9 Problems for Epiphany 9.2 Problems for section 5 9.4 Problems for section 7

9.3 Problems for section 6

Problem 72.

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

v\otimes w\in V\otimes W

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

Solution

Problem 73.

Let $V$ be a finite-dimensional representation of $\mathfrak{sl}_{2,\mathbb{C}}$ .

1.

What are the weights of the dual representation $V^{*}$ ?
2.

Deduce that $V\cong V^{*}$ .

Solution

Problem 74.

Let $(\pi,V)$ be a finite-dimensional representation of $\mathfrak{sl}_{2,\mathbb{C}}$ . Consider the Casimir element⁸⁸ 8 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}.

1.

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.
2.

What is the scalar for $V=\operatorname{Sym}^{n}(\mathbb{C}^{2})$ , the irreducible representation of highest weight $n$ ?
3.

Compute the action of $\mathcal{C}$ on the space $V$ of polynomial functions $\phi$ on $\mathbb{C}^{2}$ , with action the derivative of $(g\phi)(v)=\phi(g^{-1}v)$ (see problem 70).

Solution

Problem 75.

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

Solution

Problem 76.

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:

1.

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

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

$\operatorname{Sym}^{3}(\mathbb{C}^{2})\otimes\operatorname{Sym}^{2}(\mathbb{C}% ^{2})$ ;
4.

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

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

Solution

Problem 77.

1.

For $a\geq b$ integers, decompose the representation $\operatorname{Sym}^{a}\mathbb{C}^{2}\otimes\operatorname{Sym}^{b}\mathbb{C}^{2}$ of $\mathfrak{sl}_{2,\mathbb{C}}$ into irreducibles. (This is known as the Clebsch–Gordan formula).
2.

(+) Can you find a general expression for the highest weight vectors for the irreducible subrepresentations? What about for the weight bases?

Solution

Problem 78.

Show that the real Lie algebras $\mathfrak{sl}_{2,\mathbb{R}}$ and $\mathfrak{su}_{2}$ are not isomorphic. Hint: consider the adjoint action of an arbitrary element of $\mathfrak{su}_{2}$ .

Solution

Problem 79.

We have that $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ is the irreducible representation of $\mathrm{SU}(2)$ of dimension $n+1$ . Let $\chi_{n}$ be its character. Every conjugacy class of $\mathrm{SU}(2)$ contains an element of the form

\exp(itH)=\begin{pmatrix}e^{it}&0\\ 0&e^{-it}\end{pmatrix}.

Show that

\chi_{n}(\exp(itH))=\frac{\sin((n+1)t)}{\sin(t)}.

Solution

Problem 80.

Let $(\rho,V)$ be an irreducible representation of $\mathfrak{gl}_{2,\mathbb{C}}$ , and let $Z=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ .

1.

Show that $\rho(Z)$ is as a scalar.
2.

Show that the restriction of $V$ to $\mathfrak{sl}_{2,\mathbb{C}}$ is irreducible.
3.

Show that for every $\lambda\in\mathbb{C}$ and integer $n\geq 0$ , there is a unique irreducible representation of $\mathfrak{gl}_{2,\mathbb{C}}$ of dimension $n+1$ with $\rho(Z)=\lambda I$ .
4.

Which of these are derivatives of representations of $\operatorname{GL}_{2}(\mathbb{C})$ ? Hence classify the finite dimensional holomorphic irreducible representations of $\operatorname{GL}_{2}(\mathbb{C})$ .

Solution

The remaining problems in this section concern material that was not covered due to the strike, and are included for interest only.

Problem 81.

1.

Verify the formula

$r^{2}\Delta=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}+\ell^{2}+\ell$

as operators on $\mathcal{P}^{\ell}$ .
2.

Find the image of the Casimir element from problem 74 under our isomorphism $\mathfrak{sl}_{2,\mathbb{C}}\to\mathfrak{so}_{3,\mathbb{C}}$ , and compare to part 1.

Solution

Problem 82.

Let $\ell\geq 1$ .

1.

Verify that $(x-iy)^{\ell}$ is a highest weight vector in $\mathcal{H}^{\ell}$ .
2.

By applying the lowering operator, find weight vectors of weights $i(\ell-1)$ and $i(\ell-2)$ .
3.

Find a basis of weight vectors in $\mathcal{H}^{\ell}$ when $\ell=1$ and $\ell=2$ (see example LABEL:eg-so3-weights).

Solution

Problem 83.

1.

Prove that, for $f\in\mathcal{P}^{\ell}$ ,

$\Delta(r^{2}f)=r^{2}\Delta(f)+2(2\ell+3)f.$
2.

Find a similar formula for

$\Delta(r^{2k}f)-r^{2k}\Delta(f).$
3.

Use this to give another proof that

$\mathcal{H}^{\ell}\cap r^{2}\mathcal{P}^{\ell-2}=\{0\}.$

(Hint: if $f$ is in the intersection, let $f=r^{2k}g$ , $g$ not divisible by $r^{2}$ ).

Solution

Problem 84.

1.

Let $V$ be the standard — three-dimensional — representation of $\mathfrak{so}_{3}$ . Find a basis of weight vectors for $\operatorname{Sym}^{2}(V)$ , and decompose it into irreducible subrepresentations.
2.

Let $\mathcal{H}^{2}$ be the five-dimensional representation of $\mathrm{SO}(3)$ . Decompose $\mathcal{H}^{2}\otimes\mathcal{H}^{2}$ into irreducible representations.

Solution