Seminar: Young tableaux

Welcome to the seminar on Young tableaux in algebra and geometry.

Basic information

Our main reference will be the book Young tableaux: with applications to representation theory and geometry by William Fulton published by the Cambridge University Press (ISBN-13: 978-0521567244).

We will meet on Tuesday, 7PM–9PM in room 508 in the math department.

Final paper

You will need to write an expository paper on a topic of your choice. It will be short (no more than 5 pages) and written using LaTeX. A first draft will be due by December 1. I will return it with comments within a week and ask you to submit a revised draft by December 14. If you prefer, you can turn in your draft early, in which case I will return it early with comments.

The choice of the topic and what you want to write about it is up to you. But please keep it within 5 pages. This will restrict what you can and cannot do. For example, if a theorem has a 10 page long proof, you will have to omit it or only sketch the main idea.

Topic ideas

You can read and summarize one (or part) of the following works.

• C. Greene, Some partitions associated with a partially ordered set (Generalizes the notion of largest increasing subsequences to partially ordered sets and relates them to Young tableaux.)
• R. Stanley, Polygon dissections and standard young tableaux (A short paper showing a bijection between certain dissections of an (n+2)-gon and certain Young tableaux.)
• R. Stanley On the enumeration of skew Young tableaux (Exact and asymptotic formulas for number of skew tableaux.)
• C. Krattenthaler Nonintersecting lattice paths and oscillating tableaux (Involves variations of the RSK correspondence for so-called oscillating tableaux.)
• C. Greene, A. Nijenhuis, and H. Wilf A probabilistic proof of a formula for the number of Young tableaux of a given shape (A short probabilistic proof of the Hook length formula).
• I. Gessel and G. Viennot Binomial determinants, paths, and hook-length formulae (Gives determinantal formulae for enumerating Young tableaux).
• R. Stanley A baker's dozen of conjectures concerning plane partitions (Conjectures about enumerating Young tableaux.)

You can discuss the following theorems, formulas, or concepts.

• Hook length formula
• Jacobi-Trudi identity
• MacMohan's master theorem (a combinatorics/linear-algebraic result with lots of applications)
• Polya enumeration theorem (gives a counting formula using group actions)
• Schur-Weyl duality (relates representations of the general linear group and the symmetric group)
• Hall-Littlewood polynomials (a family of symmetric functions generalizing Schur functions)

Alternatively, you may choose to pick a related collection of exercises from Fulton's book and present their solutions. However, they must share a coherent theme and you should write them up as a unified paper rather than a series of separate problems.

Alternatively, you are free to choose any result of your choice and write about it.

Writing resources

• Obtaining LaTeX
• Introduction to LaTeX: minimal, short, not so short.
• A sample template.
• Steve Kleiman's excellent advice on writing mathematics. It is written for an old MIT writing requirement, but is applicable much more widely.

Background

Later in the semester, we will need working knowledge of groups (specifically the group of permutations S_n and the group of matrices GL_n(C).) Here are some resources:

• A crash course in group theory by Prof. Serge Richard of Nagoya University.
• Notes on Abstract Algebra by Prof. Irena Swanson of Reed College. (There is much more here than we will need, but they are a good reference.)
• Wikipedia!

Plan