Topics in Combinatorics:   Lie Theory and Combinatorics

Class meets: Tuesday, Thursday   11-12:30   room 2-102

First class: Tuesday, February 3

Instructor: Alexander Postnikov   apost at math   room 2-389

Course webpage: http://www-math.mit.edu/~apost/courses/18.318/

Synopsis:

The course is devoted to combinatorial aspects of Lie theory. Many combinatorial objects and constructions such as Young tableaux, Schur symmetric polynomials, the Bruhat order, the Littlewood-Richardson rule, and many others came originally from representation theory. We will discuss these classical topics as well as some recent advances. Representation theory provides a source of interesting combinatorial problems and constructions. On the other hand, a combinatorial approach proves helpful for many difficult problems in representation theory.

The course will include the following topics:

• Type A theory: the general linear and symmetric groups, irreducible representations, Young tableaux, Schur polynomials, Schur-Weyl duality and Robinson-Schensted-Knuth correspondence, branching rule and Gelfand-Tsetlin basis, Littlewood-Richardson rule.
• Overview of Lie theory: semisimple Lie groups and Lie algebras, root systems, Cartan-Killing classification, Weyl groups, irreducible representations, weights, characters, Kostant partition function, Weyl's character and dimension formulas, Demazure's character formula.
• Reflection groups and Coxeter groups: irreducible decompositions, Bruhat order, polynomial invariants, affine reflection groups, affine Coxeter arrangements and alcoves.
• (if time allows) ... Crystal graphs, Littlemann's paths, piecewise-linear combinatorics and polytopes, total positivity, Fomin-Zelevinsky's cluster algebras and Laurent phenomenon, chains in the Bruhat order.

The course will be accessible to first year graduate students. All required notions and definitions will be given. There are no any special prerequisites though some background in combinatorics and/or representation theory would be helpful. The course can serve as an introduction to Lie theory for combinatorialists or as an introduction to combinatorics for Lie theorists.

Course Level: Graduate

Texts: Recommended (but not required) textbooks are:

*	W. Fulton, J. Harris: Representation Theory: A First Course, Springer-Verlag, 1991.
*	J. E. Humphreys: Reflection Groups and Coxeter groups, Cambridge University Press, 1990.
*	R. P. Stanley: Enumerative Combinatorics, Vol 2, Cambridge University Press, 1999.

Problem Sets:

• Problem Set 1 (.ps file)
• Problem Set 2 (.ps file)

Useful links:

• Combinatorial representation theory, by H. Barcelo and A. Ram, appeared in the special volume in conjunction with the special year 1996-1997 in Combinatorics at MSRI in Berkeley: New perspectives in algebraic combinatorics (Berkeley, CA, 1996--97), 23--90, Math. Sci. Res. Inst. Publ., 38 , Cambridge Univ. Press, Cambridge, 1999.