Voorstellingen van groepen, 24 April 2019 up to 19 June 2019

docent: Frits Beukers

time: Monday and Wednesdays, 9:00-10:45 (lecture), 11:00-12:45 (exercises)

place: We start April 24 in BBG017 (Link to the complete schedule)

book: Representations and Characters of groups by G. James, M. Liebeck, second edition

Adressed to: math students who followed Linear Algebra and Group Theory. We treat the basic theory of representations of finite groups. These are geometric realizations of groups via linear transformations. An example is the permutation group S₄ which can be realized via rotations of the cube, but also as the symmetry rotations and reflections of the tetrahedron. Representations of groups play a crucial role in many areas of mathematics and physcics. It is also a useful preparation to more advanced courses like Representations of Lie groups.
At the end of the course there is written exam. There are no midterm exams or home work assignments, except a midterm home work assignment.

Weekly schedule

April 24
We give an overview of the necessary prerequisites in group theory and linear algebra. Followed by the definition of representation. Example: 2D-representation of general dihedral group, both as real transformations and complex diagonal matrices as rotations. Equivalence between the two. Material: Chapter 1,2,3
Exercises:1.4, 1.9, 3.2, 3.3, 3.5, 3.6
April 29
Reducible/irreducible representations with examples. FG-modules presented as vectorspaces with G-action. Connection with representations. The permutation representation of S4 and its decomposition. Maschke's theorem does not (always) hold for infinite groups. Irreducible FG-modules with examples. Representations of the cyclic group. Material: Chapter 4,5,6.
Exercises: 3.5, 3.8, 4.3, 5.3, 5.4
May 1
FG-submodules, FG-homomorphisms, FG-isomorphisms, direct sums of FG-modules. The group algebra F[G]. Maschke's theorem. Chapter 7,8
Exercises: 6.1, 6.2, 6.4, 8.1, 8.2
May 6 (cancelled)
May 8
Proof of Maschke's theorem. Schur's lemma and Prop 9.2, representations of finite abelian groups.
Exercises: 7.3, 7.6, 8.4, 8.7, 9.4, 9.5
May 13
The center of FG. Theorem 10.5, the irreducible decomposition of the regular representation contains all irreducible ones. Example, the irreducible representations of D₆.
Exercises: 10.2, 10.3, 10.4
May 15
Number of copies of an irreducible representation in the regular representation, Ch 11. Conjugacy classes in finite groups, Ch 12.
Exercises: 11.2, 11.3, 11.4, 12.2, 12.3, 12.6
May 20
Conjugacy classes, Ch 12. Character of a representation, Ch 13
Exercises: 12.2, 12.3, 12.6, 13.1, 13.2
May 22
Inner product of two characters, characters of irreducible representations form an orthonormal system, Ch 14.
Exercises: 13.5, 14.1, 14.2
Home work assignment, to be handed in before or on June 5, 9 AM.
May 27
The number of irreducible representations (characters) equals the number of conjugation classe of G. Character tables, examples, orthogonality of the columns,Ch 15, 16.
Exercises: 15.1, 15.2, 15.4, 16.2
May 29
Commutator subgroup. Lifting of characters, Chapter 17.
Exercises:16.6, 17.1, 17.4, 17.5
June 3
Tensor products and their characters, Ch 19.
Exercises: 19.3, 19.4, 19.5
June 5
Algebraic integers and applications, Ch 22 Exercises: 22.1, 22.2, 22.3, 22.5
Solutions to the home work problems.
June 10 no class, holiday
June 12
Restriction of representations, Ch20. Exercises: 22.4, 20.1, 20.3
June 17
Applications of representation theory in mathematics and maybe some physics (does not belong to the exam material), hand back of home work.
Exercises: finish exercises of past weeks, opportunity to ask questions.
In case you finished everything here is an extra exercise:
Let G be a finite subgroup of GL(n,C). Let M_n be the vector space of n times n matrices with elements in C. Let G act on M_n via: the matrix A goes to gAg^T for every g in G.
- Show that this is a representation of G.
- Show that symmetric and anti-symmetric (i.e. A^T=-A) each form a G-invariant subspace.
- Take n=2 and let G be a finite subgroup of the unitary 2 times 2 matrices. Find a complex inner product on M which is G-invariant (you need something clever here).
- Let G be the group generated by the matrices (1,-1;0,-1) and (-1,1;-1,0) (is isomorphic with S₃. Determine the irreducible decomposition of the representation on M₂.
June 25: Exam
Time: 17:00-20:00, location Edu gamma
Exam covers Chapters 1 to 22 of the book, except Chapter 21.
Here are some older exams that I found. In a number of problems there are questions about induced representations (to be recognized by the upward arrows). You must skip them, because we did not cover them in class.
Added on June 25, 8PM: Solutions of the exam problems.