Lie groups and Lie algebras: Questions 4
Grading
Remember, you'll need to do at least 4, 4, or 5 questions to get a C, B, or A respectively. For a B or an A, this will need to include 2 (respectively 3) $\alpha $ questions.
β questions
Answer as many as you want.
1. Decomposing SU(2) representations
(Depends on the video about decomposing $SU(2)$ representations).
Decompose the following $SU(2)$ representations into irreducibles.

${\mathrm{Sym}}^{2}{\mathrm{Sym}}^{3}({\mathbf{C}}^{2})$

${\mathrm{Sym}}^{3}{\mathrm{Sym}}^{2}({\mathbf{C}}^{2})$
What do you notice? Make a conjecture about the relationship between ${\mathrm{Sym}}^{i}{\mathrm{Sym}}^{j}({\mathbf{C}}^{2})$ and ${\mathrm{Sym}}^{j}{\mathrm{Sym}}^{i}({\mathbf{C}}^{2})$ more generally.
2. More decomposition
(Depends on the video about decomposing $SU(2)$ representations. The notation ${\mathrm{\Lambda}}^{n}$ is the $n$ th exterior power; this question therefore depends on some of the questions about exterior powers from Question Sheet 3)
Decompose one of:

${\mathrm{\Lambda}}^{2}{\mathrm{Sym}}^{3}({\mathbf{C}}^{2})$

${\mathrm{Sym}}^{2}{\mathrm{\Lambda}}^{2}{\mathrm{Sym}}^{3}({\mathbf{C}}^{2})$

${\mathrm{\Lambda}}^{3}{\mathrm{Sym}}^{4}({\mathbf{C}}^{2})$
3. SU(3) weight diagrams 1: Drawing diagrams
(Depends on the video about classification of $SU(3)$ representations).
Recall that ${\mathrm{\Gamma}}_{m,n}$ denotes the irreducible $SU(3)$ representation with highest weight $m{L}_{1}n{L}_{3}$ . Draw the weight diagrams of the following representations:

${\mathrm{\Gamma}}_{1,2}$

${\mathrm{\Gamma}}_{2,3}$

${\mathrm{\Gamma}}_{3,1}$
Note: It is sufficient to draw the pictures with multiplicities labelled.
α questions
Answer as many as you want. You will need to do well on at least 2 to get a B^{} and at least 3 to get an A^{}.
4. Invariants
(Depends on the video about decomposing $SU(2)$ representations. For the final part, the extra video on invariant theory will be useful.)
Decompose ${\mathrm{Sym}}^{2}{\mathrm{Sym}}^{4}({\mathbf{C}}^{2})$ and ${\mathrm{Sym}}^{3}{\mathrm{Sym}}^{4}({\mathbf{C}}^{2})$ into irreducible $SU(2)$ representations.
Consider the space of quartic polynomials $A{x}^{4}+B{x}^{3}y+C{x}^{2}{y}^{2}+Dx{y}^{3}+E{y}^{4}$ .

Does there exist a quadratic polynomial in $A,B,C,D,E$ which is invariant under the action of $SU(2)$ on $x$ and $y$ ?

Does there exist a cubic polynomial in $A,B,C,D,E$ which is invariant under the action of $SU(2)$ on $x$ and $y$ ?
NOTE: You do not need to find these invariants explicitly in terms of $A,B,C,D,E$ (if they exist). If you want to do this, copious praise will be heaped upon you, but it's not necessary.
5. Decomposing SU(3) representations
(Depends on the video about classification of $SU(3)$ representations and the video about decomposing $SU(3)$ representations).
Decompose two of the following $SU(3)$ representations into irreducibles:

${\mathrm{Sym}}^{2}({\mathbf{C}}^{3})\otimes {\mathrm{\Gamma}}_{0,1}$ .

${\mathbf{C}}^{3}\otimes {\mathrm{\Gamma}}_{2,1}$ .

${\mathrm{\Gamma}}_{1,1}\otimes {\mathrm{\Gamma}}_{1,2}$ .
6. SU(3) weight diagrams 2: Γ_{m,n} exists
(Depends on the video about classification of $SU(3)$ representations, the video about decomposing $SU(3)$ representations, and the video about duals: from this last video, it only uses the fact that the weights of the dual representation ${V}^{*}$ are minus the weights of $V$ ).
Technically, we haven't actually showed this representation exists yet. We will do so in this question.

Show that ${\mathrm{Sym}}^{n}({\mathbf{C}}^{3})\cong {\mathrm{\Gamma}}_{n,0}$ and that ${\mathrm{Sym}}^{n}{({\mathbf{C}}^{3})}^{*}\cong {\mathrm{\Gamma}}_{0,n}$ .

Deduce that ${\mathrm{Sym}}^{m}({\mathbf{C}}^{3})\otimes {\mathrm{Sym}}^{n}({\mathbf{C}}^{3})$ contains an irreducible subrepresentation isomorphic to ${\mathrm{\Gamma}}_{m,n}$ . [Hint: What is the highest weight of this representation?]
7. Classification of SU(3) representations
The proof given in the optional video about classifying $SU(3)$ representations is a bit sketchy and leaves some details to the imagination. Write out a full proof to your satisfaction, following the lines suggested in that video.
8. Irreducibility of highest weight representations
One of the gaps in the optional video on classifying $SU(3)$ representations was that I never proved that the highest weight subrepresentation (generated by $v$ and its images under the ${E}_{ij}$ ) is irreducible. Let's see if you can prove it.
Let's call the highest weight subrepresentation $V$ and write $\lambda $ for the weight of the highest weight vector $v$ . Suppose $V$ is not irreducible, and let $U\subset V$ be a subrepresentation. Let ${U}^{\u27c2}$ be its orthogonal complement with respect to an invariant Hermitian inner product.

Show that $v$ is contained in either $U$ or ${U}^{\u27c2}$ .

Deduce that either $U$ or ${U}^{\u27c2}$ contains $V$ .

Why does this mean that $V$ is irreducible?
[Hint: You may use the fact that the $\lambda $ weight space of $V$ is spanned by $v$ .]
9. SU(4)
(Depends on the video about classification of $SU(3)$ representations, the video about decomposing $SU(3)$ representations, and the video about duals: from this last video, it only uses the fact that the weights of the dual representation ${V}^{*}$ are minus the weights of $V$ ).
Let $V$ be the standard 4dimensional complex representation of $SU(4)$ , whose weight diagram is a regular tetrahedron in ${\mathbf{R}}^{3}$ . By mimicking the kinds of argument we've used for $SU(3)$ representations, can you decompose $V\otimes V$ into irreducible summands? What about $V\otimes {V}^{*}$ ?