Triangle Lectures in Combinatorics (TLC)

Third meeting: April 9, 2011, SAS Hall 1102, NCSU

Slides from the talks (as pdf files):

Vic Reiner (University of Minnesota), P-partitions revisited
Stephanie van Willigenburg (UBC), Quasisymmetric refinements of Schur functions


9:15-10am, coffee, tea and bagels (just outside SAS 1102)

10-11am, John Stembridge (University of Michigan), ``A finiteness theorem for W-graphs''

11-11:30am, coffee break

11:30am-12:30pm, Prakash Belkale (UNC Chapel Hill), ``Combinatorial questions related to the Hermitian eigenvalue problem''

2:30-3:30pm, Stephanie van Willigenburg (UBC), ``Quasisymmetric refinements of Schur functions''

3:30-4pm, coffee break

4-5pm, Vic Reiner (University of Minnesota), ``P-partitions revisited''

Evening: dinner and games night (boggle, scrabble, etc.)

Parking: You may park right outside SAS Hall for free. Here is a map of the campus with parking lots marked. On Saturdays, you can park anywhere on campus that is not specifically marked as being restricted (e.g. handicap spots are still off limits). We are hopeful that you won't need any lot except the one by SAS Hall. SAS Hall is at the right of the map, just left of the compass, and its parking lot is a large one shaded red on the map. A back-up option for parking is the coliseum parking deck.

The room: SAS Hall 1102 is the room immediately to your right when you enter from the parking lot. If you enter from the courtyard side, go down the long stairway or the elevators.

Hotel recommendations: within short walk of the math department are three hotels: Holiday Inn Brownstone (800-331-7919), Velvet Cloak Inn (919-828-0333) and Cameron Park Inn Bed and Breakfast (919-835-2171). Those with cars might also consider hotels farther away such as various choices on Wake Town Drive, which is near numerous good restaurants; some such hotels (all right next to each other) are Marriott Courtyard (919-821-3400), Hampton Inn (919-828-1813), or Extended Stay America (919-829-7271).

Talk titles and abstracts:

Prakash Belkale, ``Combinatorial questions related to the Hermitian eigenvalue problem.''

I will first review the ``classical" work on the Hermitian eigenvalue problem which characterizes the possible eigenvalues of a sum of Hermitian matrices in terms of the eigenvalues of the summands. By the work of many authors, this problem is related to two other important problems in geometry and representation theory: Schubert calculus of grassmannians (related to the famous combinatorial Littlewood-Richardson rule) and the invariant theory of GL(n). I will discuss some of the combinatorial questions that arise when we generalize to the case of arbitrary groups (such as the symplectic and orthogonal groups), and to maps between groups.

Vic Reiner, ``P-partitions revisited'' (joint work with Valentin Feray)

Counting the linear extensions of a general partially ordered set (poset) is hard. We'll explain a new product formula which works for a certain class of posets, generalizing a formula for forest posets due to Knuth, and its q-generalization by Bjorner and Wachs.

We'll also explain how this formula arises naturally when one re-examines Stanley's P-partitions from the perspective of convex cones and their affine semigroup rings.

John Stembridge, ``A finiteness theorem for W-graphs''

Given a Coxeter group W, a W-graph is a combinatorial structure that encodes a W-module, or more generally, a module for the associated Iwahori-Hecke algebra. Of special interest are the W-graphs that encode the action of the Hecke algebra on its Kazhdan-Lusztig basis, as well as the action on individual cells. Knowing the W-graph allows easy computation of the Kazhdan-Lusztig polynomials.

Previously we isolated a few basic features common to the W-graphs in Kazhdan-Lusztig theory and used these to define the class of "admissible" W-graphs. What makes this class remarkable is that it is amenable to combinatorial analysis and (we hope) classification theorems. In this talk, we will explain a surprisingly simple (but non-constructive) proof that for each finite Coxeter group W, there are only finitely many admissible W-cells (i.e., strongly connected W-graphs). We also plan to report on the feasibility of constructing the Kazhdan-Lusztig W-graph without first computing Kazhdan-Lusztig polynomials.

Stephanie van Willigenberg, ``Quasisymmetric refinements of Schur functions''

Schur functions were introduced early in the last century with respect to representation theory, and since then have become important functions in other areas such as combinatorics and algebraic geometry. They have a beautiful combinatorial description in terms of diagrams, which allows many of their properties to be determined. In this talk we introduce quasisymmetric Schur (QS) functions, which partition Schur functions in a natural way. Furthermore, we show how these QS functions also refine many well known combinatorial Schur function properties. Extending the definition of QS functions, we define skew QS functions, which likewise partition skew Schur functions. We observe how these functions arise in the study of other algebras such as NCQSym and the algebra of Poirier-Reutenauer. This is joint work with Christine Bessenrodt, Jim Haglund, Kurt Luoto and Sarah Mason.

List of pre-registered participants:

Ed Allen, Wake Forest University (NC)
Moa Apagodu, Virginia Commonwealth University (VA)
Alyssa Armstrong, NCSU (NC)
Sami Assaf, MIT (MA)
Eugenia Bakunova, NCSU (NC)
Erin Bancroft, NCSU (NC)
Cammie Smith Barnes, Sweet Briar College (VA)
Prakash Belkale, UNC Chapel Hill (NC)
Hoda Bidkhori, NCSU (NC)
Sarah Birdsong, UNC Charlotte (NC)
Abigail Bishop, NCSU (NC)
Emily Braley, UNC Chapel Hill (NC)
Ruth Davidson, NCSU (NC)
Graham Enos, UNC Charlotte (NC)
Alex Fink, NCSU (NC)
Jim Haglund, University of Pennsylvania (PA)
Allison Hedges, NCSU (NC)
Patricia Hersh, NCSU (NC)
Gabor Hetyei, UNC Charlotte (NC)
Bill Hightower, High Point University (NC)
J.T. Hird, NCSU (NC)
John Hutchens, NCSU (NC)
Austin Jones, Wake Forest University (NC)
Min Kang, NCSU (NC)
John Konvalina, NCSU (NC)
Sonja Mapes, Duke (NC)
Sarah Mason, Wake Forest University (NC)
Jed Mihalisin, Meredith College (NC)
Ezra Miller, Duke (NC)
Kailash Misra, NCSU (NC)
Walter Morris, George Mason University (VA)
Elizabeth Munch, Duke (NC)
Elizabeth Niese, Virginia Tech (VA)
Chris O'Neill, Duke (NC)
Daniel Orr, UNC Chapel Hill (NC)
Gabor Pataki, UNC Chapel Hill (NC)
Robert Proctor, UNC Chapel Hill (NC)
Nathan Reading, NCSU (NC)
Victor Reiner, Minnesota (MN)
Carla Savage, NCSU (NC)
Mike Schuster, NCSU (NC)
Anne Shiu, Duke (NC)
Michael Singer, NCSU (NC)
Rohit Sivaprasad, NCSU (NC)
Richard Stanley, MIT (MA)
John Stembridge, Michigan (MI)
Ernie Stitzinger, NCSU (NC)
Seth Sullivant, NCSU (NC)
Dewey Taylor, Virginia Commonwealth University (VA)
Chris Thunes, NCSU (NC)
Stephanie van Willigenburg, UBC (BC)
Ryan Vinroot, College of William and Mary (VA)
Gopal Viswanathan, NCSU (NC)
Kangkang Wang, Duke (NC)
Matt Willis, UNC Chapel Hill (NC)
Hangjun Xu, Duke (NC)

Organizing committee: Hoda Bidkhori (NCSU), Alex Fink (NCSU), Patricia Hersh (NCSU), Carla Savage (NCSU).