Spring
2014
Department
of Mathematics
SUNY
at Stony Brook
In this course we adopt a global approach to studying compact connected Lie groups and their representations. It should appeals to topologists and geometers.
We will follow books by Adams J.F. “Lectures on Lie groups” (Benjamin, 1969) and by Fulton W., Harris J. Representation theory. A first course (Springer, 1991)
Time/Location: TuTh 1:00pm 2:20pm Physics P127
Instructor: Mikhail
Movshev
Math Tower 4109. Phone: 6328271. email: mmovshev
at math dot sunysb dot edu
Office
Hours: TuTh 2:30  3: 30, or dropin, or by appointment.
Prerequisites: Students are expected to be familiar with most of the material of Math 530531 (Geometry/Topology III) and Math 534535 (Algebra III). Manifolds and group/algebra/module theory will be frequently used.
Requirements: There will be regular homework assignments (to be posted) due each class (see a table below), midterm and a final exam (both take home)
Grading System: The relative significance of exams and problem sets in determining final grades is as follows.
Midterm 
30.00% 
Final Exam 
40.00% 
Problem Sets/Class Participation 
30.00% 
Date 
Topic 
Homework 
Jan Tu 28 
No class 

Jan Th 30 
No class 

Feb Tu 4 
p.17 Basic definitions 

Feb Th 6 
p.712 Oneparameter subgroups 

Feb Tu 11 
p.1218 Properties of the exponential map 

Feb Th 13 
p.1925 Elementary Representation Theory. Representations over R,C and H 

Feb Tu 18 
p.2632 Restriction and extension of scalars. Tensor product of representations. 

Feb Th 20 
p.3339 Dual representation. Theory of integration over compact groups. Complete reducibility of representations. 

Feb Tu 25 
p.4046 Schur's lemma. Kfunctor. 

Feb Th 27 
p.4753 Characters. Orthogonality relation. 

Mar Tu 4 
p.5460 PeterWeyl theorem 

Mar Th 6 
p.6167 More on real and quaternionic structures on complex representations,selfconjugate representations. 

Mar Tu 11 
p.6874 Adams operations. Irr. rep. for a products of groups. Representations of compact abelian groups 

Mar Th 13 
p.7581 Maximal Tori In Lie Groups. Monogenic subgroups. Midterm(Takehome) 

Mar Tu 18 
Spring Recess 

Mar Th 20 
Spring Recess 

Mar Tu 25 
p.8288 Roots. Examples:SU,Sp,SO(2n) 

Mar Th 27 
p.8995 Examples cont.:SO(2n+1). Conjugacy of maximal tori. Weyl group W. Regular and singular elements. Action of W on roots 

Apr Tu 1 
p.96102 Subgroups U_theta, labelled by root theta. Geometry Of The Stiefel Diagram Examples of diagrams for U(2),SO(4),Sp(2) 

Apr Th 3 
p.103108 Examples cont.SU(3). Intersection of U_theta, Linear relations between roots 

Apr Tu 8 
p.109115 Action of W on roots cont. Action of W on Weyl chambers. Generators for W. Examples: SU,Sp,SO 

Apr Th 10 
p.116122 A formula for the generators. Weights. Angles comprised by roots. Series of roots.Examples SU(3),Sp(2). A fundamental Weyl chamber. Simple roots. 

Apr Tu 15 
p.123129 The Dynkin diagram.The fundamental dual Weyl chamber. Half sum of positive roots. The extended Weyl group. Description of the fundamental group in terms of the root data. 

Apr Th 17 
p.130136 The fundamental group cont.Examples of computations: SU,Sp, Groups Spin(n) 

Apr Tu 22 
p.137143 Representation Theory Weyl Integration Formula. Symmetries of characters. Characterization of antisymmetric characters. Elementary symmetric sums 

Apr Th 24 
p.144150 The elementary alternating sum. Characters of Spin(n) 

Apr Tu 29 
p.151157 Two partial orders on weight lattice. 

May Th 1 
p.158164 The maximal weight of an irr.rep. 

May Tu 5 
p.165171 K(G) is a polynomial algebra for simplyconnected G. Description of the complex representation rings K(U(n)) and K(Sp(n)). 

May Th 8 
p.172180 Computation of K(SO(n)). 

May Mon 19 
Final Exam 

Disabilities: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room 128, (631) 6326748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students requiring emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information, go to the following web site: http://www.www.ehs.stonybrook.edu/fire/disabilities.shtml