On the cardinalities of finite topologies

Following the ideas of Sharp [2,3], we will give a partial answer to the question: “Let k be an integer, k ? 2. What is the smallest integer m for which there is a topology on m points with k open sets.” We state several results in the theory of finite topologies by introducing the idea of generating topologies. Using this concept, it is possible to derive existence theorems and get numerical results in an easy manner.     

On regularity of finite reflection groups

We define a concept of “regularity” for finite unitary reflection groups, and show that an irreducible finite unitary reflection group of rank greater than 1 is regular if and only if it is a Coxeter group. Hence we get a characterization of Coxeter groups among all the irreducible finite reflection groups of rank greater than one. Received: 10 September 1999 / Revised version: 19 February 2000     

Convergence of the Poincaré series for some classical Schottky groups

The Poincaré -series for a multiply connected circular region can be either convergent or divergent absolutely. In this paper we prove a uniform convergence result for such a region.

     

On Fourier coefficients of Eisenstein series and Niebur Poincaré series of integral weight

We give a Katok-Sarnak type correspondence for Niebur type Poincaré series and Eisenstein series on the three-dimensional hyperbolic space.     

Generating sequences and Poincaré series for a finite set of plane divisorial valuations

Let V be a finite set of divisorial valuations centered at a 2-dimensional regular local ring R. In this paper we study its structure by means of the semigroup of values, SV, and the multi-index graded algebra defined by V, grVR. We prove that SV is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in V, the approximation of a reduced plane curve singularity C by families of sets V(k) of divisorial valuations, and the relationship between the value semigroup of C and the semigroups of the sets V(k), allow us to obtain the (finite) minimal generating sequences for C as well as for V.We also analyze the structure of the homogeneous components of grVR. The study of their dimensions allows us to relate the Poincaré series for V and for a general curve C of V. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincaré series of V. Moreover, the Poincaré series of C could be seen as the limit of the series of V(k), k?0.     

Reducibility of finite reflection groups

A finite (pseudo-)reflection group G naturally gives rise to a hyperplane arrangement,i.e.,its reflection arrangement.We show that G is reducible if and only if its reflection arrangement is reducible.     

Separating invariants and finite reflection groups

A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating algebra. This allows us to prove that only groups generated by reflections may have polynomial separating algebras, and only groups generated by bireflections may have complete intersection separating algebras.     

On the rigidity of sphericalt-designs that are orbits of finite reflection groups

The concept of rigid sphericalt-designs was introduced by Bannai. He conjectured that there is a functionf(t, d) such that ifX is a sphericalt design in thed-dimensional Euclidean space so that |X|>f(t, d), theX is non-rigid. Furthermore, he asked to find examples of rigid but not tight sperical designs. In the present article we shall investigate the case whenX is an orbit of a finite reflection group and prove thatX is rigid iff tight for the groupsA _n ,B _n ,C _n ,D _n ,E ₆,E ₇,F ₄,I ₃.Research was partially supported by Hungarian National Research fund Grant No. 4267.     

On the multigraded Hilbert and Poincaré-Betti series and the Golod property of monomial rings

In this paper we study the multigraded Hilbert and Poincaré-Betti series of A=S/a, where S is the ring of polynomials in n indeterminates divided by the monomial ideal a. There is a conjecture about the multigraded Poincaré-Betti series by Charalambous and Reeves which they proved in the case where the Taylor resolution is minimal. We introduce a conjecture about the minimal A-free resolution of the residue class field and show that this conjecture implies the conjecture of Charalambous and Reeves and, in addition, gives a formula for the Hilbert series. Using Algebraic Discrete Morse theory, we prove that the homology of the Koszul complex of A with respect to x₁,…,x_n is isomorphic to a graded commutative ring of polynomials over certain sets in the Taylor resolution divided by an ideal r of relations. This leads to a proof of our conjecture for some classes of algebras A. We also give an approach for the proof of our conjecture via Algebraic Discrete Morse theory in the general case.The conjecture implies that A is Golod if and only if the product (i.e. the first Massey operation) on the Koszul homology is trivial. Under the assumption of the conjecture we finally prove that a very simple purely combinatorial condition on the minimal monomial generating system of a implies Golodness for A.     

Involutions and reflection subgroups of finite Coxeter groups

We derive the classification of finite Coxeter groups in a purely algebraic manner from a simple result concerning involutions and a result of Dyer on reflection subgroups, for which we give a very short proof.     

Invariant fields of finite irreducible reflection groups

We prove the following result: If G is a finite irreducible reflection group defined over a base field k, then the invariant field of G is purely transcendental over k, even if |G| is divisible by the characteristic of k. It is well known that in the above situation the invariant ring is in general not a polynomial ring. So the question whether at least the invariant field is purely transcendental (Noether's problem) is quite natural. Received: 14 January 1998     

Poincaré series and Coxeter functors for Fuchsian singularities

We consider Fuchsian singularities of arbitrary genus and prove, in a conceptual manner, a formula for their Poincaré series. This uses Coxeter elements involving Eichler-Siegel transformations. We give geometrical interpretations for the lattices and isometries involved, lifting them to triangulated categories.     

Poincaré series and an application to Weyl algebras

Let A n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A n . By further exploring the relationship between the Poincar′e series and the dimension and the multiplicity of M , we are able to prove that the tensor product of two finitely generated modules over A n has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A n .     

Classical invariant theory for finite reflection groups

We give explicit systems of generators of the algebras of invariant polynomials in arbitrary many vector variables for the classical reflection groups (including the dihedral groups). As an application of the results we prove a generalization of Chevalley's restriction theorem for the classical Lie algebras. In the interesting case when the group is of Coxeter typeD _n (n4) we use higher polarization operators introduced by Wallach. The least upper bound for the degrees of elements in a system of generators turns out to be independent of the number of vector variables. We conjecture that this is also true for the exceptional reflection groups and then sketch a proof for the group of typeF ₄.