Math 517B - Group Theory (Spring 17)

Level

Graduate Course

University

University of Arizona

Prerequisites

group theory, representation theory of finite groups, linear algebra, algebraic geometry

Duration

16 weeks (2 lectures per week lasting 75 minutes each)

Lectures

Tue/Thu at 11:00am in MTL 124

Office Hours

Please make an appointment by email.

Exercises

• Exercise Sheet 1
• Exercise Sheet 2
• Exercise Sheet 3
• Exercise Sheet 4

Textbooks

There is not a dedicated text for this course but the following textbook will most closely follow the sturcture of the lectures:

• F. Digne and J. Michel, Representations of Finite Groups of Lie Type, vol. 21, London Mathematical Society Student Texts, Cambridge: Cambridge University Press, 1991.

Throughout the course we will need to use the theory of algebraic groups, specifically reductive algebraic groups. For an overview of this theory without too many proofs see:

• G. Malle and D. Testerman, "Linear algebraic groups and finite groups of Lie type", vol. 133, Cambridge Studies in Advanced Mathematics, Cambridge: Cambridge University Press, 2011.

The following are the standard references for algebraic groups and give a more in-depth analysis of the theory.

• T. A. Springer, "Linear algebraic groups", Modern Bikhäuser Classics, Boston, MA: Birkhäuser Boston Inc., 2009.
• J. E. Humphreys, "Linear algebraic groups", vol. 21, Graduate Texts in Mathematics, New York: Springer-Verlag, 1975.
• A. Borel, "Linear algebraic groups", second edition, vol. 126, Graduate Texts in Mathematics, New York: Springer-Verlag, 1991.
• M. Geck, "An introduction to algebraic geometry and algebraic groups", vol. 10, Oxford Graduate Texts in Mathematics, Oxford: Oxford University Press, 2003.
• B. Srinivasan, "Representations of finite Chevalley groups", vol. 764, Lecture Notes in Mathematics, New-York: Springer-Verlag, 1979.

Syllabus

We will aim to cover the following topics in the course:

• Classification and structure of reductive algebraic groups,
• Frobenius endomorphisms,
• Harish-Chandra theory,
• Alvis-Curtis duality,
• The Steinberg character,
• Deligne-Lusztig induction/restriction.

Other topics may be covered if time permits.