9 Problems for Epiphany 9.3 Problems for section 6 References

9.4 Problems for section 7

Problem 85.

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}.

Solution

Problem 86.

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 6? Solution

Problem 87.

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 $-L_{3}$ .

Solution

Problem 88.

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? Solution

Problem 89.

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.

Solution

Problem 90.

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

Problem 91.

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.

Solution

Problem 92.

(non-examinable) Let $(\rho,V)$ is 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 7.25. Solution

Problem 93.

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 $aL_{1}-bL_{3}$ .

Solution

Problem 94.

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.

Solution

Problem 95.

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 $V^{(2,1)}$ .

Solution

Problem 96.

(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 94.

Solution

Problem 97.

(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_{a}))=\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.