Math 249: Algebraic Combinatorics—Fall 2017


Announcements
Professor
Time and place
Subject matter
Textbooks
Homework and grades
Problem sets
List of lecture topics

Announcements

(8/21) Welcome to Math 249! Watch this page for further information.


Professor

Mark Haiman,
Office hours Mondays 12:00-1:30 or by appointment


Time and place

MWF 3-4 pm, 289 Cory


Subject matter

I have decided to change the focus of the course this year and take as my organizing principle the combinatorics associated with Coxeter groups and root systems.

Following this approach, we will still naturally encounter many ideas and objects of interest in classical combinatorial enumeration, such as ordinary and exponential generating functions, symmetric functions, partitions, trees, lattice paths and so on. Our aim will be to connect these things with symmetric groups, considered as Coxeter groups, and with other data associated with root systems of 'type A,' so as to inquire about deeper phenomena and generalize to other root systems.

Although we will pass more quickly over certain topics which I traditionally cover in detail (such as species), in their place we should be able to introduce Hecke algebras and Kazhdan-Lusztig polynomials, and touch on some topics of current research interest, such as k-Schur functions, LLT pollynomials, and q,t-analogs of combinatorics associated with Dyck paths.


Textbooks

There is no required textbook for this course. I will give a self-contained presentation of the material in the lectures. You may, however, find some of the following useful as reference texts.

The UC library links above should work from computers on campus. You can access library resources from off campus by using the library Proxy Server.


Homework and grades

I'll post problem sets below at irregular intervals, generally due 2-3 weeks after posting. Grades will be based on homework. There will be no exams. To get an A in the course, you should do most of the problems, not skipping the harder ones. For a B (or less), some smaller fraction of the assigned work will suffice.

You may turn in problems in class, at my office (slip them under the door if I'm not in), or by e-mail in PDF format.


Problem sets


List of lecture topics

I will update this list as the course goes on. References to Vol. I (Chapters 1-4) of Stanley's book are to the 2nd edition.

  1. (8/23) The symmetric group. Two-line, one-line and disjoint cycle notation. Conjugacy classes Cλ and their sizes.
  2. (8/25) Binomial and multinomial coefficients. Counting multisets using generating functions.
  3. (8/28) More multiset counting: bijectively, or using distributions. Multivariate generating for the |Cλ|.
  4. (8/30) Intro to finite group representations. Defining and sign representations of Sn and Bn.
  5. (9/1) G-modules versus matrix representations. Submodules, quotients, direct sums, irreducibility. Examples in char 0 and char p. Group algebra; G-modules as kG-modules.
  6. (9/6) Complete reducibility and orthogonality. Characters of V⊗W and Hom(V,W). Interpretation of inner product as dimension of HomG(V,W). Examples.
  7. (9/8) Proofs of complete reducibility and orthogonality using Reynolds operator and Schur's lemma. Character table of S3.
  8. (9/11) Center of kG, projectors, uniqueness of isotypic components in a G-module. Orthogonality of columns in the character table. Formula |G|=∑ dim(Vi)2. Character table of S4.
  9. (9/13) Symmetric functions in finite and infinitely many variables. Partitions, Young diagrams, transpose, partial ordering. Monomial and elementary bases mλ and eλ.
  10. (9/15) Complete homogeneous and power-sum bases hλ and (over Q) pλ.

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