Representation Theory IV 4 SL₃ 6 Solutions

5 Problems for Epiphany

5.1 Problems for section 1

Problem 1.

1.

Compute $\exp(X)$ for $X$ equal to $\begin{pmatrix}t&0\\ 0&s\end{pmatrix}$ , $\begin{pmatrix}0&t\\ -t&0\end{pmatrix}$ , and $\begin{pmatrix}0&t\\ t&0\end{pmatrix}$ (where $s,t\in\mathbb{R}$ ).
2.

Let $E_{a,b}$ be the elementary $n\times n$ matrix with $1$ in the $(a,b)$ -entry and $0$ elsewhere. Compute $\exp(tE_{a,b})$ for $a\neq b$ and $a=b$ .

Problem 2.

Show that

\exp(tX)\exp(tY)=\exp\left(t(X+Y)+\frac{t^{2}}{2}[X,Y]+O(t^{3})\right)

as $t\to 0$ , where

[X,Y]=XY-YX.

Solution

Problem 3.

Let $\mathfrak{n}$ be the $\mathbb{C}$ -vector space of strictly upper triangular matrices ( $0$ ’s on the diagonal) and let $N=\{g\in\operatorname{GL}_{n}(\mathbb{C})\,:\,g=I+X,X\in\mathfrak{n}\}$ .

In this problem we will see that the restriction of the exponential to $\mathfrak{n}$ is a homeomorphism onto $N$ (i.e. a continuous bijection with continuous inverse).

1.

Let $X\in\mathfrak{n}$ . Show that $X^{n}=0$ .
2.

Show that $\exp(X)\in N$ for $X\in\mathfrak{n}$ .
3.

Show that, for $g\in N$ , the logarithm $\log(g)=\sum_{k=1}^{\infty}(-1)^{k+1}\tfrac{(g-I)^{k}}{k}$ is in fact a finite sum (and hence converges).
4.

Show that $\exp|_{\mathfrak{n}}$ and $\log|_{N}$ are inverses of each other. Hint: this boils down to an identity of formal power series, which you can actually deduce from the corresponding fact over $\mathbb{R}$ .

Problem 4.

1.

Using the previous question, fill in the gaps of the proof from the notes that

$\exp:\mathfrak{gl}_{n,\mathbb{C}}\rightarrow\operatorname{GL}_{n}(\mathbb{C})$

is surjective.
2.

(+) Is the exponential map $\exp:\mathfrak{sl}_{2,\mathbb{C}}\rightarrow\operatorname{SL}_{2}(\mathbb{C})$ surjective? What about $\exp:\mathfrak{gl}_{2,\mathbb{R}}\rightarrow\operatorname{GL}^{+}_{2}(\mathbb{% R})$ ?

Problem 5.

Let $v\in\mathbb{R}^{3}$ be a unit vector and let $f:\mathbb{R}\to SO(3)$ be the map with $f(\theta)$ being rotation by $\theta$ about the axis $v$ (the angle is measured anticlockwise as you look along the vector from the origin).

Show that $f$ is a one-parameter subgroup and find its infinitesimal generator in terms of $v$ .

Problem 6.

Prove that the Lie algebra of $\mathrm{U}(n)$ is

\mathfrak{u}_{n}=\{X\in\mathfrak{gl}_{n,\mathbb{C}}:X+X^{\dagger}=0\}

and find its (real) dimension. Is it a complex vector space?

Solution

Problem 7.

Let $I_{p,q}=\begin{pmatrix}I_{p}&\\ &-I_{q}\end{pmatrix}$ , where $I_{k}$ denotes the identity matrix of size $k$ . Let $n=p+q$ . Let

\operatorname{O}(p,q)=\{g\in\operatorname{GL}_{n}(\mathbb{R})\,:\,gI_{p,q}g^{T% }=I_{p,q}\}

be the orthogonal group of signature $(p,q)$ . Let $\mathrm{SO}(p,q)=\operatorname{O}(p,q)\cap\operatorname{SL}_{n}(\mathbb{R})$ . We let $\mathfrak{o}_{p,q}$ and $\mathfrak{so}_{p,q}$ be their Lie algebras.

Show that the Lie algebra $\mathfrak{o}_{p,q}$ is given by

\mathfrak{o}(p,q)=\{X\in M_{n}(\mathbb{R})\,:\,XI_{p,q}+I_{p,q}X^{T}=0\}

and that $\mathfrak{so}_{p,q}=\mathfrak{o}_{p,q}$ .

Solution

Problem 8.

1.

Show that the Lie algebras $\mathfrak{so}_{3}$ and $\mathfrak{su}_{2}$ are isomorphic. (Later on, we will see a conceptual reason for this).

Hint: it is enough to find a basis for $\mathfrak{so}_{3}$ and a basis for $\mathfrak{su}_{2}$ which satisfy the ‘same’ Lie bracket relations. Try using the basis of $\mathfrak{so}_{3}$ consisting of infinitesimal generators for rotations around the axes, and a basis for $\mathfrak{su}_{2}$ related to the quaternions.
2.

Show that the Lie algebras $\mathfrak{so}_{2,1}$ and $\mathfrak{sl}_{2,\mathbb{R}}$ are isomorphic. See Problem 7 for the definition of $\mathfrak{so}_{2,1}$ .
3.

(+) Show that the Lie algebras $\mathfrak{so}_{3,1}$ and $\mathfrak{sl}_{2,\mathbb{C}}$ are isomorphic (as real Lie algebras). See Problem 7 for the definition of $\mathfrak{so}_{3,1}$ .

Solution

Problem 9.

Show that:

1.

If $X\in\mathfrak{sp}_{2n}$ , then $\operatorname{tr}(X)=0$ .
2.

(+) Show that, if $g\in\operatorname{Sp}(2n)$ , then $\det(g)=1$ .

Problem 10.

Prove that if $G$ is a Lie group and $G^{0}$ is the connected component of the identity, then the subgroup $G^{0}$ is normal.

Problem 11.

1.

Give a direct proof that $\mathrm{SO}(3)$ is connected, by constructing a path from an arbitrary element of $\mathrm{SO}(3)$ to the identity. Hint: every element of $\mathrm{SO}(3)$ is rotation by some angle about some axis.
2.

Prove by induction on $n$ that $SO(n)$ is connected for all $n\geq 1$ .

Problem 12.

Show that a general element of $\mathrm{SU}(2)$ may be written

\begin{pmatrix}a&-\overline{b}\\ b&\overline{a}\end{pmatrix}

for $a,b\in\mathbb{C}$ with $|a|^{2}+|b|^{2}=1$ .

Deduce that $\mathrm{SU}(2)$ is homeomorphic to the three-sphere $S^{3}=\{v\in\mathbb{R}^{4}:|v|=1\}$ .

In other words, write down a bijection $\mathrm{SU}(2)\rightarrow S^{3}$ with continuous inverse. Don’t worry about checking that the maps are continuous, just write them down. The result of this problem implies that $\mathrm{SU}(2)$ is simply-connected, because $S^{3}$ is.

Solution

Problem 13.

Show that, if $G$ is a connected (linear) Lie group with Lie algebra $\mathfrak{g}$ , then $G$ is abelian if and only if $\mathfrak{g}$ is (see Definition 1.30). Hint: for the converse, consider the adjoint map $G\to GL(\mathfrak{g})$ .

What goes wrong if $G$ is not connected?

Solution: see Proposition 2.14.

Problem 14.

If $\mathfrak{g}$ is a Lie algebra, let $\mathfrak{z}$ be its centre:

\mathfrak{z}=\{X\in\mathfrak{g}:[X,Y]=0\,:\,\text{for all $Y\in\mathfrak{g}$}.\}.

Suppose that $G$ is a connected Lie group with centre $Z$ and Lie algebra $\mathfrak{g}$ with centre $\mathfrak{z}$ .

Prove that $\mathfrak{z}$ is the Lie algebra of $Z$ .

You will always have $\operatorname{Lie}(Z)\subset\mathfrak{z}$ but the reverse inclusion requires that $G$ is connected.

Solution

Problem 15.

Solve the exercises in section 1.8

5.2 Problems for section 2

Problem 16.

1.

If $(\rho,V)$ is an irreducible finite-dimensional complex representation of $\mathfrak{g}$ and $\mathfrak{z}$ is the centre of $\mathfrak{g}$ (see problem 14), show that there is a linear map $\alpha:\mathfrak{z}\rightarrow\mathbb{C}$ such that $\rho(Z)v=\alpha(Z)v$ for all $Z\in\mathfrak{z}$ .
2.

For $\mathfrak{g}=\operatorname{GL}_{n,\mathbb{C}}$ , find $\mathfrak{z}$ . Find $\alpha$ when $V=\Lambda^{k}\mathbb{C}^{n}$ , where $\mathbb{C}^{n}$ is the standard representation and $1\leq k\leq n$ . These representations are in fact irreducible, though we haven’t proved that yet; you can just directly show that $\alpha$ exists.

Problem 17.

Prove that, for $X,Y\in\mathfrak{gl}_{n}$ ,

(\operatorname{ad}_{X})^{m}(Y)=[X,[X,\ldots,[X,Y]\ldots]]=\sum_{k=0}^{m}\binom% {m}{k}X^{k}Y(-X)^{m-k}.

Hence give a direct proof that $\exp(\operatorname{ad}_{X})=\operatorname{Ad}_{\exp(X)}$ .

Problem 18.

Consider $G=\mathrm{U}(1)$ .

1.

For $\varphi$ a continuous function on $G$ , we define its integral

$\int_{G}\varphi(g)dg=\tfrac{1}{2\pi}\int_{0}^{2\pi}\varphi(e^{it})dt.$

Note that $\int_{g}1dg=1$ . Show that

$\int_{G}\varphi(hg)dg=\int_{G}\varphi(gh)dg=\int_{G}\varphi(g)dg$

for any $h\in G$ .
2.

Let $(V,\rho)$ be a finite dimensional representation of $G$ and let $(\,,\,)$ be any Hermitian form on $V$ . Define a new Hermitian form by

$(v,w)_{\rho}=\int_{G}(\rho(g)v,\rho(g)w)dg.$

Show that $(\,,\,)_{\rho}$ is a $G$ -invariant Hermitian form on $V$ .
3.

Conclude that every finite-dimensional representation of $\mathrm{U}(1)$ is completely reducible. (This is analogous to Maschke’s theorem for finite groups.)

Problem 19.

Consider the orthogonal group $\operatorname{O}(2)$ .

1.

Show that $\mathrm{SO}(2)$ has index $2$ in $\operatorname{O}(2)$ . Deduce that every element in $\operatorname{O}(2)$ can be uniquely written as $r_{\theta}$ or $r_{\theta}s$ with $s=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$ and $r_{\theta}$ the matrix for rotation by $\theta$ . Show that

$sr_{\theta}=r_{-\theta}s.$
2.

Mimic the method we used for dihedral groups to classify all irreducible finite-dimensional representations of $\operatorname{O}(2)$ .

Solution

Problem 20.

Let $V$ be the space of functions on $\mathbb{C}^{2}$ that are polynomials in the coordinates $x$ and $y$ . Consider the (left) action of $\operatorname{GL}_{2}(\mathbb{C})$ on $V$ given by

(g\varphi)(v)=\varphi(g^{-1}v)

(here, think of $v=\begin{pmatrix}x\\ y\end{pmatrix}\in\mathbb{C}^{2}$ as a column vector).

Compute the derived action for the “standard” basis of $\mathfrak{sl}_{2}(\mathbb{C})$ given by $X=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$ , $Y=\begin{pmatrix}0&0\\ 1&0\end{pmatrix}$ , and $H=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$ . You should get something involving the partial derivatives $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ .

Solution

Problem 21.

Let $V=\mathbb{C}^{2}$ be the standard representation of $\operatorname{GL}_{2}(\mathbb{C})$ .

1.

Show that $\Lambda^{2}(V)\cong\det$ as Lie group representations.
2.

Show that $\Lambda^{2}(V)\cong\operatorname{tr}$ as representations of $\mathfrak{gl}_{2}(\mathbb{C})$ . (You could just ‘take the derivative’ of part (a), but please do it directly instead.)
3.

Find an explicit homomorphism $\rho:\operatorname{GL}_{2}(\mathbb{C})\rightarrow\operatorname{GL}_{3}(\mathbb% {C})$ corresponding to $\operatorname{Sym}^{2}(V)$ .

Solution

5.3 Problems for section 3

Problem 22.

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 $v\otimes w$ .

Problem 23.

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 $V\cong V^{*}$ .

Problem 24.

Let $(\pi,V)$ be a finite-dimensional representation of $\mathfrak{sl}_{2,\mathbb{C}}$ . Consider the Casimir element⁵⁵ 5 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 20).

Solution

Problem 25.

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 26.

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 27.

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 28.

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 $\mathfrak{su}_{2}$ .

Problem 29.

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)}.

Problem 30.

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 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 $\operatorname{GL}_{2}(\mathbb{C})$ .

Problem 31.

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 24 under our isomorphism $\mathfrak{sl}_{2,\mathbb{C}}\to\mathfrak{so}_{3,\mathbb{C}}$ , and compare to part 1.

Problem 32.

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 3.40).

Problem 33.

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 $r^{2}$ ).

Problem 34.

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.

5.4 Problems for section 4

Problem 35.

Verify that

\left[\begin{pmatrix}a_{1}&0&0\\ 0&a_{2}&0\\ 0&0&a_{3}\end{pmatrix},E_{ij}\right]=(a_{i}-a_{j})E_{ij}

and

[E_{12},E_{23}]=E_{13}.

Problem 36.

Consider, instead of $\mathfrak{sl}_{3,\mathbb{C}}$ , $\mathfrak{sl}_{2,\mathbb{C}}$ . What are the roots and root spaces? What is the relation between the weights (as linear functionals on $\mathfrak{h}$ ) and between the weights defined in section 3?

Problem 37.

Let $V=(\mathbb{C}^{3})^{*}$ be the dual of the standard representation, with basis $e_{1}^{*},e_{2}^{*},e_{3}^{*}$ dual to the standard basis.

1.

Show that the $e_{i}^{*}$ are weight vectors with weights $-L_{i}$ .
2.

Find the action of each $E_{ij}$ on $e_{3}^{*}$ and deduce that $e_{3}^{*}$ is a highest weight vector with weight $-L_{3}$ .

Problem 38.

Show that $a_{1}L_{1}+a_{2}L_{2}+a_{3}L_{3}\in\Lambda_{W}$ if and only if $a_{1}-a_{2},a_{2}-a_{3}\in\mathbb{Z}$ . Must the $a_{i}$ be integers?

Problem 39.

The root lattice $\Lambda_{R}\subset\Lambda_{W}$ is the subgroup of the weight lattice generated by the roots.

1.

Draw a picture showing the root lattice inside the weight lattice.
2.

Show that $\Lambda_{R}$ has index three in $\Lambda_{W}$ (i.e. the quotient $\Lambda_{W}/\Lambda_{R}$ has order three).
3.

What would the root lattice and weight lattice be for $\mathfrak{sl}_{2,\mathbb{C}}$ ? What is the index in this case?
4.

Let $V$ be a finite-dimensional irreducible representation of $\mathfrak{sl}_{3,\mathbb{C}}$ . Show that any two weights of $V$ differ by an element of the root lattice.

Problem 40.

Find the weights of $\operatorname{Sym}^{3}(\mathbb{C}^{3})$ and draw the weight diagram.

Problem 41.

Using weights, or otherwise, show that

\mathbb{C}^{3}\otimes(\mathbb{C}^{3})^{*}\cong\mathbb{C}\oplus\mathfrak{sl}_{3% ,\mathbb{C}}

where $\mathbb{C}$ is the trivial representation and $\mathfrak{sl}_{3,\mathbb{C}}$ is the adjoint representation.

Problem 42.

(non-examinable) Let $(\rho,V)$ be a representation of $\mathfrak{sl}_{3,\mathbb{C}}$ . As $\operatorname{SL}_{3}(\mathbb{C})$ is simply-connected $\rho$ exponentiates to a representation, $\tilde{\rho}$ , of $\operatorname{SL}_{3}(\mathbb{C})$ . Let

\sigma_{3}=\begin{pmatrix}0&1&0\\ -1&0&0\\ 0&0&1\end{pmatrix}\in\operatorname{SL}_{3}(\mathbb{C}).

Show that, for every weight $\alpha$ , $\tilde{\rho}(\sigma_{3})$ is an isomorphism

V_{\alpha}\rightarrow V_{s_{3}\alpha}.

Here $s_{3}(a_{1}L_{1}+a_{2}L_{2}+a_{3}L_{3})=a_{1}L_{2}+a_{2}L_{1}+a_{3}L_{3}$ .

Give another proof of Theorem 4.26.

Problem 43.

Let $a,b\geq 0$ be integers. Check that

e_{1}^{a}\otimes(e_{3}^{*})^{b}\in\operatorname{Sym}^{a}(\mathbb{C}^{3})% \otimes\operatorname{Sym}^{b}((\mathbb{C}^{3})^{*})

is a highest weight vector with weight $aL_{1}-bL_{3}$ .

Problem 44.

Show that, if $V$ is a finite-dimensional representation of $\mathfrak{sl}_{3,\mathbb{C}}$ with a unique highest weight vector (up to scalar multiplication), then $V$ is necessarily irreducible.

Deduce that the standard representation, its dual, and the adjoint representation are irreducible.

Problem 45.

1.

Find the weights of $\operatorname{Sym}^{2}(\mathbb{C}^{3})\otimes(\mathbb{C}^{3})^{*}$ and draw the weight diagram.
2.

Show that

$e_{1}^{2}\otimes e_{1}^{*}+e_{1}e_{2}\otimes e_{2}^{*}+e_{1}e_{3}\otimes e_{3}% ^{*}\in\operatorname{Sym}^{2}(\mathbb{C}^{3})\otimes(\mathbb{C}^{3})^{*}$

is a highest weight vector with weight $L_{1}$ .
3.

Let $v=e_{1}^{2}\otimes e_{3}^{*}$ . Calculate $E_{32}E_{21}v$ and $E_{21}E_{32}v$ and show that they are linearly independent.
4.

Show that

$\operatorname{Sym}^{2}(\mathbb{C}^{3})\otimes(\mathbb{C}^{3})^{*}\cong V^{(2,1% )}\oplus\mathbb{C}^{3}$

and find the weight diagram for $V^{(2,1)}$ .

Problem 46.

(harder!) The aim of this problem is to show that, for $n\geq 0$ ,

V^{(n,0)}=\operatorname{Sym}^{n}(\mathbb{C}^{3}).

It suffices to show that $\operatorname{Sym}^{n}(\mathbb{C}^{3})$ is irreducible with highest weight $nL_{1}$ .

1.

Show that $\operatorname{Sym}^{n}(\mathbb{C}^{3})$ has a basis of weight vectors

$\{e_{1}^{a}e_{2}^{b}e_{3}^{c}:a,b,c\geq 0,a+b+c=n\}$

and that these have distinct weights (so, every weight has multiplicity one).
2.

Show that $e_{1}^{n}$ is the unique highest weight vector in $\operatorname{Sym}^{n}(\mathbb{C}^{3})$ , up to scalar multiplication.
3.

Deduce that $\operatorname{Sym}^{n}(\mathbb{C}^{3})$ is an irreducible representation with highest weight $nL_{1}$ . See problem 44.

Problem 47.

(monster!) Let $V=\mathbb{C}^{3}$ , let $W=V^{*}$ , and let $a,b>0$ . For $v\in V,w\in W$ , define $(v,w)=w(v)$ .

Let

\phi:\operatorname{Sym}^{a}(V)\otimes\operatorname{Sym}^{b}(W)\to\operatorname% {Sym}^{a-1}(V)\otimes\operatorname{Sym}^{b-1}(W)

be defined by

\phi((v_{1}\ldots v_{a})\otimes(w_{1}\ldots w_{b}))=\sum_{i=1}^{a}\sum_{j=1}^{% b}(v_{i},w_{j})(v_{1}\ldots\hat{v}_{i}\ldots v_{a})\otimes w_{1}\ldots\hat{w}_% {j}\ldots w_{b}

where $\hat{v}_{i}$ means $v_{i}$ is omitted (and similarly for $\hat{w}_{j}$ ).

1.

Show that $\phi$ is an $\mathfrak{sl}_{3,\mathbb{C}}$ -homomorphism.
2.

Show that $\operatorname{Sym}^{a}(V)\otimes\operatorname{Sym}^{b}(W)$ has a unique highest weight vector of weight $(a-i)L_{1}-(b-i)L_{3}$ for each $0\leq i\leq\min(a,b)$ , and no other highest weight vectors.
3.

Show that the highest weight vector from the previous part is in $\ker(\phi)$ if and only if $i=0$ .
4.

Deduce that $\ker(\phi)\cong V^{(a,b)}$ is the irreducible representation of highest weight $aL_{1}-bL_{3}$ .
5.

Show that $\phi$ is surjective, and hence decompose $\operatorname{Sym}^{a}(V)\otimes\operatorname{Sym}^{b}(V^{*})$ into irreducibles.
6.

Find the dimension of $V^{(a,b)}$ . Find its weights.

This problem is hard! For a solution, see Fulton and Harris, section 13.2, but watch out for the unjustified ’clearly’ just before Claim 13.4.