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

Syllabus

Textbook for assignments: Fulton and Harris, unless otherwise indicated

All homework problems are worth 10 points unless otherwise specified.

• Due on 9/14: page 11/11, 12a.
• Due on 9/26: 2.29.
• Due on 9/28: injectivity of the map from the group algebra to the direct sum of End(V_i); 2.33 a,b; 3.1 (hint: like for S_3, the representation V is the complement of the line <(1,1,1,1,1)> in the permutation representation of S_5 on C^5; show, with detailed calculations, that V and V' are irreducible and find their characters).
• Due on 10/5: 3.23 i.
• Due on 10/12: 2.27, 3.3 (20 points).
• Due on 10/24: Identify the real quaternions of norm 1 with SU(2) and the sphere S^3 (using the description of SU(2) as those complex matrices with bar{A^T}=A^{-1}). Derive the Pauli matrices and the 2-dimensional irreducible representation of Q_8 (i-->(0 -i/-i 0), j-->(0 -1/1 0), k-->(-i 0/0 i), by rows). Then show that this representation is of quaternionic type, as follows: (a) find the Hermitian and skew-symmetric Q_8-invariant forms (check these facts); (2) find the map phi (with phi^2=-Id), and justify why it can be interpreted as "multiplication by j". Total: 20 points.
• Due on 11/9: Fulton (Young tableaux): page 86/ex. 1,3. Also, show that the Specht module S^{2,1} is isomorphic to the standard representation of S_3 (continue the calculations in class).
• Due on 11/16: Fulton (Young tableaux): page 87/ex. 4.
• Due on 11/28: Fulton (Young tableaux): page 94/ex. 9, 11(a).
• Due on 11/30: Fulton (Young tableaux): page 94/ex. 11(b). Hint: use induction on n, 11(a), and the branching formula (restriction from S_n to S_{n-1}, see page 93).
• Due on 12/7: Derive the hooklength formula from the formula in terms of l_i. Fulton (Young tableaux): page 94/8. Hint: show first that A a_T b_T and A b_T a_T are isomorphic as left S_n-modules (use the fact that a_T b_T is a projection); given this and a previous homework, it suffices to construct an isomorphism between "A a_T b_T tensor U' " and "A b_T' a_T' ", where T' is the transpose of T (why?).