18.217
Combinatorial Theory:
Schur polynomials and Schubert polynomials
Fall 2023, MIT
Instructor:
Alex Postnikov
Grader:
TBA
Class meetings:
MWF 12 pm
Room 56114
Webpage:
math.mit.edu/~apost/courses/18.217/
Course description:
Schur polynomials and Schubert polynomials play an important role in many
different areas of mathematics (such as representation theory, algebraic
geometry, etc). We will discuss beautiful properties of these
polynomials, and their various generalizations, focusing on their combinatorial
aspects. A large portion of current research in algebraic combinatorics
is devoted to the study of these polynomials. We'll discuss some of these
advances.
The course will include the following topics:
combinatorics of partitions and permutations,
Schur polynomials, Schubert polynomials, Demazure characters, Grothendieck
polynomials, quantum Schubert polynomials,
Young tablaux, RCgraphs (aka pipe dreams), RSK correspondence,
ChevalleyMonk and Pieri formulas,
BernsteinGelfandGelfand formula,
LittlewoodRichardson rule and its various generalizations,
Kashiwara's crystal graphs, alcove path model, etc.
Course Level:
The course should be accessible to first year graduate students.
Grading: Based on several problem sets.
Problem Sets:
Problem Set 1
due Friday, October 20, 2023
Problem Set 2
due Monday, December 4, 2023
Lectures:
Lecture notes by
Ilani AxelrodFreed:
Lectures 1, 2, 3, 4
Lectures 5, 6, 7, 9
Lectures 10, 11, 12, 13
Lectures 15, 16, 17
Lectures 18, 19, 22
Lecture notes by Saba Lepsveridze:
Lecture 1,
Lecture 2,
Lecture 3,
Lecture 4,
Lecture 5,
Lecture 6,
Lecture 7,
Lecture 8,
Lecture 9,
Lecture 10,
Lecture 11,
Lecture 12,
Lecture 13,
Lecture 14
List of lectures:
with links to related lecture notes from previous years

W 09/06/2023: Introduction. Schur and Schubert polynomials.
Weak Bruhat order.

F 09/08/2023: Grassmannian permutations. Basics of symmetric groups.
Wiring diagrams. Reduced decompositions. Coxeter relations and nilCoxeter relations. Divided difference operators.
Notes (pages 110)
 M 09/11/2023: Geometrical background of Schubert polynomials.
Cohomology of flag manifolds. Borel's theorem. Coinvariant algebra.
BernsteinGelfandGelfand's divided difference operators.
 W 09/13/2023: Lehmer codes of permutations. 132avoiding permutations.
Demazure operators (aka isobaric divided differences)
0Hecke relations. Demazure characters.
Formula for a Schur polynomial as a Demazure character.
Notes (pages 1116)
and
these notes (pages 13)
 F 09/15/2023: Demazure operators vs divided differences.
Semistandard Young tableaux.
Notes (pages 79)
 M 09/18/2023: Pipe dreams. Combinatorial formula for Schubert
polynomials by BilleyJockushStanley and FominStanley.
Description of pipe dreams via nilCoxeter algebra.
YangBaxter relations.

[BJS] S. C. Billey, W. Jockush, R. P. Stanley,
Some combinatorial properties of Schubert polynomials,
Journal of Algebraic Combinatorics 2 (1993), 345374.

[FS]
S. Fomin, R. P. Stanley,
Schubert polynomials and the nilCoxeter algebra,
Advances in Mathematics 103 (1994), 197207.
 W 09/20/2023: Semistandard Young tableaux as a special case of pipe
dreams. Double Schubert polynomials.
 M 09/25/2023:
Proof of the pipe dream formula for double Schubert polynomials via
YangBaxter relations.
Cauchy formula for Schur polynomials, dual Cauchy formula,
and Cauchy formula for Schubert polynomials.
Notes (pages 34)
 W 09/27/2023: RobinsonSchenstedKnuth correspondence.
Classical construction of RSK via Schensted bumping algorithm.
Symmetries of RSK.
Notes (pages 613)
and these notes (pages 110)
 F 09/29/2023: RSK via GelfandTsetlin patterns (BerensteinKirillov's
approach). Generalized RSK
as a bijection beween nonnegative matrices and reverse plane partitions.
Standard case: a correspondence between rook placements and oscillating
tableaux.
Notes (pages 813)
 M 10/02/2023: A construction of generalized RSK via toggle operations.
Notes (pages 112)
 W 10/04/2023: Proof of the hooklength formula via polytopes and
generalized RSK.
Notes (pages 1215)
 F 10/06/2023: Increasing and decreasing subsequences, Greene's theorem,
generalized Greene's theorem in terms of noncrossing lattice
paths, tropical calculus.
Notes (page 12)
 W 10/11/2023: Cauchy formula for Schubert polynomials via
YangBaxter relations, see
[Fomin, Stanley]
(section 4).
 F 10/13/2023:
ChevalleyMonk formula.
FominKirillov algebra and the Bruhat operators T_{ij}.
 M 10/16/2023: The strong Bruhat order and the weak Bruhat order.
Subwords of reduced words. MarshRietsch's lemma. Bruhat intervals and Lediagrams.
The EdelmanGreene correspondence between SYT's and reduced decompositions.
 W 10/18/2023:
More on Lediagrams. The EGcorrespondence and balanced tableaux.
Enumerations of weighted saturated chains in the weak and strong Bruhart order.
Macdonald's and Stembridge's formulas.
 F 10/20/2023: Proof of (Chevalley) Monk's formulas
based on the "Leibniz rule" for divided differences.
The generalized LRcoefficents for products of Schubert polynomials via
FominKirillov algebra.
 M 10/23/2023:
The Dpairing (the differential inner product) of polynomials.
The space of S_nharmonic polynomials. The dual Schubert polynomials.

[PS]
A. Postnikov, R. Stanley, Chains in the Bruhar order,
J. Alg. Combinatorics 29 (2009).
arXiv
 W 10/25/2023:
 F 10/27/2023:
 M 10/30/2023: The ring of symmetric functions. Bases of monomial, elementary,
complete homogeneous, and Schur symmetric functions.
The Hall inner product. Definitions skew Schur functions via the Hall inner product and via skew tableaux.
Pieri formula.
Notes
and
these notes (pages 110)
 W 11/01/2023: The LittlwoodRichardson coefficients as Schur expansion
coefficients of product of Schur functions and of skew Schur functions.
The LRcoefficients in reprentation theory of GL_n and S_N, and in geometry of
the Grassmannian Gr(k,n). The involution omega.
Notes (pages 13)
 F 11/03/2023: The LittlwoodRichardson rule. Zelevinsky's pictures.
Notes (pages 411)
 M 11/06/2023: Stembridge's "concise proof" of the LittlewoodRichardson rule
based on BenderKnuth's involutions.
Notes (pages 19)

[Stemb]
J. R. Stembridge, A concise proof of the LittlewoodRichardson rule,
Electronic Journal of Combinatorics 9 (2002).
 W 11/08/2023: (The end of) the concise proof of the LRrule.
BenderKnuth's involutions and toggles.
The BerensteinZelevinsky triangles and honeycombs.
 M 11/13/2023: KnutsonTao's honeycombs. Bijections between honeycombs
and BZtriangles, and between honecombs and LRtableaux.
Notes
 W 11/15/2023: KnutsonTao's puzzles. Bijection beween puzzles and honeycombs.
Symmetries of the LRcoefficients.
Notes
and
these notes (pages 16)
 F 11/17/2023:
Horn's problem and Klychko's cone. Relations with honeycombs.
The saturation theorem for the LRcoefficients.
Notes (pages 712)
and
these notes (pages 15).
For more details on Horn's problem, saturation theorem, honeycombs, and puzzles,
see:

[KT1]
A. Knutson, T. Tao, The honeycomb model of
GL_n(C) tensor products I: Proof of the saturation
conjecture, J. Amer. Math. Soc. 12 (1999).

[KT2]:
A. Knutson, T. Tao, Honeycombs and sums of Hermitian matrices,
Notices of Amer. Math. Soc. 48 (2001).

[KTW]
A. Knutson, T. Tao, and C. Woodward, The honeycomb model of GL_n(C)
tensor products II: Facets of the LittlewoodRichardson cone,
J. Amer. Math. Soc. 17 (2004).
 M 11/20/2023: PRV conjecture via honeycombs. Sums and difference
of permutations, dilated permutohedra, and xrays of permutations.
Notes (pages 611)
 W 11/22/2023: Schur positivity: some results and conjectures.
Notes (pages 915)
Some recommended books:
Here are some additional reading materials. The students are not required to
read all these books. But is it a good idea to take a look at them. The
course will cover some sections from them. The topics of the course are not
limited to the union of these books. When needed, we will provide additional
references.
 I. G. Macdonald, Symmetric Functions and Hall Polynomials,
Oxford University Press,
pdf.

Richard P. Stanley, Enumerative Combinatorics, Volume 2. Chapter 7.

William Fulton,
Young Tableaux: With Applications to Representation Theory and Geometry,
London Mathematical Society Student Texts, Series Number 35.

Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, Providence, R.I.
This webpage will be updated periodically. All information related to the
course, including problem sets, will be posted here.
last updated: November 26, 2023