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 3:155:15
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/graduatestudiesa?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 GL_{n}.
Lecture Plan
 Lecture 1, 23 February 2022: nperiodic 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 u_{μ}, v_{μ}, t_{μ}
 Page 4: Boxes
 Page 5: Affine coroots
 Page 6: The box greedy reduced word u_{μ}
 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: CherednikDunkl operators
 Page 3: Intertwiners
 Page 4: DAHA acts on polynomials
 Page 5: cfunctions
 Lecture 4, 16 March 2022: Symmetrizers and Eexpansions
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: H_{Y}decomposition of the polynomial representation
 Page 3: Symmetrizers
 Page 4: Eexpansions
 Page 5: Symmetrization of E_{μ}
 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 cfunctions
 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 E_{μ} and
P_{λ}

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, HallLittlewood polynomials, qWhittaker functions

Page 3: The specialization square — Iwahorispherical functions, IwahoriWhittaker 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: MacdonaldPieri operators

Page 2: Pieri rules

Page 3: Monk formulas

Page 4: ClebschGordan formulas

Page 5: LittlewoodRichardson 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 SL_{2}(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: SL_{2}(ℤ)

Page 3: The DAArt (double affine Artin group)

Page 4: The DAHA (double affine Hecke algebra)
Problem sheets

Problem sheet 1:
nperiodic permutations

Problem sheet 2:
Macdonald polynomials
Problem sheet 3:
The DAHA

Problem sheet 4:
Principal specializations and hook formulas

Problem sheet 5:
Eexpansions, 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:
nperiodic 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:
Eexpansions, 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.