# Math 6320, Spring 2017

## Prof. Allen Knutson

Book: Dummit and Foote

Notes for character theory (also treated in Dummit and Foote)

Final exam, due in my mailbox / email by noon Monday May 22. It's about as long as a long homework. Once you start, you may not discuss this class material with anyone but me and Pak-Hin until you turn in the final. Readingwise, you may use class notes, Dummit and Foote, and the links above. Topics: character theory of finite groups, derived functors (in particular of Hom), finite fields, Galois theory.

OH: directly after class on Tuesday, until 3:30.
You can also try the free-for-all on Mondays @12.
These are in my office, 515 Malott.

Pak Hin's are 4-6pm on Mondays in Rm 114.

HW#1 due Tuesday 2/20:

• Find a pair of groups G > H and a G-equivariant map G/H -> G/H that is not invertible.
• Do the exercise after thm 4.4 of the notes above.
• Find the character table of A5.
• Given one entry in a character table (and the first column), how can you guarantee that that conjugacy class is not in the center?
• Given an entire column of a character table (and the first column), how can you guarantee that that conjugacy class is in the center?
• Let v in V and f in V^* be elements of an irrep and its dual; the resulting function g |-> f(gv) on G is a matrix coefficient. Show how to use these to get a basis for Fun(G).
• Succinctly describe the character table of G x H.
• HW#2 due Thursday 3/9:
• 1. Let vi = xiyn-i be the ith basis vector of SymnC2. Recall we proved that there is a unique subrepresentation Vn strongly dominated by the same highest weight. Compute $\lim_{t\to 0} \frac{ \begin{bmatrix} 1&0 \\ t&1 \end{bmatrix} \vec v_i -\vec v_i}{t}$ and show that if vi lies in V, so does this limiting vector. Conclude that SymnC2 is irreducible.
• 2. Use the weight multiplicity diagrams to compute the decomposition of Vn tensor Vm as SL(2) representations.