Undergraduate Representation Theory: Professor Karen E. Smith

• MatS 138: Groups and their Representations

Plus an exercise session run by Lauri Kahanpaa

Course assistant: Lauri Kahanpaa

Course Objective: To give students a concrete introduction to groups theory through their representations. The course has two distinct parts: the first focuses mostly on finite groups on finite dimensional vector spaces over the complex numbers, and the second on Lie Groups. Representation theory is a standard subject all mathematicians (and physicists) should know. I hope to remedy what I see as a serious gap in the standard math curricula, which never even defines a group representation at the advanced undergraduate and even beginning graduate level!

Prerequisites: For the first half, a good linear algebra course (eg Michigan's 217), a basic algebra course (Michigan's 412 is sufficient, or some exposure to elementary number theory or some other abstract course), prior exposue to proofs. The second half is more demanding, incorporating more (basic point-set) topology (Michigan 's Math 590) and advanced calculus (essentially elementary differential geometry in R^n). *NOTE: The first half of the course is a stand-alone mini-course on the representations of finite groups.

Text: None. I hope to produce my own notes. One good text which will have much of the background material is Dummit and Foote's Algebra book. Students will be assigned some reading from that book. When we are ready, the early chapters of Fulton and Harris's Representation theory book will be our guide, but because that book is quite a bit more advanced than this course, I will of course fill in many details.

The developing course notes for the first half of the course. PLEASE TELL ME YOUR COMMENTS! Note: due to the arrival of my PhD students, the notes are now being typed by Lauri Kahanpaa (in Finnish). See his webpage.

Some topics I hope to cover: First Half: Lots of examples of groups (beginning with dihedral, then symmetric group, other symmetry groups), lots of actions of groups on sets (proving Cayley's theorem that every group is isomorphic to some transformation group for instance), orbits, class formula, subgroups and quotient groups, Cayley graphs, lots of review of linear algebra (including tensor and exterior products), and some groups of linear transformations (SL, GL, O, etc). Linear representations of groups, the regular representation, irreducible representations, complete reduciblity, Schur's lemma, characters. Classification of representations of finite groups over C. Second Half: Basics of Lie groups (including definintion of manifolds in R^N: charts and parametrizations, smooth functions, the tangent space to a point on a manifold, the lie algebra as tangent space at identity with many explicit computations of tangent space to identity matrix for small Lie groups (GL2, SL2, SO2, S^1. etc). Abstract Lie algebras, the adjoint representation of a Lie group and lie algebra, the classical and modern points of view on the exponential map. The lie algebra as invariant vector fields and derivations. The fundamental theorem that representations of a simply connected and connected group is equivalent to representations of its lie algebra. Explicit classification of the representations of SL_2.

PLANNED LECTURE TOPICS

• 1a. The dihedral group of symmetries of a square
• 1b. The definition of a group, transformation groups, matrix groups and other examples. Subgroups. Isomorphism. What is representation theory? Review of linear algebra, dihedral group as a matrix group, GL(R^n) versus group of n x n matrics.

• 2a. The symmetric group of permutations of n objects: cycle decomposition, generated by transpositions.
• 2b. Generators. cyclic group. Review of GL(2, R) as generated by elementary matrices (with linear algebra review, including multiplication by matrices on left/right corresponding to row/column ops).

• 3a. Product groups, Homomorphism. Examples: why is Z_3 x Z_2 cyclic but Z_2 x Z_2 not?
• 3b. Review of construction of modular group Z_n, Quotient group , cosets, Normal subgroups, quotient groups, Lagrange's theorem on orders of subgroups.

• 4a. Group actions on Sets, linear group actions on vector spaces. the associated representations, faithful representations.
• 4b. Cayley's theorem that all groups are transformation groups. Examples of D_4 acting tautologically on R^2, on R^4 by the permutation representation on the vertices, or on R^8 by the regular representation.

• 5 a/b. More examples of linear representations: direct sum of representations, isomorphisms, homomorphisms of G-reps.

• 6a/b General discussion of representations of groups on other interesting sets---how actions on topological spaces give rise to actions on vector spaces. The idea of a category, and of functor. How group maps G--->H allow us to pull back any H-rep to a G-rep

• 7 a. The kernel, image, and cokernel reps. Quotient reps.
• 7b. A very carefully worked example decomposing the four-dimensional vertex-permutation representation of the dihedral group D4 into irreducible representations.

• 8a. Irreducible representations, complete reducibility. Example of additive group acting on R^2 and not decomposing into irreducible subreps.
• 8b. The proof that a real or complex representation of a finite group has a complementary representation.

• 9a. Proof of the uniquness of decomposition, Schur's lemma
• 9b. Example: hands on calculation that S_3 has exactly 3 irreducible reps

• 10a. Definition of Characters of representation, explicit computation of characters of irreducible reps of S_3
• 10b. Statement of theorem that characters of irreducible reps are orthonormal. Deduction of many important consequences: number of irreducible reps bounded by number of conjugacy classes, character determine the rep,

• 11a Character of the regular rep and fact that all reps appear in reg rep.
• 11b Proof of orthonormality

SECOND PART OF COURSE: LIE GROUPS

• 12a Overview of classification for representations of Lie Groups--role of lie algebras.
• 12b Review of notions from topology: topological space, homeomorphism, continuous map, product topology, basis or generating set, Hausdorff property.

• 13a Smooth Manifolds: Classical definition (manifolds in Eucidlean space).
• 13b Smooth Manifolds: modern point of view, charts, atlas, notions of smooth functions, other categories of manifolds (complex, real analytic, algebraic). Topological groups and Lie groups

• 14a. Tangent Space
• 14b. Lie Algebra as tangent space at identity

• 15a. Detailed Example: The Lie group SL_2 and its lie algebra
• 15b. Homomorphisms of Lie Groups

• 16a. Smooth Representations: Def and Examples
• 16b. Review of Bilinear Algebra

• 17ab. More Bilinear Algebra: tensor, symmetric and exterior product, respresentations induced on products.

• 18a. Careful detailed example of decomposition of the second tensor product of the tautological representation of GL(2).

• 19: Abstract Lie Algebras: definitions and examples gl(V), Hiesenberg lie algebra,

• 20a: Why is the tangent space to identity a Lie algebra? The lie bracket as adjoint representation. Adjoint representations of lie groups and algebras.
• 20b Explicit computation of adjoint representation for gl(n) showing it agrees with "standard" lie algebra structure for matrix groups.

• 21a Exponential map: Classical definition
• 21b Explicit computation for SO(2).

• 22a Overview of the fundamental theorem: Representations of Lie groups give representations of lie algebras, and these are *equivalent* if the lie group is connected and simply connected.
• 22b Tangent bundle and vector fields

• 23 Invariant vector fields, the Lie algebra as vector fields

• 24 Definition of exponential map in general, examples

• 25 Classicification of all smooth finite dimensional representations of SL2.

• Student talks:
• Tommi Alanne: Representations of A_4 and particular physics
• Aapo Kauranen: Haar measure and representations of compact groups

HOMEWORK SETS