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Young diagrams
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All lectures will be given in real time on zoom.
The lectures will be recorded and posted on this webpage.
So you can watch them later if your time zone makes it hard to
participate in real time.
(Video recordings will be access-controlled and limited to the class.)
But I encourage everybody to participate in real-time lectures, if possible.
Please feel free to unmute yourself and ask a question or make a comment
at any moment during zoom sessions.
Course description:
The course will be about combinatorics of the symmetric group and symmetric
functions. We will discuss Young tableaux, Schur functions, the
Robinson-Schensted-Knuth correspondence, the Cauchy identity,
the Jacobi-Trudi identity, the
hook-length formula, the Littlewood-Richardson rule, the Murnaghan-Nakayama
rule, Schutzenberger's involution, jeu de taquin,
Hillman-Grassl correspondence,
Fomin's growth diagrams,
Gelfand-Tsetlin patterns,
Berenstein-Zelevinsky's triangles,
Knutson-Tao's honeycombs,
Jucys-Murphy elements,
Hecke algebras,
Okounkov-Vershik's construction,
Macdonald polynomials, etc.
We will cover classical results in this subject
and (as time allows) some recent research advances.
Course Level: Graduate.
The course should be accessible to first year graduate students.
W 09/02/2020.
Introduction. Symmetric functions. Young diagrams.
Fundamental theorem of symmetric functions.
Notes
F 09/02/2020. Relations between elementary, complete
homogeneous, and power symmetric functions. The involution omega.
Notes
M 09/07/2020. Labor Day - holiday.
W 09/09/2020.
Schur symmetric functions. Determinant formula. Semi-standard Young tableaux.
Gelfand-Tsetlin patterns.
Notes
F 09/11/2020.
Symmetric group. Wiring diagrams and reduced decompositions.
Schur polynomials as Schubert polynomials and Demazure characters.
Notes
M 09/14/2020.
Permutohedra. Divided differences vs Demazure operators.
Pieri rules.
Notes
W 09/16/2020. Cauchy identities. Robinson-Schensted-Knuth correspondence
(RSK).
Notes
F 09/18/2020. RSK (cont'd).
Insertion algorithm.
Symmetry of RSK.
Increasing and descreasing subsequences. Application: Erdos-Szekeres theorem.
Greene's theorem. Young's lattice.
Notes
M 09/21/2020.
Up and Down operators. Differential posets.
Examples: Young's and Fibonacci lattices.
Oscillating tableaux. Generalized Schensted correspondence: bijection between oscillating tableaux and rook placements.
Notes
F 09/25/2020.
Piecewise-linear RSK. Semi-standard growth diagrams.
Toggles.
Notes
M 09/28/2020.
More on toggles. Bender-Knuth involutions.
RSK vs Hillman-Grassl correspondence.
The hook length formula.
Stanley's hook-content formula.
Weyl's dimension formula.
Notes
F 10/02/2020.
Jacobi-Trudi identities. Gessel-Viennot-Lindstrom lemma.
Notes
M 10/05/2020.
Applications of Jacobi-Trudi identities.
The determinantal formulas for #SYT's vs the hook length formula.
The exponential specialization and principal specializations
of symmetric functions. q-analogs: q-factorial and q-binomial
coefficients.
Notes
W 10/07/2020. Proof of polynomiality of q-binomial coefficients
via Schubert decomposition of the Grassmannian.
Symmetry and unimodality of the Gaussian coefficients.
Sylvester's proof of the unimodality.
Weighted up and down operators.
Notes
F 10/09/2020.
Problem Set 1 is due:
Submit your solutions
Plane partitions. Counting using Lindstrom's lemma. MacMahon's formula.
RPP's vs SSYT's.
Rhombus tilings, pseudoline arrangements, and perfect matchings.
The arctic circle phenomenon.
Notes
M 10/12/2020. Columbus Day - holiday.
Tuesday 10/13/2020. (Monday schedule of classes)
Discussion of Problem Set 1. Your presentations of solutions.
(Volunteers are welcome to make short 5-10 min presentations during
the class time. Please let me know in advance if you want to present
a solution.)
W 10/14/2020.
Introduction to representation theory. Irreducible representations
of symmetric groups. Branching rule. Group algebra. Young symmetrizer.
Notes
F 10/16/2020. Vershik-Okounkov's "new approach"
to the representation theory
of the symmetric groups.
The center of the group algebra. Bratteli diagram. Gelfand-Tsetlin bases. Gelfand-Tsetlin subalgebra of the group algebra.
Notes
M 10/19/2020.
Vershik-Okounkov's approach (cont'd).
Centers and centralizers of group algebras. Jucys-Murphy elements.
The Gelfand-Tsetlin basis = the basis of common eigenvectors of JM-elements.
The degenerate affine Hecke algebra (DAHA).
Notes
A. M. Vershik, A. Yu. Okounkov.
A New Approach to the Representation Theory of the Symmetric Groups. II:
arxiv version and
journal version
W 10/21/2020.
Vershik-Okounkov's approach (cont'd).
Local analysis of the spectrum of Jucys-Murphy elements.
Abaci and content vectors of SYT's.
Young's orthogonal form.
Notes
F 10/23/2020.
Characters of representations. Character tables for S_3 and S_4.
The Murnaghan-Nakayama rule. Ribbon tableaux. Alternating permutations.
Notes
M 10/26/2020.
Multivariate generalization of the Murnaghan-Nakayama rule and scattering
amplitudes in N=4 SYM theory.
Parke-Taylor factors. Kleiss-Kuijf relations.
Ribbon and tree relations.
Notes
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. Postnikov, J. Trnka.
On-shell structures of MHV amplitudes beyond the planar limit,
J. High Energy Phys. 2015, no. 6, 179:
arXiv version.
W 10/28/2020. Proof of the (generalized) Murnaghan-Nakayama rule.
The "game" on graphs.
Non-crossing and non-nesting alternating trees and the Catalan numbers.
Chan-Robbins-Yuen conjecture (proved by Zielberger).
Kostant's partition function. Subdivisions of root polytopes.
The Orlik-Terao algebra.
The tree, forest, and Parke-Taylor S_n-submodules of the Orlik-Terao algebra.
Notes
F 10/30/2020.
Frobenius character formuala.
Proof of the Murnaghan-Nakayama rule for symmetric functions
(using the involution principle).
Notes
M 11/02/2020.
The Hall inner product on symmetric functions.
A formula for skew Schur functions.
Schutzenberger's jeu de taquin. Bender-Knuth involutions (again).
Jdt-slides vs toggle operations.
Induced representations of S_n.
The regular representation of S_n as the sum of n-ribbons.
Notes
W 11/04/2020.
The Littlewood-Richardson rule, and its friends. Zelevinsky pictures.
Notes
F 11/06/2020.
Stembridge's (cf. Berenstein-Zelevinsky) concise proof
of the Littlewood-Richardson rule via Bender-Knuth involutions.
Zelevinsky's extension of the LR-rule.
LR-rule via Gelfand-Tsetlin patterns.
Notes
M 11/09/2020.
LR rule (cont'd):
Gelfand-Tsetlin patterns,
Berenstein-Zelevinsky triangles, and Knutson-Tao honeycombs.
Notes
W 11/11/2020. Veterans Day - holiday.
F 11/13/2020.
LR rule (cont'd):
Bijections between Littlwood-Richardson tableaux,
Berenstein-Zelevinsky triangles, and Knutson-Tao honeycombs.
Puzzles.
Notes
M 11/16/2020.
Puzzles vs honeycombs. Symmetries of the LR-coefficients.
Horn's problem and the Klyachko cone.
Notes
W 11/18/2020. Saturation theorem.
Berenstein-Zelevinsky polytopes and cones.
Description of the Klyachko cone.
PRV conjecture.
Notes
F 11/20/2020.
Description of the Klyachko cone. PRV conjecture.
Sums, differences, and x-rays of permutations.
The self-dual Hopf algebra of symmetric functions.
Notes
M 11/23/2020. Thanksgiving vacation.
W 11/25/2020. Thanksgiving vacation.
F 11/27/2020. Thanksgiving vacation.
M 11/30/2020.
More on the hook length formula... The probabilistic "hook walk" proof
of Greene, Nijenhuis, Wilf. The shifted hook length formula.
Notes
W 12/02/2020.
Naruse's "excited" hook length formula for skew shapes.
Parking functions. Tree inversion polynomial. Alternating permutations.
Notes
F 12/04/2020.
Problem Set 2 is due:
pdf
More on alternating permutations & parking functions.
Representations of symmetric groups and symmetric functions associated
with parking functions. Generalized parking functions.
Strips and ribbons.
Notes
M 12/07/2020.
Generalized parking functions. Symmetric trees. A sign-reversing involution
on gamma-parking-function tableaux.
Chromatic polynomials and Stanley's chromatic symmetric functions.
Acyclic orientations of graphs and the refinement of acyclic orientations
by the number of sinks.
Notes
W 12/09/2020.
Positivity of symmetric functions: e-positivity and Schur positivity.
Claw-free graphs and (3+1)-free posets.
Stanley's conjecture on e-positivity of
chromatic symmetric functions for (3+1)-free posets.
Gasharov's Schur-positivity theorem.
Unit interval orders and the Catalan numbers.
Okounkov's Schur positivity
conjecture. LPP theorem and Schur log-concavity.
Notes
Recommended textbooks:
I. G. Macdonald, Symmetric Functions and Hall Polynomials:
pdf.
This webpage will be updated periodically. All information related to the
course
(lecture notes, recordings of zoom lectures, problem sets, etc.) will be posted
here.
