Course 3413 — Group Representations Sample Paper III

Attempt 3 questions. (If you attempt more, only the best 3 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 a representation α is simple? Determine all simple representations of S 3. from first principles. Determine the characters of S 4 induced by the simple characters of S 3. Hence or otherwise draw up the character table for S 4 2. Show that the number of simple representations of a finite group G is equal to the number s of conjugacy classes in G. Determine the conjugacy classes in A 4 (formed by the even permutations in S 4), and draw up its character table. 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.