A Unified Treatment of the Geometric Algebra of Matroids and Even Δ-Matroids

The concept of acombinatorial(W; P; U)-geometryfor a Coxeter groupW, a subsetPof its generating involutions and a subgroupUofWwithP  Uyields the combinatorial foundation for a unified treatment of the representation theories of matroids and of even Δ-matroids. The concept of a (W, P)-matroid as introduced by I. M. Gelfand and V. V. Serganova is slightly different, although for many important classes ofWandPone gets the same structures. In the present paper, we extend the concept of the Tutte group of an ordinary matroid to combinatorial (W; P; U)-geometries and suggest two equivalent definitions of a (W; P; U)-matroid with coefficients in a fuzzy ringK. While the first one is more appropriate for many theoretical considerations, the second one has already been used to show that (W; P; U)-matroids with coefficients encompass matroids with coefficients and Δ-matroids with coefficients.     

Coxeter matroid polytopes

If Δ is a polytope in real affine space, each edge of Δ determines a reflection in the perpendicular bisector of the edge. The exchange groupW (Δ) is the group generated by these reflections, and Δ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The Gelfand-Serganova Theorem and the structure of the exchange group both give us information about the matroid polytope. We then specialize this information to the case of ordinary matroids; the matroid polytope by our definition in this case turns out to be a facet of the classical matroid polytope familiar to matroid theorists. This work was supported in part by NSA grant MDA904-95-1-1056.     

On outer automorphism groups of coxeter groups

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.     

Strong Reality Properties of Normalizers of Parabolic Subgroups in Finite Coxeter Groups

Let W be a finite Coxeter group, P a parabolic subgroup of W, and N _W (P) the normalizer of P in W. We prove that every element in N _W (P) is strongly real in N _W (P), and that every irreducible complex character of N _W (P) has Frobenius-Schur indicator 1.     

A Decomposition of the Descent Algebra of a Finite Coxeter Group

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type A _n and B _n. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/W _J, where the W _J are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.     

Atomic domains in which almost all atoms are prime

ABSTRACT

Let W be a (finite or infinite) Coxeter group and W _X be a proper standard parabolic subgroup of W. We show that the semilattice made up by W equipped with the weak order is a semidirect product of two smaller semilattices associated with W _X .

     

Locally Finite Groups with All Subgroups Normal-by-(Finite Rank)

A group is said to have finite (special) rank ≤ sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/H_Ghas finite rank for all subgroupsHofG, whereH_Gdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/H_G, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3).     

On the Fully Commutative Elements of Coxeter Groups

Let W be a Coxeter group. We define an element w ε W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose members are all fully commutative. As applications of the latter, we classify all parabolic quotients with the property that (1) the Bruhat ordering is a lattice, (2) the Bruhat ordering is a distributive lattice, (3) the weak ordering is a distributive lattice, and (4) the weak ordering and Bruhat ordering coincide. Partially supported by NSF Grants DMS-9057192 and DMS-9401575.     

Elementary theories of completely simple semigroups

The connections between first-order formulas over a completely simple semigroupC and corresponding formulas over its structure groupH are found in this paper. For the case of finite sandwich-matrix the criterion of decidability of the elementary theoryT(C) is established in terms of the elementary theory ofH in the enriched signature (Theorem 1). For the general case the criterion is established in terms of two-sorted algebraic systems (Theorem 2). Sufficient conditions in terms ofH for decidability and for undecidability ofT(C) are outlined. Corollaries and examples are presented, among them an example of a completely simple semigroup with a finite structure group and with undecidable elementary theory (Theorem 3).     

The Enumeration of Coxeter Elements

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.     

On the Topology of the Combinatorial Flag Varieties

We prove that the simplicial complex Ω _n of chains of matroids (with respect to the ordering by the quotient relation) on n elements is shellable. This follows from a more general result on shellability of the simplicial complex of W -matroids for an arbitrary finite Coxeter group W , and generalises the well-known results by Solomon—Tits and Bj?rner on spherical buildings. Received January 16, 2000, and in revised form October 7, 2000, and April 16, 2001. Online publication December 21, 2001.     

Euler Characteristics of Coxeter Groups, PL-Triangulations of Closed Manifolds, and Cohomology of Subgroups of Artin Groups

The motivation for the theory of Euler characteristics of groups,which was introduced by C. T. C. Wall [21], was topology, butit has interesting connections to other branches of mathematicssuch as group theory and number theory. This paper investigatesEuler characteristics of Coxeter groups and their applications.In his paper [20], J.-P. Serre obtained several fundamentalresults concerning the Euler characteristics of Coxeter groups.In particular, he obtained a recursive formula for the Eulercharacteristic of a Coxeter group, as well as its relation tothe Poincaré series (see 3). Later, I. M. Chiswell obtainedin [10] a formula expressing the Euler characteristic of a Coxetergroup in terms of orders of finite parabolic subgroups (Theorem1). These formulae enable us to compute Euler characteristicsof arbitrary Coxeter groups. On the other hand, the Euler characteristics of Coxeter groupsW happen to be intimately related to their associated complexesF_W, which are defined by means of the posets of nontrivial parabolicsubgroups of finite order (see 2.1 for the precise definition).In particular, it follows from the recent result of M. W. Davis[13] that if F_W is a product of a simplex and a generalizedhomology 2n-sphere, then the Euler characteristic of W is zero(Corollary 3.1). The first objective of this paper is to generalizethe previously mentioned result to the case when F_W is a PL-triangulationof a closed 2n-manifold which is not necessarily a homology2n-sphere. In other words (as given below in Theorem 3), ifW is a Coxeter group such that F_W is a PL-triangulation of aclosed 2n-manifold, then the Euler characteristic of W is equalto 1–(F_W)/2.     

A subalgebra of 0-Hecke algebra

Let (W,I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [A. Berenstein, D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases, in: Quantum Groups, in: Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 13–88] constructed a monoid structure on the set of all subsets of I using unipotent χ-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.     

A Hecke Algebra Quotient and Some Combinatorial Applications

Let (W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient \-H of H and show that it has a basis parametrized by a certain subset W _cof the Coxeter group W. Specifically, W _cconsists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommuting generators in S. We determine which Coxeter groups have finite W _cand compute the cardinality of W _cwhen W is a Weyl group. Finally, we give a combinatorial application (which is related to the number of reduced expressions for w W _cof an exponential formula of Lusztig which utilizes a specialization of a subalgebra of \-H.     

Inducing W-graphs II

Let be the Hecke algebra associated with a Coxeter group W, and the Hecke algebra associated with W_J, a parabolic subgroup of W. In [5] an algorithm was described for the construction of a W-graph for an induced module where V is an -module derived from a W_J-graph. This note is a continuation of [5], and involves the following results:[] inducing ordered and bipartite W-graphs;[] the relationship between the cell decomposition of a W_J-graph and the cell decomposition of the corresponding induced W-graph;[] a Mackey-type formula for the restriction of an induced W-graph;[] a formula relating the polynomials used in the construction of induced W-graphs to Kazhdan-Lusztig polynomials.The result on cells is a version of a Theorem of M. Geck [4], dealing with cells in W (allowing unequal parameters).Mathematics Subject Classification (2000): Primary 20C     

A geometric characterization of Coxeter matroids

Coxeter matroids, introduced by Gelfand and Serganova, are combinatorial structures associated with any finite Coxeter group and its parabolic subgroup they include ordinary matroids as a specia case. A basic result in the subject is a geometric characterization of Coxeter matroids first stated by Gelfand and Serganova. This paper presents a self-contained, simple proof of a more general version of this geometric characterization.     

On the points-lines-planes conjecture

It has been conjectured that in any matroid, if W₁, W₂, W₃ denote the number of points, lines, and planes respectively, then W₂² ≥ W₁W₃. We prove this conjecture (and some strengthenings) for matroids in which no line has five or more points, thus generalizing a result of Stonesifer, who proved it for graphic matroids.     

Geometrization of 3-dimensional Coxeter orbifolds and Singer’s conjecture

Associated to any Coxeter system (W, S), there is a labeled simplicial complex L and a contractible CW-complex Σ _L (the Davis complex) on which W acts properly and cocompactly. Σ _L admits a cellulation under which the nerve of each vertex is L. It follows that if L is a triangulation of , then Σ _L is a contractible n-manifold. In this case, the orbit space, K _L := Σ _L /W, is a Coxeter orbifold. We prove a result analogous to the JSJ-decomposition for 3-dimensional manifolds: Every 3-dimensional Coxeter orbifold splits along Euclidean suborbifolds into the characteristic suborbifold and simple (hyperbolic) pieces. It follows that every 3-dimensional Coxeter orbifold has a decomposition into pieces which have hyperbolic, Euclidean, or the geometry of . (We leave out the case of spherical Coxeter orbifolds.) A version of Singer’s conjecture in dimension 3 follows: That the reduced ℓ ²-homology of Σ _L vanishes.