University of Melbourne Mathematics Department

620-619 Representation Theory Lecturer: Arun Ram

2009 Semester I

Homework Due 12 May 2009

1. Classify and construct the finite dimensional simple modules for cyclic groups.

2. Classify and construct the finite dimensional simple modules for dihedral groups.

3. Explicitly verify the Weyl character formula for the $𝔰{𝔩}_3$-crystal $B\left(\rho \right)$.

4. Explicitly decompose the $𝔰{𝔩}_3$-crystal $B\left(\rho \right)\otimes B\left(\rho \right)$.

5. Decompose the adjont representation of $S{O}_5$ as an $S{U}_3\times S{U}_2\times U_1$-module.