Combinatorial Theory: Young tableaux
Fall 2022, MIT
MWF 1-2 pm
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,
Gelfand-Tsetlin 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,
piecewise-linear combinatorics, tropical geometry, cluster algebras,
total positivity, quantum cohomology, ...
The course should be accessible to first year graduate students.
Grading: Based on several 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
Lecture Notes taken by Ilani Axelrod-Freed:
Lectures 27-30 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.
Piecewise-linear combinatorics. Toggles.
[Sagan_SG, section 3.10]
18.212 lecture 4 from 2021 (pages 8-17).
M 9/12/2022: Map φλ. Rectangular and diagonal sums. Robinson-Schensted correspondence.
[Sagan_SG, Sections 3.1, 3.2, 3.3]
W 9/14/2022: RSK. Schensted's insertion algorithm. Gelfand-Tsetlin patterns.
F 9/16/2022: RSK and GT-patterns (cont'd).
M 9/19/2022: Fomin's growth diagrams (generalized to the semi-standard 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. 0-Hecke algebra.
The permutohedron. Rado's theorem and the dominance order on partitions.
Fundamental theorem on symmetric functions.
Demazure operators vs divided differences. Basics of symmetric functions.
Fundamental theorem of symmetric functions.
Symmetric functions (cont'd).
W 10/05/2022: Specializations of Schur polynomials.
Weyl's dimension formula and Stanley's hook-content formula.
F 10/07/2022: Shifted shapes and shifted tableaux.
"Broken leg" hook-length formula for shifted tableaux.
Diagonal vectors of shifted tableaux.
W 10/12/2022: Volume of Gelfand-Tsetlin polytopes.
Newton polytopes. Associahedra. Young's lattice.
F 10/14/2022: Differential posets. Up and down operators.
M 10/17/2022: r-Differential posets.
Bijection between triangular rook placements and set partitions.
Jacobi-Trudi formulas. Lindstrom lemma and
Fomin-Zelevinsky's double wiring diagrams and the
"inverse Lindstrom lemma".
Proof of Lindstrom lemma based on a sign-reversing involution.
Back to Jacobi-Trudi formulas.
M 10/24/2022: Applications of Jacobi-Trudy 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
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.
Vershik-Okounkov's "new approach" to representations of symmetric groups.
The center of the group algebra of S_n. The Gelfand-Tsetlin subalgebra.
F 11/04/2022: "New approach" (cont'd):
The Gelfand-Tsetlin basis. Eigenvalues of Jucys-Murphy 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.
Characters of representations of S_n via Young's orthogonal form.
W 11/16/2022: Proof of Murnaghan-Nakayama rule. Multivariate
generalization of MN-rule.
F 11/18/2022: Even more general version MN-rule for tree tableaux.
Orlik-Terao algebra and shuffle identities.
M 11/21/2022: (Arnold)-Orlik-Solomon and Orlik-Terao algebras.
Hilbert series. The inreasing tree basis of OS/OT-algebra.
The "game of graphs".
W 11/23/2022: Volumes of root polytopes.
The "game of graphs" in terms of subdivisions of root polytopes.
(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.)
Richard P. Stanley,
Enumerative Combinatorics, Volume 2.
(Especially, Chapter 7.)
Young Tableaux: With Applications to Representation Theory and Geometry.
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