18.217
Combinatorial Theory: Young tableaux
Fall 2022, MIT
Instructor:
Alex Postnikov
Grader:
TBA
Class meetings:
MWF 12 pm
Room 56114
Webpage:
math.mit.edu/~apost/courses/18.217/
Course description:
The content of 18.217 varies from year to year.
This year we plan to concentrate on combinatorics of Young tableaux and their
various generalizations. We will discuss combinatorial structures that appear
in representation theory, geometry, and other areas.
We will talk about partitions, permutations, Young tableaux, Young's lattice,
GelfandTsetlin patterns, Bruhat orders, symmetric functions,
representations of symmetric and general linear groups,
Schur functions, Demazure characters,
Schubert polynomials, Grassmannian and flag manifolds, pipe
dreams, and (if time allows) crystal graphs, Littelmann path model,
piecewiselinear combinatorics, tropical geometry, cluster algebras,
total positivity, quantum cohomology, ...
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 21, 2022
Problem Set 2
due Wednesday, December 7, 2022
(optional) Problem Set 3
due Wednesday, December 14, 2022
Lectures:
Lecture Notes taken by Ilani AxelrodFreed:
Lectures 14
Lectures 67
Lecture 9
Lectures 1011
Lectures 1315
Lectures 1618
Lectures 2021
Lectures 2223
Lectures 2426
Lectures 2730 and 32
List of lectures:

W 9/7/2022: Introduction. Partitions. Young diagrams.
Standard Young tableaux. Catalan numbers.
18.212 lecture 2 from 2021 (pages 8, 9).

F 9/9/2022: Hook length formula. Polytopal proof of the hook length formula.
Piecewiselinear combinatorics. Toggles.
[Sagan_SG, section 3.10]
probabilistic proof:
18.212 lecture 4 from 2021 (pages 817).

M 9/12/2022: Map φ_{λ}. Rectangular and diagonal sums. RobinsonSchensted correspondence.
[Sagan_SG, Sections 3.1, 3.2, 3.3]

W 9/14/2022: RSK. Schensted's insertion algorithm. GelfandTsetlin patterns.

F 9/16/2022: RSK and GTpatterns (cont'd).

M 9/19/2022: Fomin's growth diagrams (generalized to the semistandard case).
Greene's theorem. Cauchy identity.

W 9/21/2022: 4 definitions of Schur polynomials: classical (in terms
of determinants), combinatorial (in terms of SSYT's), in terms of divided
difference operators, and in terms of Demazure operators.

M 9/26/2022: Schur polynomials (cont'd).

W 9/28/2022: Demazure operators. 0Hecke algebra.
The permutohedron. Rado's theorem and the dominance order on partitions.
Fundamental theorem on symmetric functions.

F 9/30/2022:
Demazure operators vs divided differences. Basics of symmetric functions.
Fundamental theorem of symmetric functions.

M 10/03/2022:
Symmetric functions (cont'd).
Involution omega.
Pieri rule.

W 10/05/2022: Specializations of Schur polynomials.
Weyl's dimension formula and Stanley's hookcontent formula.
GelfandTsetlin polytopes.

F 10/07/2022: Shifted shapes and shifted tableaux.
"Broken leg" hooklength formula for shifted tableaux.
Diagonal vectors of shifted tableaux.

W 10/12/2022: Volume of GelfandTsetlin polytopes.
Newton polytopes. Associahedra. Young's lattice.

F 10/14/2022: Differential posets. Up and down operators.
Oscillating tableaux.

M 10/17/2022: rDifferential posets.
Bijection between triangular rook placements and set partitions.

W 10/19/2022:
JacobiTrudi formulas. Lindstrom lemma and
GesselViennot method.
FominZelevinsky's double wiring diagrams and the
"inverse Lindstrom lemma".

F 10/21/2022:
Proof of Lindstrom lemma based on a signreversing involution.
Back to JacobiTrudi formulas.

M 10/24/2022: Applications of JacobiTrudy formualas.
Proof of the equivalence of the classical and combinatorial
definitions of Schur functions. Determinantal formula of
the number of SSYT's. The exponential specialization of symmetric
functions.

W 10/26/2022: Students' problem set presentations.

F 10/28/2022: More problem set presentations.

M 10/31/2022: Introduction to representation theory.
Irreducible representations of symmetric groups. Young's symmetrizer.

W 11/02/2022: More on Young's symmetrizer.
VershikOkounkov's "new approach" to representations of symmetric groups.
The center of the group algebra of S_n. The GelfandTsetlin subalgebra.
(Young)JucysMurphy elements.

F 11/04/2022: "New approach" (cont'd):
The GelfandTsetlin basis. Eigenvalues of JucysMurphy elements.
Degenerate Affine Hecke Algebra (DAHA).

M 11/07/2022: "New approach" (cont'd):
Local analysis of the spectrum.

W 11/09/2022: "New approach" (cont'd):
Allowed transpositions. Spec(n) and Cont(n).
Abacus and Young diagrams. Young's orthogonal form.

M 11/14/2022:
Characters of representations of S_n via Young's orthogonal form.
Ribbon tableaux.
MurnaghanNakayama rule.

W 11/16/2022: Proof of MurnaghanNakayama rule. Multivariate
generalization of MNrule.

F 11/18/2022: Even more general version MNrule for tree tableaux.
OrlikTerao algebra and shuffle identities.

M 11/21/2022: (Arnold)OrlikSolomon and OrlikTerao algebras.
Hilbert series. The inreasing tree basis of OS/OTalgebra.
The "game of graphs".

W 11/23/2022: Volumes of root polytopes.
The "game of graphs" in terms of subdivisions of root polytopes.
Recommended books:
(The students are not required to have these books.
The material of the course has a nonempty intersection with the union of
these three books. But we might present the material in a different order.)

[Stanley_EC2]
Richard P. Stanley,
Enumerative Combinatorics, Volume 2.
(Especially, Chapter 7.)

[Fulton_YT]
William Fulton,
Young Tableaux: With Applications to Representation Theory and Geometry.

[Sagan_SG]
Bruce E. Sagan, The Symmetric Group:
Representations, Combinatorial Algorithms, and Symmetric Functions.
This webpage will be updated periodically. All information related to the
course, including problem sets, will be posted here.
last updated: November 27, 2022