18.315   M.I.T.   Fall 2010

Combinatorial Theory

Class meets: Tuesday and Thursday, 2:30-4 pm, room 2-135

Instructor: Alexander Postnikov

Office hour: Tuesday 4-5 pm

Grader: Karola Mészáros

This course is an introduction to algebraic combinatorics. This area of mathematics applies combinatorial (enumerative, bijective) techniques to problems arising in algebra (for example, in representation theory), and vise versa. We will discuss classical matherial, such as combinatorics of Young tableaux, as well as material from some recent research papers. The course will include the following topics: the symmetric group and its representations, symmetric functions, Young tableaux, Schur functions, Gelfand-Tsetlin patterns, Vershik-Okounkov's "new approach", the Robinson-Schensted-Knuth correspondence, the hook-length formula (and its generalizations), the Littlewood-Richardson rule (and its various reincarnations), Berenstein-Zelevinsky's triangles, Knutson-Tao's puzzles, hives, and honeycombs, integer lattice points of polytopes and piecewise linear combinatorics (including piecewise linear versions of RSK and LR rule), the Murnaghan-Nakayama rule, divided difference operators, Schubert polynomials, Pieri formula, the weak and the strong Bruhat orders, reduced decompositions and wiring diagrams, Grassmannians, matroids, Schur positivity, the Hecke algebra, the nil Hecke algebra, quantum Schubert calculus, etc.

The class will be accessible to first year graduate students.

Course Level: Graduate

Recommended textbooks:
*  R. P. Stanley: Enumerative Combinatorics, Volumes 1 and 2, Cambridge University Press, 1996 and 1999.
*  L. Manivel: Symmetric Functions, Schubert Polynomials and Degeneracy Loci, SMF/AMS Texts and Monographs, Vol 6 and Cours Specialises Numero 3, 1998.
*  I. G. Macdonald: Symmetric Functions and Hall polynomials, 2nd Edition, Clarendon Press, Oxford, 1995.
*  W. Fulton: Young Tableaux, Cambridge University Press, 1997.

Grading: Based on several problem sets

Problem Sets:

last updated: November 30, 2010