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\centerline{\bf Math 652 Syllabus, spring 2001}
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{\bf Course=Topics in Algebra, Math 652}
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Instructor=Sam Evens
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Time=class is scheduled for MWF, 11:45--12:35, but I would be happy
to discuss changing the time.
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We will study the relations between representation theory of groups and
geometry and topology. In particular, we will use ideas from algebraic/complex
geometry to construct irreducible representations of groups like GL(n,C)
and GL(n,R), and use ideas from topology to construct and study irreducible
representations of the symmetric group, and more generally Weyl groups.
Topics I intend to cover include the Borel-Weil theorem, D-module
constructions of Verma modules, and Springer representations. Other
topics may include Kazhdan-Lusztig polynomials, Hecke algebra representations,
and Lie algebra cohomology.
My intention is to base lectures on relevant papers, survey articles,
and sections of books, which I can copy and distribute to the audience.
The book by Chriss and Ginzburg should be useful at times.
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\par\noindent{\bf Prerequisites}:
The prerequisites should include some background in Lie theory. Certainly
my class Math 651 is sufficient, but a knowledge of the group GL(n)
and its finite dimensional representations will be adequate for many
purposes. The level of geometry/topology assumed will depend on the
audience. I won't prove all results from geometry/topology that are
needed, but will try to explain ideas and how they work in examples.
Optional homework will be given.
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\par\noindent{\bf Optional Texts}
Representation Theory and Complex Geometry by N. Chriss and V. Ginzburg,
Birhauser.
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Representations and Invariants of the Classical Groups
by Roe Goodman and Nolan R. Wallach. Cambridge University Press.
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Introduction to Lie Algebras and Representation Theory,
by James Humphreys, Springer Graduate Text, 9.
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