Math 620 - Representation Theory of Finite Groups: Syllabus and Homework

Textbook for assignments: Fulton and Harris, unless otherwise indicated.

All homework problems are worth 10 points unless otherwise specified.

Due on 2/14: page 11/11, 12a -- without the use of characters! Hint: Use the explicit realization of V in lectures. For 12a, first find a copy of the trivial and the sign representations (denoting the elements of S_3 by 1, sigma, sigma', sigma", tau, tau^2, find the relevant linear combinations). Then find elements v,v' of the form 1+a tau+b tau^2, which have eigenvalues omega and omega^2 upon the action of tau. Use them to find two copies of V inside the regular representation.
Due on 2/21: 2.29 Hint:. Use the regular representation.
Due on 3/2: 3.23 i. Hint: Use Frobenius reciprocity, by proceeding in two ways. Method 1: Find the restrictions of the irreducibles of S_4 to the given subgroup. Determine the eigenvalues of (1234) in the given irreducibles, using the character table. Method 2: compute the inner products (with respect to the given subgroup) of the character of U_1 with the characters of the restrictions of the irreducibles.
Due on 3/13: 2.27, 3.3 (20 points).