Math. 841 (Fall 2003)

FALL 2003
Advanced Topics in Algebra (Math. 841)
Jeb F. Willenbring

Meeting: 10:00-10:50 Monday, Wednesday and Friday in room E416 (EMS building)
Office Hours: 11:00-12:00 MWF in E494

Our official textbook will be:

Manivel, Laurent. Symmetric functions, Schubert polynomials and degeneracy loci.
Translated from the 1998 French original by John R. Swallow.
SMF/AMS Texts and Monographs, 6. Cours Spécialisés [Specialized Courses], 3.
American Mathematical Society, Providence, RI;
Société Mathématique de France, Paris, 2001.
viii+167 pp. \$44.00. ISBN 0-8218-2154-7

However, there are several books which overlap considerably with Manivel's book. For example:

Fulton, William. Young tableaux. With applications to representation theory and geometry.
London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997.
x+260 pp. \$59.95; \$19.95 paperbound. ISBN 0-521-56144-2; 0-521-56724-6

The goal of the course will be to cover the material in the book by Manivel. This will introduce us to certain important mathematical topics at the intersection of algebraic geometry, representation theory and combinatorics. Beyond the textbook, we will try to supplement the topics with enough Lie theory to understand how the results related to the symmetric and general linear groups can be put into a more general context. This will involve (at a minimum) a presentation of the theory of root systems, weights and the Weyl groups as they relate to the representation theory of complex semisimple linear algebraic groups. References for this supplementary material will be given as needed.

A partial list of topics (in no particular order) included is:

representation theory of S_n, symmetric functions, RSK correspondence, root systems, Weyl groups, weights, representations theory of GL_n and other classical groups, Weyl's character formula, weights, standard and semi-standard tableaux, Schur polynomials, Kostka numbers, plane partitions, Kostka-Foulkes Polynomials, Littlewood-Richardson rule, Jacobi-Trudi identity, tensor products of representations, Grassmannians, flag varieties, Pieri and Giambelli's formulas, fundamental classes, Schubert varieties, degree of a projective variety, Bruhat order, Schubert polynomials, Standard monomials, and intersection theory.

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