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Grad Studies A

Lectures on
Macdonald Polynomials

Arun Ram

Semester I 2022

Lectures

Lecturer: Arun Ram, 174 Peter Hall, email: aram@unimelb.edu.au

Time and Location of the Lectures: Wednesday 3:15-5:15
       In person lecture: Russell Love Theatre, Peter Hall Building
       Announcements and Zoom link: Subscribe to the email list at https://lists.unimelb.edu.au/subscribe/graduate-studies-a?previous_action=info
       If that doesn't work then please send an email to aram@unimelb.edu.au to request the Zoom link.

The plan is to make each lecture independent so that one does not need to attend (or remember) any previous lectures to follow any particular day.

The necessary background is some linear algebra (i.e., vector spaces, bases, eigenvalues) what the symmetric group is and what polynomials in n variables are.

Subject Overview

The goal of this lecture series is to provide a review of the theory of Macdonald polynomials with a focus on examples.

References

The course will mostly work through of the content and examples of the results of the following texts. My style differs, so it will not look the same, but the main results are mostly the same -- I will add details, explicit examples and specific computations. Several of the proofs that I will present are different. and I will present a few additional combinatorial formulas. The texts below are written in the setting of general affine root systems, for the lectures I shall focus primarily on type GLn.

  • I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki 1996 Vol. 1994/1995, Astérisque No. 237, Exp. No. 797, 4, 189-207.
  • I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge University Press 2003 ISBN: 9780511542824 DOI: https://doi.org/10.1017/CBO9780511542824.
    If you are at University of Melbourne try this link for this book: Find It@UniMelb

Lecture Plan

  • Lecture 1, 23 February 2022: n-periodic permutations: Tentative lecture notes for Lecture 1
    Zoom recording Passcode: Tf?D2pTR
    Handwritten notes for Lecture 1
    Tutorial sheet for Week 1
    Problem sheet for Week 1
    • Page 1: The affine Weyl group
    • Page 2: Inversions
    • Page 3: The elements uμ, vμ, tμ
    • Page 4: Boxes
    • Page 5: Affine coroots
    • Page 6: The box greedy reduced word uμ


  • Lecture 2, 2 March 2022: Macdonald polynomials Tentative lecture notes for Lecture 2
    Zoom recording Passcode: M*cdRW5R
    Handwritten notes for Lecture 2
    • Page 1: Nonsymmetric Macdonald polynomials
    • Page 2: Symmetric Macdonald polynomials
    • Page 3: Fermionic Macdonald polynomials
    • Page 4: Eigenvalues
    • Page 5: Creation formulas


  • Lecture 3, 9 March 2022: The double affine Hecke algebra (DAHA) Tentative lecture notes for Lecture 3
    Tutorial sheet for Week 3
    Handwritten notes for Lecture 3
    Zoom recording Passcode: 0VT*yNC^
    • Page 1: Presentation of the DAHA
    • Page 2: Cherednik-Dunkl operators
    • Page 3: Intertwiners
    • Page 4: DAHA acts on polynomials
    • Page 5: c-functions


  • Lecture 4, 16 March 2022: Symmetrizers and E-expansions Tentative lecture notes for Lecture 4
    Tutorial sheet for Week 4
    Handwritten notes for Lecture 4
    • Page 1: Nonsymmetric, relative, symmetric and fermionic Macdonald polynomials
    • Page 2: HY-decomposition of the polynomial representation
    • Page 3: Symmetrizers
    • Page 4: E-expansions
    • Page 5: Symmetrization of Eμ
    • Page 6: KZ families


  • Lecture 5, 23 March 2022: Principal specializations and hook formulas Tentative lecture notes for Lecture 5
    Handwritten notes for Lecture 5
    Tutorial sheet for Week 5
    Zoom recording Passcode: !Rps@X3N
    • Page 1: Principal specialization formulas by c-functions
    • Page 2: A hook formula for the symmetric case
    • Page 3: hook formula for the nonsymmetric case
    • Page 4: Elliptic, quantum and ordinary formulas


  • Lecture 6, 30 March 2022: Alcove walks, set valued tableaux and column strict tableaux Tentative lecture notes for Lecture 6
    Handwritten notes for Lecture 6
    Zoom recording Passcode: MJh2q#w@
    • Page 1: Creation formulas
    • Page 2: Alcove walks formula
    • Page 3: Set valued tableaux formula
    • Page 4: Nonattacking fillings formula
    • Page 5: Column strict tableaux formula


  • Lecture 7, 6 April 2022: The Boson Fermion correspondence and the Weyl character formula Tentative lecture notes for Lecture 7
    Handwritten notes for Lecture 7
    Tutorial sheet for Week 7
    Zoom recording Passcode: cxyb!Ej9
    • Page 1: Geometric Satake
    • Page 2: Symmetrizers and the polynomial representation
    • Page 3: The inner product (,)q,t
    • Page 4: The inner product charcterization of Eμ and Pλ
    • Page 5: Goping up a level from t to qt
    • Page 6: The Weyl character formula for Macdonald polynomials


  • Lecture 8, 13 April 2022: Orthogonality Tentative lecture notes for Lecture 8
    Handwritten notes for Lecture 8
    Zoom recording Passcode: #?0D$cCx
    • Page 1: Definition of the inner products
    • Page 2: adjoints
    • Page 3: recursions for norms
    • Page 4: Macdonald constant term conjectures
    • Page 5: Norm formulas


  • Lecture 9, 27 April 2022: Specializations of Macdonald polynomials
    Handwritten notes for Lecture 9
    Zoome recording Passcode: !Z#i0h9M
    • Page 1: Intertwiners, Demazure operators and Hecke operators
    • Page 2: The specialization square — Schur functions, Hall-Littlewood polynomials, q-Whittaker functions
    • Page 3: The specialization square — Iwahori-spherical functions, Iwahori-Whittaker functions, Demazure characters
    • Page 4: Comparison of operators
    • Page 5: Grothendieck polynomials and Schubert polynomials


  • Lecture 10, 4 May 2022: Products of Macdonald polynomials
    Handwritten notes for Lecture 10
    Zoom recording Passcode: 4EC3#Q?S
    • Page 1: Macdonald-Pieri operators
    • Page 2: Pieri rules
    • Page 3: Monk formulas
    • Page 4: Clebsch-Gordan formulas
    • Page 5: Littlewood-Richardson rules


  • Lecture 11, 11 May 2022: Koornwinder polynomials
    Handwritten notes for Lecture 11
    Zoom recording Passcode: 4%7x&3H3
    • Page 1: 11 non exceptional type affine root systems
    • Page 2: type C∨C and specialzations
    • Page 3: affine Weyl group, inversions and boxes
    • Page 4: Some formulas


  • Lecture 12, 18 May 2022: The DAWG, DAArt, DAHA and SL2(ℤ)
    Handwritten notes for Lecture 12
    Zoom recording Passcode: 1N0n3jP@
    Zoom recording Passcode: p%=C2c=k
    • Page 1: The DAWG (double affine Weyl group)
    • Page 2: SL2(ℤ)
    • Page 3: The DAArt (double affine Artin group)
    • Page 4: The DAHA (double affine Hecke algebra)


Problem sheets

  • Problem sheet 1: n-periodic permutations
  • Problem sheet 2: Macdonald polynomials
    • Problem sheet 3: The DAHA
  • Problem sheet 4: Principal specializations and hook formulas
  • Problem sheet 5: E-expansions, relative Macdonald polynomials and KZ families
  • Problem sheet 6: Alcove walks, set valued tableaux and column strict tableaux
  • Problem sheet 7: The Boson Fermion correspondence and the Weyl character formula
  • Problem sheet 8: Orthogonality
  • Problem sheet 9: Specializations of Macdonald polynomials
  • Problem sheet 10: Products of Macdonald polynomials
  • Problem sheet 11: Koornwinder polynomials
  • Problem sheet 12: Modified Macdonald polynomials

Tutorials and Tutorial sheets

If tutorials take place, they will be organized by the students.

  • Tutorial sheet 1: n-periodic permutations
  • Tutorial sheet 2: Macdonald polynomials
  • Tutorial sheet 3: The DAWG and the DAHA
  • Tutorial sheet 4: Principal specializations and hook formulas
  • Tutorial sheet 5: E-expansions, relative Macdonald polynomials and KZ families
  • Tutorial sheet 6: Alcove walks, set valued tableaux and column strict tableaux
  • Tutorial sheet 7: The Boson Fermion correspondence and the Weyl character formula
  • Tutorial sheet 8: Orthogonality
  • Tutorial sheet 9: Specializations of Macdonald polynomials
  • Tutorial sheet 10: Products of Macdonald polynomials
  • Tutorial sheet 11: Koornwinder polynomials
  • Tutorial sheet 12: Modified macdonald polynomials

Discussion

A key part of Lecture time is the Ask me a question time. Generally this starts on the hour. Since this is a series of "training lectures" and not a formal course with credits, the Ask me a Question time may organically expand accomodate more of this very important part of the learning process.

Consultation hours: Send me an email to schedule a time.

Student Representatives: ??? email: ???@student.unimelb.edu.au and ??? email: ???@student.unimelb.edu.au

Facebook group: https://www.facebook.com/groups/???????/

Assignments

The homework assignment is to write up your own set of notes for the course, as if you were teaching it yourself, including tutorial sheets, homework assignments, sample exams, online video resources, etc.