We recall Lusztig's construction of the asymptotic Hecke algebra J of a Coxeter system (W, S) via the Kazhdan-Lusztig basis of the corresponding Hecke algebra. The algebra J has a direct summand J E for each two-sided Kazhdan-Lusztig cell of W , and we study the summand J C corresponding to a particular cell C called the subregular cell. We develop a combinatorial method involving truncated Clebsch-Gordan rules to compute J C without using the Kazhdan-Lusztig basis. As applications, we deduce some connections between J C and the Coxeter diagram of W , and we show that for certain Coxeter systems J C contains subalgebras that are free fusion rings in the sense of [3], thereby connecting the subalgebras to compact quantum groups arising from operator algebra theory. 1 2 TIANYUAN XUis the ring referred to in the title of the paper. We study J C and subalgebras J s of J C that correspond to the generators s ∈ S of W . Thanks to a result of Lusztig in [24], the cell C can be characterized as the set of non-identity elements in W with unique reduced words. The main theme of the paper is to exploit this combinatorial characterization and study J C and J s (s ∈ S) without reference to Kazhdan-Lusztig bases. This is desirable since a main obstacle in understanding J for arbitrary Coxeter systems lies in the difficulty of understanding Kazhdan-Lusztig bases.Remark 1.1. It is worth mentioning that the algebra J, as well as the subalgebra J E where E is an arbitrary two-sided cell of W , do not generally have units in the usual sense. More specifically, we need to consider the set D of distinguished involutions of W (see Equation 6) and J has a unit element, namely d∈D t d , only when D is finite. On the other hand, when D is infinite, the set {t d : d ∈ D} may be viewed as a generalized unit element of J in the sense that Section 18.3). Similarly, J E has unit d∈E∩D t d if the set E ∩ D is finite while otherwise the set {t d : d ∈ E ∩ D} may be viewed as a generalized unit of J E . For the subregular cell C, the set C ∩ D turns out to be the generating set S of W , therefore J C is unital (as we will assume S is finite). For each s ∈ S, the algebra J s mentioned above will be unital as well, with the element t s as its unit (see Proposition 3.5 and Remark 3.2). Proposition 3.1 ([23], Lemma 8.2). Let y ∈ W . Then (1) the set H ≤Ly := ⊕ x:x≤Ly Ac x is a left ideal of H; (2) the set H ≤Ry := ⊕ x:x≤Ry Ac x is a right ideal of H; (3) the set H ≤LRy := ⊕ x:x≤LRy Ac x is a two-sided ideal of H. Proof. The definition of ≺ L guarantees that H ≤y is closed under left multiplication by c s for each s ∈ S. Since the elements c s generate H by Proposition 2.6, it follows that H ≤y is a left ideal, proving (1). The proofs of (2) and (3) are similar. Next, we recall two compatibility results on cells and the inverse map on W . Proposition 3.2 ([23], Section 8.1). The map w → w −1 takes left cells in W to right cells, right cells to left cells, and 2-sided cells to 2-sided cells. Proposition 3.3 ([23], Conjecture 14.2). For any w ∈ W , we ...

We consider Lusztig's a-function on Coxeter groups (in the equal parameter case) and classify all Coxeter groups with finitely many elements of a-value 2 in terms of Coxeter diagrams.

There is a strong relationship between proposed frameworks for autonomic computing, such as the IBM Blueprint for Autonomic Computing, and the Web Services Distributed Management (WSDM) standard proposed by OASIS. We examine this relationship through a description of our efforts to implement an autonomic Web service using WSDM. The example autonomic Web service is based on our Autonomic Web Service Environment (AWSE) framework. We explain how WSDM is used to implement the AWSE components and evaluate the results of the exercise. We present the lessons we learned in carrying out the implementation effort and draw general conclusions concerning the relationship between autonomic computing and WSDM.

A Coxeter group is said to be a(2)-finite if it has finitely many elements of a-value 2 in the sense of Lusztig. In this paper, we give explicit combinatorial descriptions of the left, right, and two-sided Kazhdan-Lusztig cells of a-value 2 in an irreducible a(2)-finite Coxeter group. In particular, we introduce elements we call stubs to parameterize the one-sided cells and we characterize the one-sided cells via both star operations and weak Bruhat orders. We also compute the cardinalities of all the one-sided and two-sided cells.

We consider the subgroup of the abelian sandpile group of the grid graph consisting of configurations of sand that are symmetric with respect to central vertical and horizontal axes. We show that the size of this group is (i) the number of domino tilings of a corresponding weighted rectangular checkerboard; (ii) a product of special values of Chebyshev polynomials; and (iii) a double-product whose factors are sums of squares of values of trigonometric functions. We provide a new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of a 2m × 2n rectangular checkerboard and a new way of counting the number of domino tilings of a 2m × 2n checkerboard on a Möbius strip.