IOWA STATE UNIVERSITY

Mathematics 618: Representation Theory

Instructor: 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).

Click here for information about special accommodations.

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.

Communication devices must remain switched off during the class periods.

Additional Mathematics course policies

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.

Assignments

4/27 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  EndA(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. Show that, working over the complex numbers, each square matrix is similar to an upper triangular matrix.
  2. Give a justified example of a real square matrix of degree 2 which is not similar to a real upper triangular matrix.
  3. Show that the trace of the product of two square matrices (of the same degree) is independent of the order of the factors in the product.

1/29 for 2/1: Read Serre, Section 1.6. Also, let  F  be a field of characteristic zero. For commuting indeterminates  X1 , . . . , Xn , consider the ring
 P  =  F [ X1 , . . . , Xn ]
of multivariate polynomials with coefficients in  F . For each natural number  r , let  Pr  be the subspace of  P  consisting of homogeneous polynomials of degree  r, i.e., linear combinations of monomials of total degree  r .
  1. Show that the vector space  P  is an infinite direct sum of the subspaces  Pr .
  2. Let  V  be the  F-vector space with basis { X1 , . . . , Xn }. Show that  P2  is linearly isomorphic with  Sym2( V ) .

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  p2 = 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  A5 = I5. 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).