Andrew Berget continued his discussion of letter place algebras today, first reviewing the basic construction as a super algebra on a set of letters and a set of places, and then getting into new constructions. The Whitney Algebra of a matroid was described in detail, culminating in a recent result on the images of tableaux in the Whitney Algebra of a realizable matroid in characteristic zero. A brief description of the Schubert Matroid was then given as an illustration of the theorem.

Andrew Berget presented an introduction to Letter Place Algebras, which can be approached from a number of different angles.

In the abstract, a letter place algebra is the result of taking two signed, ordered alphabets L and P and constructing a new alphabet whose letters are (e|p) for e in L and p in P, where the sign of (e|p) is the sum of the signs of e and p. Then the Letter-Place algebra is the super Lie algebra on this new alphabet, Super(L|P). This structure is naturally a Hopf Algebra.

Letter Place Algebras generalize tableaux in the following sense. Let T be a filling of a Young diagram with letters in L and S be a filling of a Young Diagram with letters in P. Then the pair (T|S) can be considered as a product of elements in Super(L|P); in fact, the Hopf Algebra structure dictates that this product will be zero if the Young Diagrams are not of the same shape. This is a useful construction, because it turns out that pairs (T|S) give an integral basis of Super(L|P). Specializing to the case where both L and P are the set of natural numbers, then these pairs of tableaux become pairs of semistandard Young Tableaux.

Andrew then described the Whitney Algebra of a matroid as a quotient of a letter-place algebra by a suitable quotient describing the dependent sets of the matroid. Next week, an application will be given to the Schubert Matroid, constructed from the Schubert cells in the complex Grassmannian.

Tom Denton presented work from the last couple of months on a formula for a family of orthogonal idempotents in the Zero-Hecke Algebra of the symmetric group.

He has finally managed to prove the formula that was conjectured following computer experimentation in the Spring, and demonstrated a branching rule for moving from the idempotents for the H_0(S_N) to idempotents for H_0(S_{n+1}).

The formula begins with a 'signed Dynkin Diagram,' obtained by attaching a plus or minus sign to each of the nodes of the Dynkin Diagram for S_N. Then a fairly simple recipe can be applied to obtain an 'idempotent candidate' associated to any given signed Dynkin Diagram.

The formula is still not quite perfect: each candidate decomposes into an idempotent part and a nilpotent part, such that the candidate must be raised to some power in order to obtain the actual idempotent. Some progress has been made on determining to what power the candidate must be raised (it's a number between 1 and N-2), but there is not an obvious way to read off this exponent from the diagram in most cases.

Brant Jones continued his discussion from last week by giving the explicit construction of the mask matching for computing R-polynomials. The process involves first choosing an element and reduced expression, and then finding the unique `constant mask' associated to the pair. For another element sitting under the first in Bruhat order, the uniqueness of the first mask allows the computation of a mask for the new element, derived from the first mask. The process creates a function from masks to masks, which turns out to be a matching.

After Brant's presentation, new postdoc Andrew Berget gave a very brief introduction to Letter Place Algebras, which will be continued in two weeks.

Brant Jones presented his work on assembling a new proof for an oft-proven theorem: For an interval [y,w] in the Bruhat order of a Coxeter system (W,S), the mobius function is given by (-1)^(l(w)-l(x)). The proof relies on providing a 'special matching' on the Hasse diagram of the Coxeter Group's Bruhat order.

The matching is derived by using 'masks' to embed the Bruhat order into a lattice, and then creating a special matching in the lattice that descends to the Bruhat order.

This new matching is potentially of some further significance: a theorem of Brenti states that for any special matching on the interval [id,w], then a particular recurrence in terms of the matching holds on the R polynomials. What's an R polynomial? It's a polynomial indexed by an interval in the Bruhat order which is useful for explicitly computing the Kazhdan-Lusztig polynomials. Brenti's result shows that the Kazhdan-Luztig polynomials for intervals [id,w] depend only on the poset structure of the Bruhat order. The 'Combinatorial Invariance Conjecture' conjects that all K-L polynomials depend only on the poset structure.

The new matching allows the computation of R polynomials for arbitrary [x,w] intervals, and thus gives a way to compute polynomials that may or may not be the K-L polynomials, depending on the truth of the Combinatorial Invariance Conjecture. However, computational evidence for S_N with N<=7 suggests that the new matching is indeed giving K-L polynomials.

Various props were given to the work of Drew Armstrong, whose work on expanding the Bruhat order is closely related to these results.

Steve Pon gave a talk describing his work on Affine Stanley Symmetric Functions. He developed a combinatorial description based on the notions of weak strips and weak tableaux, which are defined as sequences of weak strips. A weak strip is a pair of elements (u,v) such that u<=v in the weak Bruhat order and vu^{-1} is a cyclically decreasing element of the affine symmetric group. The weak tableau is then an eventually-stable sequence of weak strips

He then reviewed another combinatorial description of Affine Stanley Symmetric Functions in terms of Grassmannian elements and k-tableaux. These methods can be found in the lecture notes from Prof. Schilling's Schubert Calculus course notes from last year.

Steve is currently working on a formulation of k-tableaux for affine type C, in hopes of creating an analogous definition for Affine Stanley Symmetric Functions in that context.

Qiang Wang gave a presentation on recent work with Cyclic Sieving Phenomenon (CSP) and the promotion operator. A CSP consists of a cyclic group G of order n, a set S being acted upon by the group, and a polynomial P chosen according to the following conditions. Let z be a primitive nth root of unity and g a generator of G. Then we have a function P0 such that P(k) is the number of fixed points of g^k under the action of G on S. If P0 extends to a polynomial P, then the triple (G,S,P) is a CSP. A particularly good CSP includes an additional function \mu such that P(x)=sum( [ x^(\mu(s) for s in S ] ). (Please forgive the mixing of Sage and Latex code!)

As an example, the promotion operator, the set of rectangular MxN standard Young tableaux, and \mu=maj(T)+constant(sh(T)) determine a CSP.

Wang has been working on identifying a CSP for tableaux of staircase shape. An outcropping of this work includes a very interesting formula for the number of rectangular tableau of a given shape in terms of specialization of a product of cyclotomic polynomials. Wang is also working on proving this formula and finding an extension to the case of staircase shape.

Cory Brunson, Mark Shimozono's student from Virginia Tech presented work on describing the homology of the affine Grassmannian. This involves taking a Schubert variety in the affine Grassmannian, embedding it into the infinite Grassmannian, identifying that embedding with a variety in a finite Grassmannain, and then looking at a matrix representative of that variety. Then working in the other direction, he identifies the equivariant cohomology of a point, and pushes the homological information back through the prior chain of maps to obtain a k-Schur function corresponding to the variety we started with at the start of this paragraph.

A nice combinatorial result of this sequence of embeddings is that the finite-dimensional Grassmannian variety that one obtains from an affine Grassmannian variety is indexed by the same n-core associated to its affine permutation.

Nicolas Borie presented a plan of research for finding a generating set for invariant polynomials in N variables under the action of an arbitrary subroup G of the permutation group S_N. A polynomial is invariant if it is fixed by the action of G by permutation of the variables. Noether had a result placing an upper bound on the number of generating polynomials needed; later Stanley showed that one needs N! over the order of G 'secondary invariants' in addition to the usual symmetric polynomials in N variables.

A major hurdle in computing invariants is the multiplication of long, multivariable polynomials, which is very slow in general. However, a kind of Fourier Transform-like operation should allow one to see whether a given polynomial is a product of lower degree polynomials or not, and thus save a huge amount of computation time.

Tom Denton presented recent work on non-decreasing parking functions (NDPF) in types A and B, which can be realized as a certain quotient of the zero-Hecke Algebra. In particular, computer evidence strongly suggests that the fibers of the quotient in Type A contain a unique [231]-avoiding permutation, and that certain fibers in the Type B case contain a unique [4321]-avoiding permutation.

A brief overview of a paper by Nathan Reading and David Speyer was also given. In their paper, they construct a map from an arbitrary element of a Weyl group to a unique 'c-sorting' element below it in the weak Bruhat order. In the Type A case, with a particular choice of c, this corresponds to finding a unique [231]-avoiding permutation. However, this initially seems not to be the same correspondence suggested by the NDPF quotient. Further work may reconcile the differences, though.

Illustration: Cayley Graph of the Type B3 NDPF.

Anne Schilling presented recent ideas on connections between the Affine Local Plactic Algebra and the Killirov-Reshsetikhin crystals of Type A. The starting point was the phase model from physics, which consists of N labelled 'sites' on a circular path, where some number of particles sits at each site. She then introduce the 'hopping operators,' which move a particle from site i to site i+1 mod N. At this point, we identify the phase model on N sites with the Fock space, and establish that these hopping operators satisfy the relations of the Affine Local Plactic Algebra (ALPA). A theorem of Postnikov, Stoppel, and Koff gives a relationship between this algebra and the 'level-k fusion coefficients.' Anne then goes on to describe an isomorphism between the action of the ALPA on the Fock Space and certain Kirillov-Reshetikhin crystals.

Brant Jones presents recent work with Anne Schilling concerning the construction of affine crystals for types E6 and E7. This is accomplished using symmetries of the Dynkin Diagams for the affine types, combined with branching rules that restrict what can happen to the zero-arrows in these affine crystals. The other key ingredient is a process for building up arbitrary classical crystals of Type E6 or E7 using one of a few fundamental crystals. For E6 there are two fundamental crystals, connected by an arrow-reversing duality, and for E7 there is just one fundamental crystal. Open questions abound, and the work continues apace.

Nicolas Thiery (with occasional shouts from Florent Hivert) outlined work in implementing mathematical categories in Sage. This rather large undertaking would involve altering the structure of Sage itself, creating a system of Categories working alongside the existing object oriented structure. This new structure would eliminate a good deal of redundant code, and would possibly streamline the coercion system and perhaps even make things run faster. The downside is that it uses a new Python feature - dynamic classes - which hasn't been fully vetted upstream.

Tom Denton will present recent work in identifying orthogonal idempotents in the 0-Hecke Algebra. A basis for the algebra H_0(N) is indexed by permutations of N, and can be realized with the same generators and relations as the symmetric group, but with the simple transpositions made to act as idempotents. Then the generators can be described as the 'bubble anti-sorting operators.'

From the representation theory of monoids, it is easy to show that the monoid generated by these operators contains 2^N idempotent elements, which can be used to obtain the indecomposable projective modules, and then the simple modules in turn. Looking next to the representation theory of the 0-Hecke algebra, constructions exist for the projective modules, but there is no explicit construction of the idempotents that generate these modules. Tom believes he has found a good candidate for an explicit formula for a set of 2^N orthogonal idempotents, which will be presented in this talk.

Steve Pon presented a paper by Richard Stanley on extending promotion and evacuation operators from the setting of tableau to general posets. In particular, one can label a poset with an arbitrary linear extension of the poset, analagous to the linear extension of a poset associated to a Standard Young Tableau. Defining evacuation and dual evacuation, one obtains a transitive action of an infinite Coxeter group on the set of linear extensions of the poset. For special tableau shapes, there are results describing this group action, as well as the action of promotion. However, one can ask whether results can be extended to other tableau shapes or classes of posets.

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."