Over a couple weeks, Tom Denton presented Chari and Pressley's classification of finite dimensional irreps for the quantum group assoiated to affine sl_2. Their construction relies heavily on a realization of the quantum group of the affine algebra U_q(g^) as a space of maps from U_q(g) to C^x, and extends to any untwisted infinite dimensional Lie algebra. Using this construction and some direct computation, it was shown that a particular combinatorial crystal for U_q(sl_2^) could not arise from an actual module. This analysis is a small step towards completing the result of Bandlow, Schilling, and Thiery classifying the combinatorial crystals of Killirov-Reshitikin modules for U_q(sl_n^) by ruling out exceptional cases arising from non-standard promotion operators.

References:

Chari, Pressley, "Quantum Affine Algebras and their Representations."

Bandlow, Schilling, Thiery, "On the uniqueness of promotion operators on tensor products of type A crystals."

Professor Schilling presented recent collaboarative work with Okado on combinatorial models of K-R Crystals using automorphisms of Dynkin Diagrams and virtual crystals. The diagram automorphisms were presented, and the Type A case (realized by the promotion operator on crystals of rectangular tableaux) was explicitly constructed. A link to the paper will be made available when it hits the Arxiv.

On September 2nd and 8th, Brant Jones presented some new combinatorics for the Kazhdan-Lusztig polynomials involving pattern-avoiding permutations and the 'heap' of an element, used in to construct the Lascoux--Schutzenberger trees, which in turn are used to calculate the K-L polynomials. He then presented a conjecture by Alex Woo on how to compute the mask-sets that determine the polynomial using the same heap construction.

References:

Introductory material for the Kazhdan--Lusztig polynomials cab be found in Humphreys (Ch. 7) and Bjorner--Brenti (Ch. 5).

Deodhar, "A combinatorial setting for questions in Kazhdan--Lusztig theory." Geom. Dedicata 36 (1990) no. 1, pg. 95-119.

Brenti, "Kazhdan--Lusztig and R-polynomials from a combinatorial point of view." Discrete Mathematics 193 (1998), pg. 93-116.

August 19&26: Tom Denton presented material from Hong and Kang (Ch 10) on perfect crystals and the path model for constructing highest-weight modules of the quantum enveloping algebra of an affine Lie groups.

Spring 2008 saw two Sage-Combinat development parties, one hosted at MSRI and the second here in Davis. Jason Bandlow helped us navigate the Mercurial patch system, documented here. Participants made significant progress developing Sage code relating to crystal bases, root systems, and Hecke algebras. Participants included Jason Bandlow, Nicolas Borie, Tom Denton, Mike Hansen, Florent Hivert, Brant Jones, Anne Schilling, Mark Shimozono, Steve Pon, and Qiang Wang.

Jason Bandlow presented joint work on the promotion operator, which crystallized into the following paper:

Bandlow, Schilling, Thiery: On the uniqueness of promotion operators on tensor products of type A crystals.

Nicolas Borie presented computational work and a conjecture on the structure of Hecke algebras at q equal to a root of unity. (Slides (Fr).)

Reference:

Hivert, Schilling, Thiery: Hecke group algebras as degenerate affine Hecke algebras

Steve Pon presented work on the Affine Stanley Symmetric functions, and refined his quals talk.

References:

T. Lam: Affine Stanley symmetric functions; math.CO/0501335 2005.

T. Lam: Schubert Polynomials for the affine Grassmannian; math.CO/0603125 2006.

T. Lam, A. Schilling, M. Shimozono: Schubert Polynomials for the affine Grassmannian of the symplectic group; math.CO/07102720 2007.

Tom Denton presented two papers on Macdonald polynomials, comparing and contrasting the combinatorial formula developed by Ram and Yip with that of Mark Haiman.

References:

Haiman: "Cherednik algebras, Macdonald polynomials and combinatorics" (from Haiman's website.)

Ram, Yip: "A combinatorial formula for Macdonald polynomials"

Ram: "Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux"

Ram: "Seminormal representations of Weyl groups and Iwahori-Hecke algebras"

Michelle Snider, student of Allen Knutson at San Diego, gave a presentation entitled, 'Positivity of K-Theoretic Littlewood-Richardson Numbers: Unifying Two Approaches.'

Abstract: "We consider Buch’s rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Combining geometric results of Brion and Ramanathan, one sees that Buch’s coefficients are related to Mobius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials."