18.315 M.I.T. Fall 2006
Combinatorial Theory
Class meets: Tuesday and Thursday, 2:304 pm,
room 12142
Instructor:
Alexander Postnikov
Office hours: TR 45pm in Room 2389.
Grader: Pavlo Pylyavskyy
Description for the Catalog:
Topics for Fall 2006 include enumerative and algebraic
combinatorics related to representations of the symmetric group, symmetric
functions, and Young tableaux.
Synopsis:
This course is an introduction to algebraic combinatorics that came from
representation theory of the symmetric group.
The class will cover the classical topics such as
Specht modules, Young symmetrizers,
Young tableaux,
the branching rule,
GelfandTzetlin bases,
Schur functions,
the RobinsonSchenstedKnuth correspondence,
the JacobiTrudi identity,
the hooklength formula,
the LittlewoodRichardson rule,
the MurnaghanNakayama rule,
Schützenberger's involution and jeu de taquin,
etc.
If time allows, some of the following more recent topics
will be included: Hecke algebras,
OkounkovVershik construction,
JucysMurphy elements,
Schur positivity, connections with Horn's Problem and
with tropical geometry, Macdonald polynomials,
noncommutative symmetric functions, etc.
The class should be accessible for first year graduate students.
The prerequisites are minimal.
Course Level: Graduate
Recommended textbook:
*
Bruce E. Sagan: The Symmetric Group,
2nd Edition,
Graduate Texts in Mathematics, Springer, 2001.
Additional reading:
*
Richard P. Stanley:
Enumerative Combinatorics, Volumes 1 and 2,
Cambridge University Press, 1996 and 1999.
*
I. G. Macdonald:
Symmetric Functions and Hall polynomials,
2nd Edition, Clarendon Press, Oxford, 1995.
*
William Fulton:
Young Tableaux,
Cambridge University Press, 1997.
*
William Fulton, Joe Harris:
Representation Theory, A First Course,
Graduate Texts in Mathematics, Springer, 1991.
*
Gordon James, Adalbert Kerber:
The Representation Theory of the Symmetric Group,
Cambridge University Press, 1984.
Grading: Based on several problem sets
Problem Sets:
Lectures:
 R 09/07/2006. Introduction.
 T 09/12/2006. Representations of S_n and the Young lattice.
Young tableaux as paths. Up and down operators.
Representation theory basics. Group algebra.
Maschke's theorem.
 R 09/14/2006. Schur's lemma and its corollaries.
Characters. Conjugacy classes. Orthogonality of characters.
 T 09/19/2006. Characters (cont'd). Character table for S_4.
Wiring diagrams and Coxeter relations.
OkounkovVershik construction:
Branching graph and GelfandTsetlin bases.
For more details, see:

A. Vershik, A. Okounkov: A new approach to the representation theory
of the symmetric groups, II.
math.RT/0503040

A. Vershik:
A new approach to the representation theory of the symmetric groups, III:
Induced representations and the FrobeniusYoung correspondence.
math.RT/0609258
 P. Py: On representation theory of symmetric groups.
PDF
 R 09/21/2006. OkounkovVershik construction (cont'd).
The center of the group algebra. GelfandTsetlin subalgebra.
JucysMurphy elements.
 T 09/26/2006. OkounkovVershik construction (cont'd).
Example: Standard (n1)dimensional representation of S_n.
JucysMurphy elements satisfy degenerate Hecke algebra relations.
Content vectors.
 R 09/28/2006. OkounkovVershik construction (end).
Hooklength formula and hookwalks.
For more details, see:

C. Greene, A. Nijenhuis, H. S. Wilf, A probabilistic proof of a formula
for the number of Young tableaux of a given shape.
Adv. in Math. 31 (1979), no. 1, 104109.
 T 10/03/2006. Schur functions via semistandard tableaux. Cauchy identity.
RobinsonSchenstedKnuth (RSK) correspondence.
 R 10/05/2006. Properties of RSK. Fomin's growth diagrams.
T 10/10/2006. Columbus Day Vacation
 R 10/12/2006. RSK (cont'd). Longest increasing subsequences.
GelfandTsetlin patterns.
 T 10/17/2006. qbinomials coefficients. Unimodality of
Gaussian coefficients.
 R 10/19/2006.
Problem set review.
 T 10/24/2006.
The ring of symmetric functions and its various bases.
 R 10/26/2006. Lindstrom's lemma and JacobiTrudi identity.
The classical definition of Schur polynomials.
The determinantal formula for the number of SYT's.
 T 10/31/2006.
qanalogue of the determinantal formula and of the hooklength
formula. Reverse plane partitions.
HillmanGrassl correspondence.
 R 11/02/2006.
Problem set review.
 T 11/07/2006.
Plane partitions, noncrossing paths, rhombus tilings, perfect matchings, and
pseudoline arrangements.
Viennot's shadow construction for RSK.
 R 11/09/2006. Green's theorem. Pequivalence and Knuth's
equivalence. Schutzenberger's jeu de taquin.
 T 11/14/2006. The LittlewoodRichardson rule via jeu de taquin.
 R 11/16/2006. The classical LRrule (via lattice words).
 T 11/21/2006. Variants of the LRrule: Zelevinsky's pictures,
BerensteinZelevinsky's triangles, KnutsonTao's honeycombs and
puzzles. Symmetries of the LRcoefficients.
R 11/23/2006. Thanksgiving Vacation
 T 11/28/2006. The MurnaghanNakayama rule.
The Frobenius characteristic map. The characters of the symmetric group.
 R 11/30/2006. Problem Set review.
For more details on Problems 7 and 8(b) (noncommutative Schur functions,
the EdelmanGreene correspondence, and related stuff),
see the following papers:

S. Fomin, C. Greene: Noncommutative Schur functions and their applications,
Discrete Mathematics 193 (1998), 179200.

R. Stanley: On the number of reduced decompositions of elements of
Coxeter groups,” European. J. Combin. 5 (1984), 359–372.

P. Edelman, C. Greene: Balanced tableaux, Adv. Math. 63 (1987), 4299.
 T 12/05/2006. Guest lecture by Richard Stanley.