# Lie groups and Lie algebras: Questions 4

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

Decompose the following $SU(2)$ representations into irreducibles.

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

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 $\Lambda^{n}$ 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:

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

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

3. $\Lambda^{3}\mathrm{Sym}^{4}(\mathbf{C}^{2})$

### 3. SU(3) weight diagrams 1: Drawing diagrams

Recall that $\Gamma_{m,n}$ denotes the irreducible $SU(3)$ representation with highest weight $mL_{1}-nL_{3}$ . Draw the weight diagrams of the following representations:

1. $\Gamma_{1,2}$

2. $\Gamma_{2,3}$

3. $\Gamma_{3,1}$

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 $Ax^{4}+Bx^{3}y+Cx^{2}y^{2}+Dxy^{3}+Ey^{4}$ .

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

2. 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

Decompose two of the following $SU(3)$ representations into irreducibles:

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

• $\mathbf{C}^{3}\otimes\Gamma_{2,1}$ .

• $\Gamma_{1,1}\otimes\Gamma_{1,2}$ .

### 6. SU(3) weight diagrams 2: Γm,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.

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

2. Deduce that $\mathrm{Sym}^{m}(\mathbf{C}^{3})\otimes\mathrm{Sym}^{n}(\mathbf{C}^{3})$ contains an irreducible subrepresentation isomorphic to $\Gamma_{m,n}$ . [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 $U^{\perp}$ be its orthogonal complement with respect to an invariant Hermitian inner product.

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

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

3. 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 4-dimensional 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 $V\otimes V^{*}$ ?