Lie groups and Lie algebras: Questions 4

1. Decomposing SU(2) representations

(Depends on the video about decomposing $SU(2)$ representations).

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

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

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

What do you notice? Make a conjecture about the relationship between ${\mathrm{Sym}}^i{\mathrm{Sym}}^j({𝐂}^2)$ and ${\mathrm{Sym}}^j{\mathrm{Sym}}^i({𝐂}^2)$ 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:

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

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

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

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:

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

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

3. ${\mathrm{\Gamma }}_{3,1}$

Note: It is sufficient to draw the pictures with multiplicities labelled.

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 ${\mathrm{Sym}}^2{\mathrm{Sym}}^4({𝐂}^2)$ and ${\mathrm{Sym}}^3{\mathrm{Sym}}^4({𝐂}^2)$ into irreducible $SU(2)$ representations.

Consider the space of quartic polynomials $A{x}^4+B{x}^{3}y+C{x}^2{y}^2+Dx{y}^3+E{y}^4$ .

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

(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:

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

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

• ${\mathrm{\Gamma }}_{1,1}\otimes {\mathrm{\Gamma }}_{1,2}$ .

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.

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

2. Deduce that ${\mathrm{Sym}}^m({𝐂}^3)\otimes {\mathrm{Sym}}^n({𝐂}^3)$ 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 $U^{⟂}$ be its orthogonal complement with respect to an invariant Hermitian inner product.

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

2. Deduce that either $U$ or $U^{⟂}$ 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 ${𝐑}^3$ . By mimicking the kinds of argument we've used for $SU(3)$ representations, can you decompose $V\otimes V$ into irreducible summands? What about $V\otimes V^{*}$ ?