Course 424

Group Representations

Dr Timothy Murphy

GMB ??  Friday, ?? 2005  ??:00–??:00

Attempt 7 questions. (If you attempt more, only the best 7 will be counted.) All questions carry the same number of marks. Unless otherwise stated, all groups are compact (or finite),, and all representations are of finite degree over ℂ.

1. Define a group representation. What is meant by saying that 2 representations α, β are equivalent? Find all representations of S_{3} of degree 2 (up to equivalence).

What is meant by saying that a representation α is simple? Find all simple representations of D_{4} from first principles.

2. What is meant by saying that a representation α is semisimple?

Define a measure on a compact space. State carefully, and outline the main steps in the proof of, Haar’s Theorem on the existence of an invariant measure on a compact group.

Prove that every representation of a compact group is semisimple.

3. Define the character χ_{α} of a representation α, and show that it is a class function (ie it is constant on conjugacy classes).

Define the intertwining number I(α, β) of 2 representations α, β of a group G, and show that if G is compact then

I(α, β) = ∫_{G} \overline{χ_{α}(g)}χ_{β}(g) dg.

Prove that a representation α is simple if and only if I(α, α) = 1.

4. Draw up the character table for S_{4}.

Determine also the representation-ring for this group, ie express the product αβ of each pair of simple representation as a sum of simple representations.

5. Show that the number of simple representations of a finite group G is equal to the number s of conjugacy classes in G.

Show also that if these representations are σ_{1}, …, σ_{s} then

dim^{2} σ_{1} + ⋯ + dim^{2} σ_{s} = |G|.

Determine the dimensions of the simple representations of S_{5}, stating clearly any results you assume.

6. Determine the conjugacy classes in SU(2); and prove that this group has just one simple representation of each dimension.

Find the character of the representation D(j) of dimensions 2j + 1 (where j = 0, \frac{1}{2}, 1, \frac{3}{2}, …).

Express each product D(i)D(j) as a sum of simple representations D(k).

7. Define the exponential e^{X} of a square matrix X.

Determine e^{X} in each of the following cases:

X = ( \begin{matrix}1 & 0 \\ 0 & -1 \end{matrix} ) , X = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , X = ( \begin{matrix}0 & -1 \\ 1 & 0 \end{matrix} ) , X = ( \begin{matrix}1 & -1 \\ 1 & 1 \end{matrix} ) .

Show that if X has eigenvalues λ, μ then e^{X} has eigenvalues e^{λ}, e^{μ}.

Which of the above 5 matrices X are themselves expressible in the form X = e^{Y} for some real matrix Y? (Justify your answers in all cases.)

8. Define a linear group, and a Lie algebra; and define the Lie algebra ℒ G of a linear group G, showing that it is indeed a Lie algebra.

Define the dimension of a linear group; and determine the dimensions of each of the following groups:

O(n), SO(n), U(n), SU(n), GL(n, ℝ), SL(n, ℝ), GL(n, ℂ), SL(n, ℂ)?

9. Determine the Lie algebras of SU(2) and SO(3), and show that they are isomomorphic.

Show that the 2 groups themselves are not isomorphic.

10. Define a representation of a Lie algebra ℒ. What is meant by saying that such a representation is (a) simple, (b) semisimple?

Determine the Lie algebra of SL(2, ℝ), and find all the simple representations of this algebra.

Show that every representation of the group SL(2, ℝ) is semisimple, stating carefully but without proof any results you need.