.2) for $1\le i,j\le n$ completely determine the multiplication in $A\text{.}$ The $c_{ij}^k$ are called structure constants. The center of an algebra $A$ is the subalgebra

$$Z\left(A\right)=\{b\in A\mid ab=ba\text{for all}a\in A\}\text{.}$$

A nonzero element $p\in A$ such that $pp=p$ is called an idempotent. Two idempotents $p_1,p_2\in A$ are orthogonal if $p_1{p}_2=p_2{p}_1=0\text{.}$ A minimal idempotent is an idempotent $p\in A$ that cannot be written as a sum $p=p_1+p_2$ of orthogonal idempotents $p_1,p_2\in A\text{.}$

For each positive integer $d$ we denote the algebra of $d\times d$ matrices with entries from $ℂ$ and ordinary matrix multiplication by $M_d\left(ℂ\right)\text{.}$ We denote the $d\times d$ identity matrix in $M_d\left(ℂ\right)$ by $I_d\text{.}$ For a general algebra $A,$ $M_d\left(A\right)$ denotes $d\times d$ matrices with entries in $A\text{.}$ We denote the algebra of matrices of the form

$$\left(\begin{array}{cccc}a& 0& \dots & 0\\ 0& a& \dots & 0\\ \dots & \dots & \dots & \dots \\ 0& 0& \dots & a\end{array}\right),a\in A,$$

by $I_n\left(A\right)\text{.}$ Note that $I_n\left(A\right)\cong A,$ as algebras. The trace, $\text{tr}\left(a\right),$ of a matrix $a=\mid a_{ij}\mid$ is the sum of the diagonal entries of $a,$ $\text{tr}\left(a\right)=\sum _i{a}_{ii}\text{.}$

An algebra homomorphism of an algebra $A$ into an algebra $V$ is a $ℂ\text{-linear}$ map $f:A\to B$ such that for all $a_1,a_2\in A,$

$$\begin{array}{cc}\begin{array}{c}f\left(1\right)=1,\\ f\left(a_1{a}_2\right)=f\left(a_1\right)f\left(a_2\right)\text{.}\end{array}& \text{(1.3)}\end{array}$$

A representation of an algebra $A$ is an algebra homomorphism

$$V:A⟶M_d\left(ℂ\right)\text{.}$$

The dimension of the representation $V$ is $d\text{.}$ The image $V\left(A\right)$ of the representation $V$ is a finite dimensional algebra of $d\times d$ matrices which we call the algebra of the representation $V\text{.}$ It is a subalgebra of $M_d\left(ℂ\right)\text{.}$ A faithful representation is a representation which is injective. In this case the algebra $V\left(A\right)$ is called a faithful realization of $A$ and $A\cong V\left(A\right)$ The character of the representation $V$ of $A$ is the function ${\chi }_V:A\to ℂ$ given by

$$\begin{array}{cc}{\chi }_V\left(a\right)=\text{tr}\left(V\left(a\right)\right)\text{.}& \text{(1.4)}\end{array}$$

An anti-representation of an algebra $A$ is a $ℂ\text{-linear}$ map $V^{\prime }:A\to M_d\left(ℂ\right)$ such that for all $a_1,a_2\in A,$

$$\begin{array}{c}{V}^{\prime }\left(1\right)=I_d,\\ V^{\prime }\left(a_1{a}_2\right)=V^{\prime }\left(a_2\right)V^{\prime }\left(a_1\right)\text{.}\end{array}$$

As before the dimension of the anti-representation is $d$ and the image, $V^{\prime }\left(A\right),$ of the anti-representation is an algebra of matrices called the algebra of the anti-representation.

The group algebra $ℂG$ of a group $G$ is the algebra of formal finite linear combinations of elements of $G$ where the multiplication is given by the linear extension of the multiplication in $G\text{.}$ The elements of $G$ constitute a basis of $ℂG\text{.}$ A representation of the group $G$ is a representation of its group algebra.

Let $A$ be an algebra. An $A\text{-module}$ is a vector space $V$ with an $A$ action $A\times V\to V$ such that for all $a,a_1,a_2\in A,$ $v,v_1,v_2\in V,$ and $c_1,c_2\in ℂ,$

$$\begin{array}{cc}\begin{array}{ccc}1v& =& v,\\ a_1\left(a_{2}v\right)& =& \left(a_1{a}_2\right)v,\\ (a_1+a_2)v& =& a_{1}v+a_{2}v,\\ a(c_1{v}_1+c_2{v}_2)& =& c_1\left(a{v}_1\right)+c_2\left(a{v}_2\right)\text{.}\end{array}& \text{(1.5)}\end{array}$$

An $A\text{-module}$ homomorphism is a $ℂ\text{-linear}$ map $f:V\to V^{\prime }$ between $A\text{-modules}$ $V$ and $V^{\prime }$ such that for all $a\in A$ and $v\in V,$

$$\begin{array}{cc}f\left(av\right)=af\left(v\right)\text{.}& \text{(1.6)}\end{array}$$

An $A\text{-module}$ isomorphism is a bijective $A\text{-module}$ homomorphism.

By condition 3 of (1.5) the action of $a\in A$ on $V$ is a linear transformation $V\left(a\right)$ of $V\text{.}$ If we specify a basis $B$ of $V$ then the linear transformation $V\left(a\right)$ can be written as a $d\times d$ matrix, where $\text{dim}V=d\text{.}$ In this way we associate to every element of $A$ a $d\times d$ matrix. This gives a representation of $A$ which we shall also denote by $V\text{.}$

Conversely, if $T$ is a $d$ dimensional representation of $A$ and $V$ is a $d$ dimensional vector space with basis $B$ then we can define the action of an element $a$ in $A$ by the action of the linear transformation on $V$ determined by the matrix $T\left(a\right)$ so that for all $v\in V,$

$$av=T\left(a\right)v\text{.}$$

In this way $V$ becomes an $A\text{-module.}$ Thus the notion of A $A\text{-module}$ is equivalent to the notion of representation. When we view the $A\text{-module}$ we are focusing on the vector space and when we view the representation we are focusing on the linear transformations (matrices).

Let $V$ be an $A\text{-module}$ with basis $B$ and let $B^{\prime }$ be another basis of $V$ and denote the change of basis matrix by $P\text{.}$ Let $a\in A$ and let $V\left(a\right),$ $V^{\prime }\left(a\right)$ be the matrices, with respect to the bases $B$ and $B^{\prime }$ respectively, of the linear transformation on $V$ induced by $a\text{.}$ Then by elementary linear algebra we have that

$$\begin{array}{cc}{V}^{\prime }\left(a\right)=PV\left(a\right)P^{-1}\text{.}& \text{(1.7)}\end{array}$$

This leads us to the following definition. Two $d$ dimensional representations $V$ and $V^{\prime }$ of an algebra $A$ are equivalent if there exists an invertible $d\times d$ matrix $P$ such that (1.7) holds for all $a\in A\text{.}$ Isomorphic modules define equivalent representations.

The direct sum $V_1\oplus V_2$ of two $A\text{-modules}$ $V_1$ and $V2$ is the $A\text{-module}$ of all pairs $(v_1,v_2),$ $v_1\in V_1$ and $v_2\in V_2,$ with the $A$ action given by

$$a(v_1,v_2)=(a{v}_1,a{v}_2),$$

for all $a\in A\text{.}$ The direct sum $V_1\oplus V_2$ of two representations $V_1$ and $V_2$ of $A$ is the representation $V$ of $A$ given by

$$\begin{array}{cc}V\left(a\right)=\left(\begin{array}{cc}{V}_1\left(a\right)& 0\\ 0& V_2\left(a\right)\end{array}\right)\text{.}& \text{(1.8)}\end{array}$$

Direct sums of $n>2$ representations or $A\text{-modules}$ are defined analogously. We denote $V\oplus V\oplus \dots \oplus V,$ $n$ factors, by $V^{\oplus n}\text{.}$ Note that the algebra of the representation $V^{\oplus n},$ $V^{\oplus n}\left(A\right),$ is $I_n\left(V\left(A\right)\right)\text{.}$

An $A\text{-invariant}$ subspace of an $A\text{-module}$ $V$ is a subspace $V^{\prime }$ of $V$ such that

$$\{a{v}^{\prime }\mid a\in A,v^{\prime }\in V^{\prime }\}=A{V}^{\prime }\subseteq V^{\prime }\text{.}$$

An $A\text{-invariant}$ subspace of $V$ is just a submodule of $V\text{.}$ Note that the intersection $V^{\prime }\cap V^{\prime \prime }$ of any two invariant subspaces $V^{\prime },$ $V^{\prime \prime }$ of $V$ is also an invariant subspace of $V\text{.}$

An $A\text{-module}$ with no submodules is a simple module. An irreducible representation is a representation that is not equivalent to a representation of the form

$$\begin{array}{cc}V\left(a\right)=\left(\begin{array}{cc}{V}^{\prime }\left(a\right)& *\\ 0& *\end{array}\right),& \text{(1.9)}\end{array}$$

where $V^{\prime }$ is also representation of $A\text{.}$ If $V^{\prime },$ $V^{\prime \prime }$ are invariant subspaces of a representation $V$ and $V^{\prime }$ is irreducible then $V^{\prime }\cap V^{\prime \prime }$ is either equal to 0 or $V^{\prime }\text{.}$ A completely decomposable representation is a representation that is equivalent to a direct sum of irreducible representations. An algebra $A$ is called completely decomposable if every representation of $A$ is completely decomposable.

The centralizer of an algebra $A$ of $d\times d$ matrices is the algebra $\bar{A}$ of $d\times d$ matrices $\bar{a}$ such that for all matrices $a\in A,$

$$\begin{array}{cc}\bar{a}a=a\bar{a}\text{.}& \text{(1.10)}\end{array}$$

The centralizer of a representation $V$ of an algebra $A$ is the algebra $\overline{V\left(A\right)}\text{.}$

Examples

1. Let $A$ be an algebra of $d\times d$ matrices. Since all matrices in $A$ commute with all elements of $A,$

$$A\subseteq \stackrel{\overline{‾}}{A}\text{.}$$

Also,

$$\begin{array}{c}\overline{I_n\left(A\right)}=M_n\left(\bar{A}\right)\text{and}\\ \overline{M_n\left(A\right)}=I_n\left(\bar{A}\right)\text{.}\end{array}$$

Hence,

$$\stackrel{\overline{‾}}{I_n\left(A\right)}=I_n\left(\stackrel{\overline{‾}}{A}\right)\text{.}$$

2. Schur's lemma. Let $W_1$ and $W_2$ be irreducible representations of $A$ of dimensions $d_1$ and $d_2\text{.}$ If $B$ is a $d_1\times d_2$ matrix such that

$$W_1\left(a\right)B=B{W}_2\left(a\right),\text{for all}a\in A,$$

then either

1. $W_1\ncong W_2$ and $B=0,$ or
2. $W_1\cong W_2$ and if $W_1=W_2$ then $B=c{I}_{d_1}$ for some $c\in ℂ\text{.}$

		Proof.
	$B$ determines a linear transformation $B:W_1\to W_2\text{.}$ Since $Ba=aB$ for all $a\in A$ we have that $$B\left(aw1\right)=Baw1=aBw1=aB\left(w1\right),$$ for all $a\in A$ and $w_1\in W_1$ Thus $B$ is an $A\text{-module}$ homomorphism. $\text{ker}B$ and $\text{im}B$ are submodules of $W_1$ and $W_2$ respectively and are therefore either 0 or equal to $W_1$ or $W_2$ respectively. If $\text{ker}B=W_1$ or $\text{im}B=0$ then $B=0\text{.}$ In the remaining case $B$ is a bijection, and thus an isomorphism between $W_1$ and $W_2$ In this case we have that $d_1=d_2\text{.}$ Thus the matrix $B$ is square and invertible. Now suppose that $W_1=W_2$ and let $c$ be an eigenvalue of $B\text{.}$ Then the matrix $c{I}_{d_1}-B$ is such that $W_1\left(a\right)(c{I}_{d_1}-B)=(c{I}_{d_1}-B)W_1\left(a\right)$ for all $a\in A\text{.}$ The argument in the preceding paragraph shows that $c{I}_{d_1}-B$ is either invertible or 0. But if $c$ is an eigenvalue of $B$ then $\text{det}(c{I}_{d_1}-B)=0\text{.}$ Thus $c{I}_{d_1}-B=0\text{.}$ $\square$

3. Suppose that $V$ is a completely decomposable representation of an algebra $A$ and that $V\cong {\oplus }_{\lambda }{W}_{\lambda }^{\oplus m_{\lambda }}$ where the $W_{\lambda }$ are nonisomorphic irreducible representations of $A\text{.}$ Schur's lemma shows that the $A$-homomorphisms from $W_{\lambda }$ to $V$ form a vector space $${\mathrm{Hom}}_A(W_{\lambda },V)\cong {ℂ}^{\oplus m_{\lambda }}\text{.}$$ The multiplicity of the irreducible representation $W_{\lambda }$ un $V$ is $$m_{\lambda }=\dim {\mathrm{Hom}}_A(W_{\lambda },V)\text{.}$$

4. Suppose that $V$ is a completely decomposable representation of an algebra $A$ and that $V\cong {\oplus }_{\lambda }{W}_{\lambda }^{\oplus m_{\lambda }}$ where the $W_{\lambda }$ are nonisomorphic irreducible representations of $A$ and let $\dim W_{\lambda }=d_{\lambda }\text{.}$ Then $$V\left(A\right)\cong {\oplus }_i{W}_{\lambda }^{\oplus m_{\lambda }}\left(A\right)\cong {\oplus }_{\lambda }{I}_{m_{\lambda }}\left(W_{\lambda }\left(A\right)\right)\cong {\oplus }_{\lambda }{W}_{\lambda }\left(A\right)\text{.}$$ If we view elements of ${\oplus }_{\lambda }{I}_{m_{\lambda }}{W}_{\lambda }\left(A\right)$ as block diagonal matrices with $m_{\lambda }$ blocks of size $d_{\lambda }\times d_{\lambda }$ for each $\lambda$, then by using Ex 1 and Schur's lemma we get that $$\begin{array}{ccc}V\left(A\right)\cong {\oplus }_{\lambda }{I}_{m_{\lambda }}\left(W_{\lambda }\left(A\right)\right)& =& {\oplus }_{\lambda }{M}_{m_{\lambda }}\left(W_{\lambda }\left(A\right)\right)\\ & =& {\oplus }_{\lambda }{M}_{m_{\lambda }}\left(I_{d_{\lambda }}\left(ℂ\right)\right)\text{.}\end{array}$$

5. Let $V$ be an $A$-module and let $p$ be an idempotent of $A\text{.}$ Then $pV$ is a subspace of $V$ and the action of $p$ on $V$ is a projection from $V$ to $pV\text{.}$ If $p_1,p_2\in A$ are orthogonal idempotents of $A$ then $p_{1}V$ and $p_{2}V$ are mutually orthogonal subspaces of $V,$ since if $p_{1}v=p_{2}v\text{'}$ for some $v,v\text{'}\in V$ then $p_{1}v=p_1{p}_{1}v=p_1{p}_{2}v\text{'}=0\text{.}$ So $V=p_{1}V\oplus p_{2}V\text{.}$

6. Let $p$ be an idempotent in $A$ and suppose that for every $a\in A,pap=kp$ for some constant $k\in ℂ\text{.}$ If $p$ is not minimal then $p=p_1+p_2,$ where $p_1,p_2\in A$ are idempotents such that $p_1{p}_2=p_2{p}_1=0\text{.}$ Then $p_1=p{p}_{1}p=kp$ for some constant $k\in ℂ\text{.}$ This implies that $p_1=p_1{p}_1=k{p}_1{p}_1=k{p}_1,$ giving that either $k=1$ or $p_1=0\text{.}$ So $p$ is minimal.

7. Let $A$ be a finite dimensional algebra and suppose that $z\in A$ is an idempotent of $A\text{.}$ If $z$ is not minimal then $z=p_1+p_2$ where $p_1$ and $p_2$ are orthogonal idempotents of $A\text{.}$ If any idempotent in this sum is not minimal we can decompose it into a sum of orthogonal idempotents. We continue this process until we have decomposed $z$ as a sum of minimal orthogonal idempotents. At any particular stage in this process $z$ is expresed as a sum of orthogonal idempotents, $z={\sum }_i{p}_i\text{.}$ So $zA={\sum }_i{p}_{i}A\text{.}$ None of the spaces $p_{i}A$ is 0 since $p_i=p_i\text{.}1\in p_{i}A$ and the spacers $p_{i}A$ are all mutually orthogonal. Thus, since $zA$ is finite dimensional it will only take a finite number of steps to decompose $z$ into minimal idempotents. A partition of unity is a decomposition of 1 into minimal orthogonal idempotents.

Notes and References

This is an excerpt from the unpublished first chapter of Arun Ram's dissertation entitled Representation Theory, written July 4, 1990.

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