R. M. Green

GENERALIZED TEMPERLEY–LIEB ALGEBRAS AND DECORATED TANGLES

We give presentations, by means of diagrammatic generators and relations, of the analogues of the Temperley–Lieb algebras associated as Hecke algebra quotients to Coxeter graphs of type B and D. This generalizes Kauffmans diagram calculus for the Temperley–Lieb algebra.

On the Affine Temperley–Lieb Algebras

The paper describes the cell structure of the affine Temperley–Lieb algebra with respect to a monomial basis. A diagram calculus is constructed for this algebra.

Tabular algebras and their asymptotic versions

We introduce tabular algebras, which are simultaneous generalizations of cellular algebras (in the sense of Graham–Lehrer) and table algebras (in the sense of Arad–Blau). We show that if a tabular algebra is equipped with a certain kind of trace map then the algebra has a corresponding asymptotic version whose structure can be explicitly determined. We also study various natural examples of tabular algebras.

On 321-Avoiding Permutations in Affine Weyl Groups

We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type An − 1 by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in W coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of W (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations.Using Shis characterization of the Kazhdan–Lusztig cells in the group W, we use our main result to show that the fully commutative elements of W form a union of Kazhdan–Lusztig cells. This phenomenon has been studied by the author and J. Losonczy for finite Coxeter groups, and is interesting partly because it allows certain structure constants for the Kazhdan–Lusztig basis of the associated Hecke algebra to be computed combinatorially.We also show how some of our results can be generalized to a larger group of permutations, the extended affine Weyl group associated to GLn(ℂ)

Completions of cellular algebras

We introduce procellular algebras, so called because they are inverse limits of finite dimensional cellular algebras as defined by Graham and Lehrer. A procellular algebra is defined as a certain completion of an infinite dimensional cellular algebra whose cell datum is of “proflnite type”. We show how these notions overcome some known obstructions to the theory of cellular agebras in infinite dimensions.

A projection property for Kazhdan-Lusztig bases.

We compare the canonical basis for a generalized Temperley-Lieb algebra of type A or B with the Kazhdan-Lusztig basis for the corresponding Hecke algebra.

Acyclic Heaps of Pieces, I

Heaps of pieces were introduced by Viennot and have applications to algebraic combinatorics, theoretical computer science and statistical physics. In this paper, we show how certain combinatorial properties of heaps studied by Fan and by Stembridge are closely related to the properties of a certain linear map ∂E associated to a heap E. We examine the relationship between ∂E and ∂F when F is a subheap of E. This approach allows neat statements and proofs of results on certain associative algebras (generalized Temperley–Lieb algebras) that are otherwise tricky to prove. The key to the proof is to interpret the structure constants of the aforementioned algebras in terms of the maps ∂.

The Veldkamp Space of the Smallest Slim Dense Near Hexagon

We give a detailed description of the Veldkamp space of the smallest slim dense near hexagon. This space is isomorphic to PG(7,2) and its 28 - 1 = 255 Veldkamp points (that is, geometric hyperplanes of the near hexagon) fall into five distinct classes, each of which is uniquely characterized by the number of points/lines as well as by a sequence of the cardinalities of points of given orders and/or that of (grid-)quads of given types. For each type we also give its weight, stabilizer group within the full automorphism group of the near hexagon and the total number of copies. The totality of (255 choose 2)/3 = 10,795 Veldkamp lines split into 41 different types. We give a complete classification of them in terms of the properties of their cores (i.e. subconfigurations of points and lines common to all the three hyperplanes comprising a given Veldkamp line) and the types of the hyperplanes they are composed of. These findings may lend themselves into important physical applications, especially in view of recent emergence of a variety of closely related finite geometrical concepts linking quantum information with black holes.

On rank functions for heaps

Motivated by work of Stembridge, we study rank functions for Viennots heaps of pieces. We produce a simple and sufficient criterion for a heap to be a ranked poset and apply the results to the heaps arising from fully commutative words in Coxeter groups.

Positivity properties for q-Schur algebras.

We prove Dus positivity conjecture for the canonical basis of the q-Schur algebra, using elementary arguments and the positivity result for Lusztigs canonical basis for U+q(sln). We also describe a family of subalgebras of the q-Schur algebra, each of which is spanned by the canonical basis elements it contains.