18.217   Combinatorial Theory:
Schur polynomials and Schubert polynomials

Fall 2023, MIT

Instructor: Alex Postnikov
Grader: TBA    
Class meetings: MWF 1-2 pm     Room 56-114
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, RC-graphs (aka pipe dreams), RSK correspondence, Chevalley-Monk and Pieri formulas, Bernstein-Gelfand-Gelfand formula, Littlewood-Richardson 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 Set 1  due Friday, October 20, 2023

Problem Set 2  due Monday, December 4, 2023

Lectures:

Lecture notes by Ilani Axelrod-Freed:   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

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

2. F 09/08/2023: Grassmannian permutations. Basics of symmetric groups. Wiring diagrams. Reduced decompositions. Coxeter relations and nilCoxeter relations. Divided difference operators. Notes (pages 1-10)

3. M 09/11/2023: Geometrical background of Schubert polynomials. Cohomology of flag manifolds. Borel's theorem. Coinvariant algebra. Bernstein-Gelfand-Gelfand's divided difference operators.

4. W 09/13/2023: Lehmer codes of permutations. 132-avoiding permutations. Demazure operators (aka isobaric divided differences) 0-Hecke relations. Demazure characters. Formula for a Schur polynomial as a Demazure character. Notes (pages 11-16) and these notes (pages 1-3)

5. F 09/15/2023: Demazure operators vs divided differences. Semi-standard Young tableaux. Notes (pages 7-9)

6. M 09/18/2023: Pipe dreams. Combinatorial formula for Schubert polynomials by Billey-Jockush-Stanley and Fomin-Stanley. Description of pipe dreams via nilCoxeter algebra. Yang-Baxter relations.
   • [BJS] S. C. Billey, W. Jockush, R. P. Stanley, Some combinatorial properties of Schubert polynomials, Journal of Algebraic Combinatorics 2 (1993), 345-374.

   • [FS] S. Fomin, R. P. Stanley, Schubert polynomials and the nilCoxeter algebra, Advances in Mathematics 103 (1994), 197-207.

7. W 09/20/2023: Semistandard Young tableaux as a special case of pipe dreams. Double Schubert polynomials.

8. M 09/25/2023: Proof of the pipe dream formula for double Schubert polynomials via Yang-Baxter relations. Cauchy formula for Schur polynomials, dual Cauchy formula, and Cauchy formula for Schubert polynomials. Notes (pages 3-4)

9. W 09/27/2023: Robinson-Schensted-Knuth correspondence. Classical construction of RSK via Schensted bumping algorithm. Symmetries of RSK. Notes (pages 6-13) and these notes (pages 1-10)

10. F 09/29/2023: RSK via Gelfand-Tsetlin patterns (Berenstein-Kirillov's approach). Generalized RSK as a bijection beween non-negative matrices and reverse plane partitions. Standard case: a correspondence between rook placements and oscillating tableaux. Notes (pages 8-13)

11. M 10/02/2023: A construction of generalized RSK via toggle operations. Notes (pages 1-12)

12. W 10/04/2023: Proof of the hook-length formula via polytopes and generalized RSK. Notes (pages 12-15)

13. F 10/06/2023: Increasing and decreasing subsequences, Greene's theorem, generalized Greene's theorem in terms of non-crossing lattice paths, tropical calculus. Notes (page 12)

14. W 10/11/2023: Cauchy formula for Schubert polynomials via Yang-Baxter relations, see [Fomin, Stanley] (section 4).

15. F 10/13/2023: Chevalley-Monk formula. Fomin-Kirillov algebra and the Bruhat operators T_{ij}.

16. M 10/16/2023: The strong Bruhat order and the weak Bruhat order. Subwords of reduced words. Marsh-Rietsch's lemma. Bruhat intervals and Le-diagrams. The Edelman-Greene correspondence between SYT's and reduced decompositions.

17. W 10/18/2023: More on Le-diagrams. The EG-correspondence and balanced tableaux. Enumerations of weighted saturated chains in the weak and strong Bruhart order. Macdonald's and Stembridge's formulas.

18. F 10/20/2023: Proof of (Chevalley) Monk's formulas based on the "Leibniz rule" for divided differences. The generalized LR-coefficents for products of Schubert polynomials via Fomin-Kirillov algebra.

19. M 10/23/2023: The D-pairing (the differential inner product) of polynomials. The space of S_n-harmonic polynomials. The dual Schubert polynomials.
    • [PS] A. Postnikov, R. Stanley, Chains in the Bruhar order, J. Alg. Combinatorics 29 (2009). arXiv

20. W 10/25/2023:

21. F 10/27/2023:

22. 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 1-10)

23. W 11/01/2023: The Littlwood-Richardson coefficients as Schur expansion coefficients of product of Schur functions and of skew Schur functions. The LR-coefficients in reprentation theory of GL_n and S_N, and in geometry of the Grassmannian Gr(k,n). The involution omega. Notes (pages 1-3)

24. F 11/03/2023: The Littlwood-Richardson rule. Zelevinsky's pictures. Notes (pages 4-11)

25. M 11/06/2023: Stembridge's "concise proof" of the Littlewood-Richardson rule based on Bender-Knuth's involutions. Notes (pages 1-9)
    • [Stemb] J. R. Stembridge, A concise proof of the Littlewood-Richardson rule, Electronic Journal of Combinatorics 9 (2002).

26. W 11/08/2023: (The end of) the concise proof of the LR-rule. Bender-Knuth's involutions and toggles. The Berenstein-Zelevinsky triangles and honeycombs.

27. M 11/13/2023: Knutson-Tao's honeycombs. Bijections between honeycombs and BZ-triangles, and between honecombs and LR-tableaux. Notes

28. W 11/15/2023: Knutson-Tao's puzzles. Bijection beween puzzles and honeycombs. Symmetries of the LR-coefficients. Notes and these notes (pages 1-6)

29. F 11/17/2023: Horn's problem and Klychko's cone. Relations with honeycombs. The saturation theorem for the LR-coefficients. Notes (pages 7-12) and these notes (pages 1-5).

    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 Littlewood-Richardson cone, J. Amer. Math. Soc. 17 (2004).

30. M 11/20/2023: PRV conjecture via honeycombs. Sums and difference of permutations, dilated permutohedra, and x-rays of permutations. Notes (pages 6-11)

31. W 11/22/2023: Schur positivity: some results and conjectures. Notes (pages 9-15)

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