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MA3413: Group Representations I Tutorial questions, February 11, 2015 The main goal of this tutorial is to describe characters of all complex irreducible representations of S

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MA3413: Group Representations I Tutorial questions, February 11, 2015

The main goal of this tutorial is to describe characters of all complex irreducible representations of S4. We already know that there are two one- dimensional representations, the trivial representation and the sign represen- tation.

1.Describe conjugacy classes of S4.

2. Recall that the representation of S4 in C4 by permutations of basis vectors is isomorphic to a direct sum of the trivial representation and a three-dimensional irreducible representation, that we shall denote by (V, ρ).

Compute the character of V.

3. Show that the representation(V, ρ⊗sign) where each element g acts by sign(g)ρ(g) is irreducible and not isomorphic to (V, ρ).

4.Find a surjective homomorphism from S4 toS3. Explain how to use it to construct a two-dimensional irreducible representation U.

5.Write down the character table for S4.

6. (If there is time left) Show that V is an irreducible representation of the group A4 of even permutations of four elements, and describe all other complex irreducible representations of A4 and their characters.

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