Multiderivations of Coxeter arrangements

Let V be an ℓ-dimensional Euclidean space. Let G⊂O(V) be a finite irreducible orthogonal reflection group. Let ? be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H∈? choose α _H ∈V ^* such that H=ker(α _H ). For each nonnegative integer m, define the derivation module D ^(m) (?)={θ∈Der _S |θ(α _H )∈Sα ^m _H }. The module is known to be a free S-module of rank ℓ by K. Saito (1975) for m=1 and L. Solomon-H. Terao (1998) for m=2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D ^(m) (?). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m−1)h/2)+m _i (1≤i≤ℓ) (when m is odd). Here m ₁≤···≤m _ℓ are the exponents of G and h=m _ℓ+1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G). Some new results concerning the primitive derivation D are obtained in the course of proof of the main result. Oblatum 27-XI-2001 & 4-XII-2001?Published online: 18 February 2002     

A primitive derivation and logarithmic differential forms of Coxeter arrangements

Let W be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case.     

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.     

Modules of Differential Operators of Order 2 on Coxeter Arrangements

The collection of reflection hyperplanes of a finite reflection group is called a Coxeter arrangement. A Coxeter arrangement is known to be free. K. Saito has constructed a basis consisting of invariant elements for the module of derivations on a Coxeter arrangement. We study the module of \(\mathcal{A}\) -differential operators as a generalization of the study of the module of \(\mathcal{A}\) -derivations. In this article, we prove that the modules of differential operators of order 2 on Coxeter arrangements of types A, B and D are free, by exhibiting their bases. We also prove that the modules cannot have bases consisting of only invariant elements. Two keys for the proof of freeness are the “Cauchy-Sylvester theorem on compound determinants” and the “Saito-Holm criterion for freeness.”     

Chordal Coxeter groups

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.      

Derivation Simple Color Algebras and Semisimple Lie Color Algebras

Let K be an algebraically closed field of arbitrary characteristic and Γ an abelian multiplicative group equipped with a bicharacter ε: Γ × Γ → K*. It is proved that, for any finite-dimensional derivation simple color algebra A over K, there exists a simple color algebra S and a color vector space V such that A? S? S^ε(V), where S^ε(V) is the ε-symmetric algebra of V. As an application of this result, a necessary and sufficient condition such that a Lie color algebra is semisimple is obtained.     

Homology representations arising from the half cube, II

In a previous work, we defined a family of subcomplexes of the n-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least k, and we proved that the reduced homology of such a subcomplex is concentrated in degree k−1. This homology module supports a natural action of the Coxeter group W(Dn) of type D. In this paper, we explicitly determine the characters (over C) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group Sn by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of Sn agree (over C) with the representations of Sn on the (k−2)-nd homology of the complement of the k-equal real hyperplane arrangement.     

The non-projective part of the Lie module for the symmetric group

The Lie module of the group algebra F\mathfrakS_n{{F\mathfrak{S}_n}} of the symmetric group is known to be not projective if and only if the characteristic p of F divides n. We show that in this case its non-projective summands belong to the principal block of F\mathfrakS_n{{F\mathfrak{S}_n}} . Let V be a vector space of dimension m over F, and let L ⁿ (V) be the n-th homogeneous part of the free Lie algebra on V; this is a polynomial representation of GL _m (F) of degree n, or equivalently, a module of the Schur algebra S(m, n). Our result implies that, when m ≥ n, every summand of L ⁿ (V) which is not a tilting module belongs to the principal block of S(m, n), by which we mean the block containing the n-th symmetric power of V.     

Semiprime Lie Rings of (Anti-)Symmetric Derivations of Commutative Rings

Let R be a 2-torsion free commutative ring with involution, and δ a nonzero derivation of R. Let S be the set of symmetric elements in R, and let K be the set of anti-symmetric elements in R. In this article, we investigate the semiprimeness of the Lie rings Sδ when δ is symmetric and Kδ when δ is anti-symmetric.     

Characterizations of trees with equal domination parameters

Let G = (V, E) be a graph. A set S ⊆ V is a restrained dominating set, if every vertex not in S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a weak dominating set of G if, for every u in V − S, there exists a v ∈ S such that uv ∈ E and deg u ≥ deg v. The weak domination number of G, denoted by γ_w(G), is the minimum cardinality of a weak dominating set of G. In this article, we provide a constructive characterization of those trees with equal independent domination and restrained domination numbers. A constructive characterization of those trees with equal independent domination and weak domination numbers is also obtained. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 142–153, 2000     

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 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.     

Inducing <Emphasis Type="Italic">W</Emphasis>-graphs

 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     

Module structure of the free Lie ring on three generators

Let Lⁿ denote the homogeneous component of degree n in the free Lie ring on three generators, viewed as a module for the symmetric group S₃ of all permutations of those generators. This paper gives a Krull-Schmidt Theorem for the LⁿL^n: if n > 1n>1 and Lⁿ is written as a direct sum of indecomposable submodules, then the summands come from four isomorphism classes, and explicit formulas for the number of summands from each isomorphism class show that these multiplicities are independent of the decomposition chosen.¶A similar result for the free Lie ring on two generators was implicit in a recent paper of R.M. Bryant and the second author. That work, and its continuation on free Lie algebras of prime rank p over fields of characteristic p, provide the critical tools here. The proof also makes use of the identification of the isomorphism types of \Bbb Z \Bbb Z -free indecomposable \Bbb Z S ₃\Bbb Z S _3-modules due to M. P. Lee. (There are, in all, ten such isomorphism types, and in general there is no Krull-Schmidt Theorem for their direct sums.)     

On packing 3‐vertex paths in a graph

Let G be a connected graph and let eb(G) and λ(G) denote the number of end‐blocks and the maximum number of disjoint 3‐vertex paths Λ in G. We prove the following theorems on claw‐free graphs: (t1) if G is claw‐free and eb(G) ≤ 2 (and in particular, G is 2‐connected) then λ(G) = ⌊| V(G)|/3⌋; (t2) if G is claw‐free and eb(G) ≥ 2 then λ(G) ≥ ⌊(| V(G) | − eb(G) + 2)/3 ⌋; and (t3) if G is claw‐free and Δ^*‐free then λ(G) = ⌊| V(G) |/3⌋ (here Δ^* is a graph obtained from a triangle Δ by attaching to each vertex a new dangling edge). We also give the following sufficient condition for a graph to have a Λ‐factor: Let n and p be integers, 1 ≤ p ≤ n − 2, G a 2‐connected graph, and |V(G)| = 3n. Suppose that G − S has a Λ‐factor for every S ⊆ V(G) such that |S| = 3p and both V(G) − S and S induce connected subgraphs in G. Then G has a Λ‐factor. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 175–197, 2001     

Reducible Families of Curves with Ordinary Multiple Points on Surfaces in IP3 C

For a Coxeter system (G, S) the multi-parametric alternating subalgebra H ⁺(G) of the Hecke algebra and the alternating subgroup ?⁺(G) of the braid group are defined. Two presentations for H ⁺(G) and ?⁺(G) are given; one generalizes the Bourbaki presentation for the alternating subgroups of Coxeter groups, another one uses generators related to edges of the Coxeter graph.     

Involutive automorphism of symmetric groups

Let (𝔖_n,S) be a Coxeter system of the symmetric group, we show that the set of automorphisms of 𝔖_n which are involutions and leave S stable is a finite group of order less than 3.     

The actions ofS n+1 andS n on the cohomology ring of a coxeter arrangement of typeA n−1

Let be the complexified Coxeter arrangement of hyperplanes of typeA _n−1. In this paper we construct anS _n+1 extension of the naturalS _n action on the complex cohomology ring of the complement ofA _n−1. Recurrence formulas connecting characters with respect to theS _n and theS _n+1 action are given.