Groups, Representations and Physics

Despite the title's emphasis on physics, this is really a mathematics course. It will introduce the main ideas and techniques in representation theory of finite groups and Lie groups. Applications will be discussed, to the extent that my understanding of them will permit, as a way to add some perspective to the mathematical theory.

An ideal text for the course would be "Group theory and physics" by S. Sternberg (Cambridge University Press), which unfortunately is out-of-print. I do plan to use it a great deal in preparing for the lectures, so it may be useful, but not essential, to have a copy of it. The text by H. F. Jones, "Group, representations and Physics" has in broad outline the material I plan to cover, and it is available at the campus bookstore. Please note, though, that this is only a recommended text. Since I will be providing some reading material at the first day of classes, you can safely leave the decision whether to buy it after the semester gets started.

Occasionally, material from the following texts may be used:

1. J.-P. Serre, "Linear Representations of Finite Groups" (mainly pages 1-44);
2. Michael Artin, "Algebra" (for general group theory and linear algebra);
3. W. Fulton and J. Harris, "Representation Theory - A First Course" (for a number of examples);
4. D. H. Sattinger and O. L. Weaver, "Lie Groups and algebras with applications to physics, geometry, and mechanics" (when discussing general manifolds, Lie groups and algebras).

Here is a list of topics we plan to cover:

1. Examples of groups (finite rotation groups, matrix Lie groups, the symmetric group);
2. Linear algebra (inner product spaces, duals, tensor product, function spaces, the spectral theorem for normal operators, some special decompositions, Hilbert spaces);
3. Representations of finite groups (character theory, the symmetric group);
4. Basics of differentiable manifolds, Lie groups and Lie algebras;
5. Representations of compact Lie groups (Peter-Weyl theorem);
6. Finite dimensional representations of SO(3), SL(n,C), U(n) (emphasis on dimensions 2 and 3), semidirect products and the Poincare group;
7. The basic structure of semisimple Lie groups and their representations.

Among the possible applications to choose from:

1. Molecular vibrations and spectra;
2. The hydrogen atom and the periodic table;
3. Elements of gauge theory.

Evaluation will be based on biweekly homework assignments. The following is a tentative grade scale:

• A - 100-85
• B - 70-84
• C - 55-69
• D - 40-54
• F below 39

This scale will be applied with enough flexibility to account for what I hope will be a diverse mixture of student backgrounds. Undergraduate and graduate students, from math, physics, chemistry, or other sciences, are all welcome.

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