Course 424

Group Representations

Dr Timothy Murphy

G.M.B.  Friday, 22 May 1995  14:00–16:00

Answer as many questions as you can; all carry the same number of marks. Unless otherwise stated, all representations are finite-dimensional over ℂ.

1. Define a group representation. What is meant by saying that 2 representations α, β are equivalent? What is meant by saying that the representation α is simple?

Determine all simple representations of D_{4} (the symmetry group of a square) up to equivalence, from first principles.

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

Prove that every finite-dimensional representation α of a finite group over ℂ is semisimple.

Define the character χ_{α} of a representation α.

Define the intertwining number I(α, β) of 2 representations α, β. State without proof a formula expressing I(α, β) in terms of χ_{α}, χ_{β}.

Show that the simple parts of a semisimple representation are unique up to order.

3. Draw up the character table of S_{4}.

Determine also the representation ring of S_{4}, ie express each product of simple representations of S_{4} as a sum of simple representations.

4. 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.

5. Explain the division of simple representations of a finite group G over ℂ into real, essentially complex and quaternionic. Give an example of each (justifying your answers).

Show that if α is a simple representation with character χ then the value of

∑_{g∈G} χ(g^{2})

determines which of these 3 types α falls into.

6. Define a measure on a compact space. State carefully, but without proof, Haar’s Theorem on the existence of an invariant measure on a compact group. To what extent is such a measure unique?

Prove that every representation of a compact group is semisimple.

Which of the following groups are (a) compact, (b) connected:

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

(Justify your answer in each case.)

7. Determine the conjugacy classes in SU(2).

Prove that SU(2) has just one simple representation of each dimension 1, 2, …; and determine the character of this representation.

If D(j) denotes the simple representation of SU(2) of dimension 2j + 1, for j = 0, 1/2, 1, …, express the product D(j)D(k) as a sum of D(j)’s.

8. 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}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})

Which of these 5 matrices X are themselves expressible in the form X = e^{Y}, with Y real? (Justify your answers in all cases.)

9. Define a linear group, and a Lie algebra.

Define the Lie algebra ℒ G of a linear group G, and outline the proof that it is indeed a Lie algebra.

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

10. Define a representation of a Lie algebra; and show how each representation α of a linear group G gives rise to a representation ℒ α of ℒ G.

Determine the Lie algebra of SL(2, ℝ); and show that this Lie algebra slg(2, ℝ) has just 1 simple representation of each dimension 1, 2, 3, . . ..