University of Melbourne Mathematics Department

620-619 Representation Theory Lecturer: Arun Ram

2010 Semester I

Homework Due 19 April 2010

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. Classify and construct the finite dimensional simple modules for $U_q𝔰{𝔩}_2$, where $U_q𝔰{𝔩}_2$ is the algebra generated by $E$, $F$, $K^{\pm 1}$, with relations

   ${\displaystyle KE{K}^{-1}=q^{2}E,KF{K}^{-1}=q^{-2}F,\text{and}EF-FE=\frac{K-K^{-1}}{q-q^{-1}}.}$

4. Define the symmetric group (via permutations).

5. Show that $S_k$ is generated by $s_1,\dots ,s_{k-1}$ with relations

   $s_i^2=1,s_i{s}_{i+1}{s}_i=s_{i+1}{s}_i{s}_{i+1},\text{and}{s}_i{s}_j=s_j{s}_i\text{for}j\ne i,i\pm 1.$

6. In the group algebra of the symmetric group $ℂS_k$ define

   $m_j=s_{1j}+s_{2j}+\cdots +s_{j-1,j}\hspace{0.17em},$

   where $s_{ij}$ is the transposition that switches $i$ and $j$. Let $m_1=0$.
   
   
   1. Show that $m_1+\cdots +m_k$ is an element of the center of $ℂS_k$.
      
      
   2. Show that $m_i{m}_j=m_j{m}_i$ for all $1\le i,j\le k$.