Mathematics 618: Representation TheoryInstructor: Jonathan Smith, 496 Carver, 4-8172 (voice mail) e-mail: jdhsmithATiastateDOTedu (substitute punctuation) Office Hours: Mon. 11 - 11.50 am, 3.10 - 4 pm; Wed. 11 - 11.50 am; Fri. 11 - 11.50 am (subject to change)
Grading: based on five graded homework assignments (last due Wed., 4/27).
Textbook: J.-P. Serre, Linear Representations of Finite Groups, Springer, 1977. ISBN 978-1-4684-9460-0. The book is also available online (10MB).
Study Plan: reserve 1 - 2 hours for homework between each pair of classes. Successful performance in the class depends critically on completion of the homework assignments.
Syllabus: Fundamentals of the representation and character theory of groups, corresponding to material selected from Chapters 1 - 8 of the text. If time allows, some supplementary topics will be considered. Assignments4/27 for 4/29: Complete reading of Serre, Section 8.2.Final graded homework (due 4/27, 10 am - strict deadline!): Best three of the following four exercises from Serre: 5.4 (in Section 5.7), 6.4, 6.7, 7.2. 4/25 for 4/27: Read Serre, Section 8.2 through statement of Prop. 25. 4/22 for 4/25: Read Serre, Section 7.4. 4/20 for 4/22: Read Serre, Section 7.3. 4/18 for 4/20: Read Serre, Section 7.2. 4/15 for 4/18: Read Serre, Section 7.1. 4/13 for 4/15: Read Serre, Section 6.5 through Corollary 2. 4/11 for 4/13: Read Serre, Section 6.4. Fourth graded homework (due 4/6, 10 am - strict deadline!): Best three of the following four exercises from Serre, Chapter 6: 6.1, 6.2, 6.3(a)(b), 6.3(c)(d). 3/30 for 4/1: Read Serre, Section 6.3. 3/28 for 3/30: Complete reading of Serre, Section 6.2. 3/25 for 3/28: Read Serre, Section 6.1, and 6.2 through Proposition 10. 3/23 for 3/25: Let A be a unital, Artinian ring. Show that A is semisimple if and only if each right A-module is a direct sum of simple modules. 3/21 for 3/23: Let A be a unital ring. If M is a simple (left) A-module, show that End_{A}(M) is a skewfield. Third graded homework (due 3/9, 10 am - strict deadline!): Best three of Serre, Exercises 3.1 - 3.4. 3/7 for 3/9: Read Serre, Section 5.7. 3/4 for 3/7: Complete reading of Serre, Section 3.3. 3/2 for 3/4: Read Serre, Section 3.3 through proof of Lemma 1. 2/29 for 3/2: Read Serre, Section 3.2. 2/26 for 2/29: Read Serre, first paragraph of Section 3.3, and all of Section 3.1. 2/22 for 2/24: Complete reading of Serre, Section 2.5. Second graded homework (due 2/22): Best three of the following four exercises from Serre, Chapter 2: 2.5, 2.6(a), 2.6(b)(c), 2.7. 2/19 for 2/22: Read Serre, Section 2.5 through Theorem 6. 2/17 for 2/19: Read Serre, Section 2.4. 2/15 for 2/17: Complete reading of Serre, Section 2.3. 2/10 for 2/12: Read Serre, Section 2.3 through Theorem 3. First graded homework (due 2/10): Best three of four problems from the exercise set of Serre, Section 2.1. 2/8 for 2/10: Complete reading of Serre, Section 2.2. 2/5 for 2/8: Read Serre, Section 2.2 through Corollary 1. 2/3 for 2/5: Complete reading of Serre, Section 2.1. Also, show that each unitary matrix is diagonalizable. 2/1 for 2/3: Read Serre, Section 2.1 through the proof of Proposition 2. Also:
1/29 for 2/1: Read Serre, Section 1.6. Also, let F be a field of characteristic zero. For commuting indeterminates X_{1} , . . . , X_{n} , consider the ring P = F [ X_{1} , . . . , X_{n} ]of multivariate polynomials with coefficients in F . For each natural number r , let P_{r} be the subspace of P consisting of homogeneous polynomials of degree r, i.e., linear combinations of monomials of total degree r .
Zeroth graded homework (due 1/29). 1/27 for 1/29: Read Serre, Section 1.5. 1/25 for 1/27: Read Serre, Section 1.4. 1/22 for 1/25: Complete reading of Serre, Section 1.3. 1/20 for 1/22: Read Serre, Section 1.3 through the proof of Theorem 1. Also prove that "projections give complements", i.e., if p is a linear operator on V with p^{2} = p , then V is the direct sum of the image Im p and the kernel Ker p . 1/15 for 1/20: Read Serre, Sections 1.1, 1.2. Also, show that sending a real number x to the matrix [ cosh(x) sinh(x) ] [ sinh(x) cosh(x) ] gives a representation of the additive group of real numbers. 1/13 for 1/15: Let A be a 5 x 5 complex matrix with A^{5} = I_{5}. Find the eigenvalues and eigenspaces of A. 1/11 for 1/13: Read Serre, p.3. Then: Let f(x) be an even function. Determine f '(0). |