The Natural Embedding of Positive Singular Artin Monoids

We explore some combinatorial properties of singular Artin monoids and invoke them to prove that a positive singular Artin monoid of arbitrary Coxeter type necessarily injects into the corresponding singular Artin monoid. This is an extension of L. Pari' result that positive Artin monoids embed in the correpsonding Artin groups: Adjoining inverses of the generators does not produce any new identities between words that do not involve those inverses.     

Universal Upper Bound for the Growth of Artin Monoids

In this paper we study the growth rates of Artin monoids, and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given.     

On the classifying space of Artin monoids

A theorem proved by Dobrinskaya [9 Dobrinskaya, N. E. (2006). Configuration spaces of labeled particles and finite Eilenberg-MacLane complexes. Proc. Steklov Inst. Math. 252(1):30–46.[Crossref] , [Google Scholar]] shows that there is a strong connection between the K(π,1) conjecture for Artin groups and the classifying spaces of Artin monoids. More recently Ozornova obtained a different proof of Dobrinskaya’s theorem based on the application of discrete Morse theory to the standard CW model of the classifying space of an Artin monoid. In Ozornova’s work, there are hints at some deeper connections between the above-mentioned CW model and the Salvetti complex, a CW complex which arises in the combinatorial study of Artin groups. In this work we show that such connections actually exist, and as a consequence, we derive yet another proof of Dobrinskaya’s theorem.     

Artin monoids inject in their groups

We prove that the natural homomorphism from an Artin monoid to its associated Artin group is always injective. Received: March 14, 2002     

A poset connected to Artin monoids of simply laced type

Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representations of the corresponding Artin group A. The poset generalizes many properties of the usual order on positive roots of W given by height. In this paper, a linear representation of the positive monoid of A is defined by use of the poset.     

Relative Hyperbolicity and Artin Groups

Let G=⟨ a₁,&ldots; , a _n | a _i a _j a _i &ldots; = a_ja_ia_j,&ldots; ,i>j⟩ be an Artin group and let m _ij =m _ji be the length of each of the sides of the defining relation involving a _i and a _j . We show if all m_ij ⩾ 7 then G is relatively hyperbolic in the sense of Farb with respect to the collection of its two-generator subgroups a _i , a _j for which m_ij >&infty;.     

An introduction to right-angled Artin groups

Recently, right-angled Artin groups have attracted much attention in geometric group theory. They have a rich structure of subgroups and nice algorithmic properties, and they give rise to cubical complexes with a variety of applications. This survey article is meant to introduce readers to these groups and to give an overview of the relevant literature.      

Lorenzen Monoids: A Multiplicative Approach to Kronecker Function Rings

Using the formalism of module systems on a commutative cancellative monoid, we generalize the classical concept of Lorenzen monoids to obtain a multiplicative model for the semistar Kronecker function ring introduced by Fontana and Loper. We prove a universal mapping property and investigate the generalized Lorenzen monoid from a valuation-theoretic and an ideal-theoretic point of view.     

Presentations for subgroups of Artin groups

Recently, M. Bestvina and N. Brady have exhibited groups that are of type but not finitely presented. We give explicit presentations for groups of the type considered by Bestvina-Brady. This leads to algebraic proofs of some of their results.

     

Relative hyperbolicity and Artin groups

This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a two-dimensional Artin group the Deligne complex is Gromov hyperbolic precisely when the corresponding Davis complex is Gromov hyperbolic, that is, precisely when the underlying Coxeter group is a hyperbolic group. For Artin groups of FC type we give a sufficient condition for hyperbolicity of the Deligne complex which applies to a large class of these groups for which the underlying Coxeter group is hyperbolic. The key tool in the proof is an extension of the Milnor-Svarc Lemma which states that if a group G admits a discontinuous, co-compact action by isometries on a Gromov hyperbolic metric space, then G is weakly hyperbolic relative to the isotropy subgroups of the action.      

On a Class of Factorizable Inverse Monoids Associated with Braid Groups

We prove that a constructible nilpotent-by-abelian group G can be obtained from a polycyclic group by forming d successive properly ascending HNN-extensions if and only if d is the dimension of the linear subspace of Hom(G, R) spanned by the geometric invariant Σ¹ (G, Z) ^c . We also obtain a result on the finiteness properties “type FP _m ” of certain subgroups of G     

On the Analytic Type of Artin Algebras

In this paper we give a complete analytic classification of Artin Gorenstein almost stretched algebras, i.e., Artin Gorenstein algebras with Hilbert function {1, n, 2,…, 2, 1,…, 1}.     

On the Artin exponent of some rational groups

A finite group G is called rational if all its irreducible complex characters are rational valued. In this paper, we show that if G is a direct product of finitely many rational Frobenius groups then every rationally represented character of G is a generalized permutation character. Also we show that the same assertion holds when G is a solvable rational group with a Sylow 2-subgroup isomorphic to the dihedral group of order 8 and an abelian normal Sylow 3-subgroup.     

退化矩阵Liouville,Beta,Dirichlet分布

该文引进和讨论了退化矩阵Liouville分布，由此导出退化矩阵Beta分布、退化矩阵Dirichlet分布．推广了文献［1］关于退化Wishart分布和秩为1的退化矩阵Beta分布的结果。     

On the -rank Artin Conjecture

We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which can be generated by given multiplicatively independent numbers. In the case when the given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown-Zassenhaus (J. Number Theory 3 (1971), 306-309). Finally, in some examples, we compare the densities approximated with the natural densities calculated with primes up to .

     

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type and with non-trivial local coefficients. The main result

is an explicit computation of the cohomology of the Artin group of type with coefficients over the module Here the first standard generators of the group act by -multiplication, while the last one acts by -multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type as well as the cohomology of the classical braid group with coefficients in the -dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

     

Subalgebras and finitistic dimensions of Artin algebras

Let A be an Artin algebra.We investigate subalgebras of A with certain conditions and obtain some classes of algebras whose finitistic dimensions are finite.