3.13. Lecture 13

3.13.1. Highest weights

The following is our first version of the fundamental weight calculation.

Lemma 3.13.1.

Let $(\rho,V)$ be a complex-linear representation of $\mathfrak{sl}_{2,\mathbb{C}}$ . Let $\alpha$ be a weight of $V$ and let ${\bf v}\in V_{\alpha}$ . Then

X{\bf v}\in V_{\alpha+2}

and

Y{\bf v}\in V_{\alpha-2}.

Thus we have three maps:

	$\displaystyle H$	$\displaystyle:V_{\alpha}\longrightarrow V_{\alpha}$
	$\displaystyle X$	$\displaystyle:V_{\alpha}\longrightarrow V_{\alpha+2}$
	$\displaystyle Y$	$\displaystyle:V_{\alpha}\longrightarrow V_{\alpha-2}.$

Proof.

We have, for ${\bf v}\in V_{\alpha}$ ,

	$\displaystyle\rho(H)\rho(X){\bf v}$	$\displaystyle=[\rho(H),\rho(X)]v+\rho(X)\rho(H){\bf v}$
		$\displaystyle=\rho([H,X]){\bf v}+\rho(X)\rho(H){\bf v}$
		$\displaystyle=2\rho(X){\bf v}+\alpha\rho(X){\bf v}$
		$\displaystyle=(\alpha+2)\rho(X){\bf v}.$

So $\rho(X){\bf v}\in V_{\alpha+2}$ as required.

The claim about the action of $Y$ is proved similarly. ∎

Definition 3.13.2.

A vector ${\bf v}\in V_{\alpha}$ is a highest weight vector if it is a weight vector and if

X{\bf v}=0.

In this case we call the weight of ${\bf v}$ a highest weight.

Lemma 3.13.3.

Any finite-dimensional complex linear representation $V$ of $\mathfrak{sl}_{2,\mathbb{C}}$ has a highest weight vector.

Proof.

Indeed, let $\alpha$ be the numerically greatest weight of $V$ (there must be one, as $V$ is finite-dimensional) and let ${\bf v}$ be a weight vector of weight $\alpha$ . Then $Xv$ has weight $\alpha+2$ by the fundamental weight calculation, so must be zero as $\alpha$ was maximal. ∎

Example 3.13.4.

Let $V=\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ . Then the highest weight vectors are ${\bf e}_{1}\otimes{\bf e}_{1}$ and ${\bf e}_{1}\otimes{\bf e}_{-1}-{\bf e}_{-1}\otimes{\bf e}_{1}$ .

These are easily checked to be highest weight vectors — the first is killed by $X$ since $X{\bf e}_{1}=0$ , the second becomes

{\bf e}_{1}\otimes{\bf e}_{1}-{\bf e}_{1}\otimes{\bf e}_{1}=0.

It is left to you to check that there are no further highest weight vectors.

3.13.2. Classification of irreducible representations of $\mathfrak{sl}_{2,\mathbb{C}}$ .

The key point here is that a highest weight vector must have non-negative integer weight $n$ , and it then generates an irreducible representation of dimension $n+1$ whose isomorphism class is determined by $n$ and which has a very natural basis of weight vectors.

Let $(\rho,V)$ be a complex-linear representation of $\mathfrak{sl}_{2,\mathbb{C}}$ .

Lemma 3.13.5.

Suppose that $V$ has a highest weight vector ${\bf v}$ of weight $n$ . Then the subspace $W$ spanned by the vectors

{\bf v},Y{\bf v},Y^{2}{\bf v}=Y(Y{\bf v}),\dots

is an $\mathfrak{sl}_{2,\mathbb{C}}$ -invariant subspace of $V$ .

Moreover, for $n\geq 0$ , the dimension of $W$ is $n+1$ , with basis ${\bf v},Y{\bf v},\ldots,Y^{n}{\bf v}$ , where $Y^{n+1}{\bf v}=0$ .

Proof.

Let $W$ be the span of the $Y^{k}{\bf v}$ . Since the $Y^{k}{\bf v}$ are weight vectors, their span is $H$ -invariant. It is clearly also $Y$ -invariant. So we only need to check the invariance under the $X$ -action. We claim that for all $m\geq 1$ ,

XY^{m}{\bf v}=m(n-m+1)Y^{m-1}{\bf v}.

(3.13.6)

The proof is by induction. The case $m=1$ is:

XY{\bf v}=([X,Y]+YX){\bf v}=H{\bf v}+Y(X{\bf v})=H{\bf v}=n{\bf v}.

If the formula holds for $m$ , then

$\displaystyle XY^{m+1}{\bf v}$	$\displaystyle=([X,Y]+YX)Y^{m}{\bf v}$
	$\displaystyle=HY^{m}{\bf v}+Y(XY^{m}{\bf v})$
	$\displaystyle=(n-2m)Y^{m}{\bf v}+m(n-m+1)Y^{m}{\bf v}$	(induction hypothesis)
	$\displaystyle=(m+1)(n-m)Y^{m}{\bf v}$

as required.

If $n$ is not a nonnegative integer, then $m(n-m+1)\neq 0$ for all $m$ . So

Y^{m-1}{\bf v}\neq 0\implies XY^{m}{\bf v}\neq 0\implies Y^{m}{\bf v}\neq 0,

whence $Y^{m}{\bf v}\neq 0$ for all $m$ . As these are weight vectors with distinct weights, they are linearly independent and so span an infinite dimensional subspace.

If $Y^{i}{\bf v}=0$ for some $0<i\leq n$ , then

0=XY^{i}{\bf v}=i(n-i+1)Y^{i-1}{\bf v}

and so $Y^{i-1}{\bf v}=0$ . Repeating gives that ${\bf v}=Y^{0}{\bf v}=0$ , a contradiction.

Now,

XY^{n+1}{\bf v}=(n+1)(n-n){\bf v}=0

and so $Y^{n+1}{\bf v}$ is either zero or a highest weight vector of weight $-(n+2)<0$ . We have already seen that the second possibility cannot happen, so $Y^{n+1}{\bf v}=0$ .

Thus $W$ is spanned by the (non-zero) weight vectors ${\bf v},Y{\bf v},\ldots,Y^{n}{\bf v}$ with distinct weights, which are therefore linearly independent and so a basis for $W$ . ∎

Remark 3.13.7.

If we didn’t assume that $V$ was finite-dimensional, then the first part of the previous lemma would still be true.

Corollary 3.13.8.

In the situation of the previous lemma, $W$ is irreducible.

Proof.

Suppose that $W^{\prime}\subseteq W$ is a non-zero subrepresentation. Then it has a highest weight vector, which must be (proportional to) $Y^{i}v$ for some $0\leq i\leq n$ . But then

XY^{i}v=i(n-i+1)Y^{i-1}v\neq 0

if $i>0$ and so $i=0$ , meaning that ${\bf v}\in W^{\prime}$ . But then $Y^{i}v\in W^{\prime}$ for all $i$ , so $W^{\prime}=W$ as required. ∎

The weights of $W$ are illustrated in Figure 3.13.2.

Theorem 3.13.9.

Suppose $V$ is an irreducible finite-dimensional complex-linear representation of $\mathfrak{sl}_{2,\mathbb{C}}$ . Then:

(i)

There is a unique (up to scalar) highest weight vector, with highest weight $n\geq 0$ .
(ii)

The weights of $V$ are $n,n-2,\ldots,2-n,-n$ .
(iii)

All weight spaces $V_{\alpha}$ are one-dimensional (we say that $V$ is ‘multiplicity free’).
(iv)

The dimension of $V$ is $n+1$ .
(v)

For every $n\geq 0$ , the unique irreducible complex-linear representation of $\mathfrak{sl}_{2,\mathbb{C}}$ with highest weight $n$ is $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ (up to isomorphism).

Proof.

Let $V$ be an irreducible finite-dimensional complex-linear representation of $\mathfrak{sl}_{2,\mathbb{C}}$ . Let ${\bf v}\in V$ be a highest weight vector. By Lemma 3.13.5 its weight is a nonnegative integer $n$ , and by Corollary 3.13.8 the vectors ${\bf v},Y{\bf v},\ldots,Y^{n}{\bf v}$ span an irreducible subrepresentation of $V$ of dimension $n+1$ , which must therefore be the whole of $V$ . The claims (i) to (iv) follow immediately.

Moreover, the actions of $X$ , $Y$ and $H$ on this basis are given by explicit matrices depending only on $n$ , so the isomorphism class of $V$ is determined by $n$ . It remains only to show that a representation with highest weight $n$ exists for all $n\geq 0$ . Consider $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ . The vector ${\bf e}_{1}^{n}$ is a highest weight vector of weight $n$ , so we are done. In fact, in this case $Y^{a}{\bf e}_{1}^{n}$ is proportional to ${\bf e}_{1}^{n-a}{\bf e}_{-1}^{a}$ , so the irreducible representation generated by ${\bf e}_{1}^{n}$ is in fact the whole of $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ . ∎

In fact, if $(\rho,V)$ is the irreducible representation with highest weight $n$ , highest weight vector ${\bf v}$ , then the matrices of $H$ , $Y$ , and $X$ with respect to the basis

\{{\bf v},Y({\bf v}),\ldots,Y^{n}({\bf v})\}

are respectively

	$\displaystyle\rho(H)$	$\displaystyle=\begin{pmatrix}n&&&\\ &n-2&&\\ &&\ddots&\\ &&&-n\end{pmatrix},$
	$\displaystyle\rho(Y)$	$\displaystyle=\begin{pmatrix}0&&&\\ 1&0&&\\ &\ddots&\ddots&\\ &&1&0\end{pmatrix},$
and
	$\displaystyle\rho(X)$	$\displaystyle=\begin{pmatrix}0&n&&&&\\ &0&2(n-1)&&&\\ &&0&3(n-2)&&\\ &&&\ddots&\ddots&\\ &&&&\ddots&n\\ &&&&&0\end{pmatrix}.$

Theorem 3.13.10.

Every finite-dimensional irreducible complex-linear representation of $\operatorname{SL}_{2}(\mathbb{C})$ or $\mathfrak{sl}_{2,\mathbb{C}}$ is isomorphic to $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ , the symmetric product of the standard representation.

Proof.

We already proved this for $\mathfrak{sl}_{2,\mathbb{C}}$ , Theorem 3.13.9(v). If $V$ is a representation of $\operatorname{SL}_{2}(\mathbb{C})$ , then its derivative will be isomorphic to $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ . Since $\operatorname{SL}_{2}(\mathbb{C})$ is connected, this implies that $V\cong\operatorname{Sym}^{n}(\mathbb{C}^{2})$ . ∎

3.13.3. Exercises

Problem 31. Let $V$ , $W$ be representations of $\mathfrak{sl}_{2,\mathbb{C}}$ . Let ${\bf v}$ and ${\bf w}$ be two weight vectors of $V$ and $W$ respectively with respective weights $\alpha$ and $\beta$ . Show that

{\bf v}\otimes{\bf w}\in V\otimes W

is a weight vector with weight $\alpha+\beta$ , and that if ${\bf v}$ and ${\bf w}$ are highest weight vectors then so is ${\bf v}\otimes{\bf w}$ .

Problem 32. If $\lambda\in\mathbb{C}$ , show that there is a (possibly infinite dimensional!) representation of $\mathfrak{sl}_{2,\mathbb{C}}$ with highest weight $\lambda$ .

Problem 33. Consider $V=\operatorname{Sym}^{n}(\mathbb{C}^{2})$ , the irreducible representation of highest weight $n$ of $\mathfrak{sl}_{2,\mathbb{C}}$ . Decompose the following representations into irreducibles, and find highest weight vectors for the irreducible constituents:

(a)

$\operatorname{Sym}^{2}(\operatorname{Sym}^{2}(\mathbb{C}^{2}))$ ;
(b)

$\bigwedge^{2}\left(\operatorname{Sym}^{2}(\mathbb{C}^{2})\right)$ ;
(c)

$\operatorname{Sym}^{3}(\mathbb{C}^{2})\otimes\operatorname{Sym}^{2}(\mathbb{C}% ^{2})$ ;
(d)

$\operatorname{Sym}^{3}\left(\operatorname{Sym}^{2}(\mathbb{C}^{2})\right)$ .

For the third example, find bases for the irreducible subrepresentations.

Problem 34. Let $(\pi,V)$ be a finite-dimensional representation of $\mathfrak{sl}_{2,\mathbb{C}}$ . Consider the Casimir element¹¹ 1 Conventions differ; it might be more usual to call $1+2\mathcal{C}$ the Casimir.

\mathcal{C}=\pi(X)\pi(Y)+\pi(Y)\pi(X)+\frac{1}{2}\pi(H)^{2}.

(a)

Show $\mathcal{C}$ commutes with the action of $\mathfrak{sl}_{2,\mathbb{C}}$ . Conclude that if $V$ is irreducible then $\mathcal{C}$ acts as a scalar.
(b)

What is the scalar for $V=\operatorname{Sym}^{n}(\mathbb{C}^{2})$ , the irreducible representation of highest weight $n$ ?
(c)

Compute the action of $\mathcal{C}$ on the space $V$ of polynomial functions $\phi$ on $\mathbb{C}^{2}$ , with action the derivative of $(g\phi)({\bf v})=\phi(g^{-1}v)$ (see problem 2.11.2).

3.13. Lecture 13

3.13.1. Highest weights

Lemma 3.13.1.

Proof.

Definition 3.13.2.

Lemma 3.13.3.

Proof.

Example 3.13.4.

3.13.2. Classification of irreducible representations of 𝔰⁢𝔩2,ℂ\mathfrak{sl}_{2,\mathbb{C}}.

Lemma 3.13.5.

Proof.

Remark 3.13.7.

Corollary 3.13.8.

Proof.

Theorem 3.13.9.

Proof.

Theorem 3.13.10.

Proof.

3.13.3. Exercises

3.13.2. Classification of irreducible representations of $\mathfrak{sl}_{2,\mathbb{C}}$ .