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.

  1. Sym 3 of Sym 3 C 2

  2. Sym 3 of Sym 2 C 2

What do you notice? Make a conjecture about the relationship between Sym i of Sym j C 2 and Sym j of Sym i C 2 more generally.

2. More decomposition

(Depends on the video about decomposing SU(2) representations. The notation \Lambda^n is the nth exterior power; this question therefore depends on some of the questions about exterior powers from Question Sheet 3)

Decompose one of:

  1. Lambda 2 of Sym 3 of C 2

  2. Sym 2 of Lambda 2 of Sym 3 C 2

  3. Lambda 3 of Sym 4 C 2

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

(Depends on the video about classification of SU(3) representations).

Recall that Gamma_{m, n} denotes the irreducible SU(3) representation with highest weight m L_1 minus n L_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 Sym 2 of Sym 4 C 2 and Sym 3 of Sym 4 C 2 into irreducible SU(2) representations.

Consider the space of quartic polynomials A x to the 4 + B x cubed y + C x squared y squared + D x y cubed + E y to the 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:

  • Sym 3 C 3 tensor Gamma_{0, 1}.

  • C 3 tensor Gamma_{2, 1}.

  • Gamma_{1, 1} tensor 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 Sym n C 3 is isomorphic to Gamma_{n, 0} and that Sym n of C 3 dual is isomorphic to Gamma_{0, n}.

  2. Deduce that Sym m C 3 tensor Sym n 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_{i j}) 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 inside 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 R 3. By mimicking the kinds of argument we've used for SU(3) representations, can you decompose V tensor V into irreducible summands? What about V tensor V dual?