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1.
We study a new monoid structure for Artin groups associated with finite Coxeter systems. Like the classical positive braid monoid, the new monoid is a Garside monoid. We give several equivalent constructions: algebraically, the new monoid arises when studying Coxeter systems in a “dual” way, replacing the pair (W,S) by (W,T), with T the set of all reflections; geometrically, it arises when looking at the reflection arrangement from a certain basepoint. In the type A case, we recover the monoid constructed by Birman, Ko and Lee.  相似文献   

2.
A W-graph for a Coxeter group W is a combinatorial structure that encodes a module for the group algebra of W, 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. In previous work, 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. The main result of this paper resolves one of the basic question about admissible W-graphs: there are only finitely many admissible W-cells (i.e., strongly connected admissible W-graphs) for each finite Coxeter group W. Ultimately, the finiteness depends only on the fact that admissible W-graphs have nonnegative integer edge weights. Indeed, we formulate a much more general finiteness theorem for “cells” in finite-dimensional algebras which in turn is fundamentally a finiteness theorem for nonnegative integer matrices satisfying a polynomial identity.  相似文献   

3.
《Journal of Algebra》2003,259(2):613-627
In [J. Amer. Math. Soc. 5 (1992) 805–851] Stanley introduced the concept of a P-kernel for any locally finite partially ordered set P. In [Proc. Sympos. Pure Math., Vol. 56, AMS, 1994, pp. 135–148] Du introduced, for any set P, the concept of an IC basis. The purpose of this article is to show that, under some mild hypotheses, these two concepts are equivalent, and to characterize, for a given Coxeter group W, partially ordered by Bruhat order, the W-kernel corresponding to the Kazhdan–Lusztig basis of the Hecke algebra of W. Finally, we show that this W-kernel factorizes as a product of other W-kernels, and that these provide a solution to the Yang–Baxter equations for W.  相似文献   

4.
Associated with any Coxeter group is a Coxeter monoid, which has the same elements, and the same identity, but a different multiplication. (Some authors call these Coxeter monoids 0-Hecke monoids, because of their relation to the 0-Hecke algebras—the q=0 case of the Hecke algebra of a Coxeter group.) A Coxeter group is defined as a group having a particular presentation, but a pair of isomorphic groups could be obtained via non-isomorphic presentations of this form. We show that when we have both the group and the monoid structure, we can reconstruct the presentation uniquely up to isomorphism and present a characterisation of those finite group and monoid structures that occur as a Coxeter group and its corresponding Coxeter monoid. The Coxeter monoid structure is related to this Bruhat order. More precisely, multiplication in the Coxeter monoid corresponds to element-wise multiplication of principal downsets in the Bruhat order. Using this property and our characterisation of Coxeter groups among structures with a group and monoid operation, we derive a classification of Coxeter groups among all groups admitting a partial order.  相似文献   

5.
 Let ℋ be the Hecke algebra associated with a Coxeter group W. Many interesting ℋ-modules can be described using the concept of a W-graph, as introduced in the influential paper [4] of Kazhdan and Lusztig. In particular, Kazhdan and Lusztig showed that the regular representation of ℋ has an associated W-graph. The purpose of this note is to show that if W J is a parabolic subgroup of W and V is a module for the corresponding Hecke algebra ℋ J , then a W J -graph structure for V gives rise to a W-graph structure for the induced module ℋ⊗ ℋJ V. In the case that W J is the identity subgroup and V has dimension 1, our construction coincides with that given by Kazhdan and Lusztig for the regular representation. For arbitrary J and V of dimension 1 we recover the constructions of Couillens [1] and Deodhar [3]. Received: 14 June 2002; in final form: 13 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 20C08  相似文献   

6.
For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I2(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G2, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in Bn-, Dn- and G2-bracket algebras, and as an application, discover a Pieri-type formula in the Bn-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary Bn-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.  相似文献   

7.
In 1979 Kazhdan and Lusztig defined, for every Coxeter group W, a family of polynomials, indexed by pairs of elements of W, which have become known as the Kazhdan-Lusztig polynomials of W, and which have proven to be of importance in several areas of mathematics. In this paper, we show that the combinatorial concept of a special matching plays a fundamental role in the computation of these polynomials. Our results also imply, and generalize, the recent one in [Adv. in Math. 180 (2003) 146-175] on the combinatorial invariance of Kazhdan-Lusztig polynomials.  相似文献   

8.
We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type A n – 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 Shi's 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 GL n ()  相似文献   

9.
Let (W,S, ) be a Coxeter system: a Coxeter group W with S its distinguished generator set and its Coxeter graph. In the present paper, we always assume that the cardinality l=|S| ofS is finite. A Coxeter element of W is by definition a product of all generators s S in any fixed order. We use the notation C(W) to denote the set of all the Coxeter elements in W. These elements play an important role in the theory of Coxeter groups, e.g., the determination of polynomial invariants, the Poincaré polynomial, the Coxeter number and the group order of W (see [1–5] for example). They are also important in representation theory (see [6]). In the present paper, we show that the set C(W) is in one-to-one correspondence with the setC() of all acyclic orientations of . Then we use some graph-theoretic tricks to compute the cardinality c(W) of the setC(W) for any Coxeter group W. We deduce a recurrence formula for this number. Furthermore, we obtain some direct formulae of c(W) for a large family of Coxeter groups, which include all the finite, affine and hyperbolic Coxeter groups.The content of the paper is organized as below. In Section 1, we discuss some properties of Coxeter elements for simplifying the computation of the value c(W). In particular, we establish a bijection between the sets C(W) andC() . Then among the other results, we give a recurrence formula of c(W) in Section 2. Subsequently we deduce some closed formulae of c(W) for certain families of Coxeter groups in Section 3.  相似文献   

10.
In our recent paper (Douglass et al. (2011)), we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik–Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note, we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.  相似文献   

11.
The growth series W(t) of a Coxeter system (W, S) is always a rational function. We prove that for a very general class of infinite Coxeter groups, this function satisfies W(t –1)=±W(t).Partially supported by NSF grant DMS-8905378.  相似文献   

12.
Yunchuan Yin 《代数通讯》2013,41(2):547-565
ABSTRACT

The “W-graph” concept was introduced by Kazhdan and Lusztig in their influential article Kazhdan and Lusztig (1979 Kazhdan , D. , Lusztig , G. ( 1979 ). Representations of Coxeter groups and Hecke algebras . Invent. Math. 53 : 165184 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] [Google Scholar]). If W is a Coxeter group, then a W-graph provides a method for constructing a matrix representation of the Hecke algebra ? associated with W (the degree of the representation being the number of vertices of the W-graph). The aim of this note is to explicitly construct all the irreducible representations of ? when W is of type D 4 and D 5.  相似文献   

13.
Given a finite Coxeter system (W,S) and a Coxeter element c, or equivalently an orientation of the Coxeter graph of W, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan Fc, settling a conjecture of Reading that this is possible. We call this polytope the c-generalized associahedron. Our approach generalizes Loday's realization of the associahedron (a type A c-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the c-singleton cones, the cones in the c-Cambrian fan which consist of a single maximal cone from the Coxeter fan.Moreover, if W is a Weyl group and the vertices of the permutahedron are chosen in a lattice associated to W, then we show that our realizations have integer coordinates in this lattice.  相似文献   

14.
Summary It is shown that the outer automorphism group of a Coxeter groupW of finite rank is finite if the Coxeter graph contains no infinite bonds. A key step in the proof is to show that if the group is irreducible andΠ 1 andΠ 2 any two bases of the root system ofW, thenΠ 2 = ±ωΠ 1 for some ω εW. The proof of this latter fact employs some properties of the dominance order on the root system introduced by Brink and Howlett. This article was processed by the author using the Springer-Verlag TEX PJour1g macro package 1991.  相似文献   

15.
A solution of the isomorphism problem is presented for the class of Coxeter groups W that have a finite set of Coxeter generators S such that the underlying graph of the presentation diagram of the system (W,S) has the property that every cycle of length at least four has a chord. As an application, we construct counterexamples to two conjectures concerning the isomorphism problem for Coxeter groups.   相似文献   

16.
Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.  相似文献   

17.
Koji Nuida 《代数通讯》2013,41(7):2559-2595
In this article, we prove that any irreducible Coxeter group of infinite order, which is possibly of infinite rank, is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W, the centralizers in W of the normal subgroups of W that are generated by involu-tions. As a consequence, the problem of deciding whether two general Coxeter groups are isomorphic is reduced to the case of irreducible ones. We also describe the automorphism group of a general Coxeter group in terms of those of its irreducible components.  相似文献   

18.
19.
Let X* be a free monoid over an alphabet X and W be a finite language over X. Let S(W) be the Rees quotient X*/I(W), where I(W) is the ideal of X* consisting of all elements of X* that are not subwords of W. Then S(W) is a finite monoid with zero and is called the discrete syntactic monoid of W. W is called finitely based if the monoid S(W) is finitely based. In this paper, we give some sufficient conditions for a monoid to be non-finitely based. Using these conditions and other results, we describe all finitely based 2-limited words over a three-element alphabet. Furthermore, an explicit algorithm is given to decide that whether or not a 2-limited word in which there are exactly two non-linear letters is finitely based.  相似文献   

20.
We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated. We describe in terms of our model the crystal structure and R-matrix of the affine geometric crystal of products of symmetric and dual symmetric powers of type A. Local realizations of the R-matrix and crystal actions are used to construct a double affine geometric crystal on a torus, generalizing the commutation result of Kajiwara et al. (Lett Math Phys, 60(3):211–219, 2002) and an observation of Berenstein and Kazhdan (MSJ Mem, 17:1–9, 2007). We show that our model on a cylinder gives a decomposition and parametrization of the totally non-negative part of the rational unipotent loop group of GL n .  相似文献   

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