Barbara Baumeister

A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements

In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W lie as well in this parabolic subgroup.

The Universal Covers of Certain Semibiplanes

The universal covers ofc.c*-geometries constructed from the length two orbits of certain involutory automorphism ofPG(2,q)are determined. It particular, we answer a question from 5.

On Flat Flag-Transitive c.c^* -Geometries

We study flat flag-transitive c.c*-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometry with automorphism group G = 22n· Ln(2) and covered by the truncated Coxeter complex of type D2n. The non-canonical ways give us geometries with smaller automorphism group (G ≤ 22n· (2n−1)n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.

Two New Sporadic Semibiplanes Related to M22

We construct a c.c*-geometry admitting M22 as flag transitive automorphism group. Furthermore, we classify all flag transitive c.c*-geometries supposing that the residue of a circle is isomorphic to the complete graph K15 . We use coset enumeration to determine the universal 2-cover of the example related to M22.

On the Hurwitz action in finite Coxeter groups

Abstract We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups.

Simple dual braids, noncrossing partitions and Mikado braids of type Dn

We show that the simple elements of the dual Garside structure of an Artin group of type Dn are Mikado braids, giving a positive answer to a conjecture of Digne and the second author. To this end, we use an embedding of the Artin group of type Dn in a suitable quotient of an Artin group of type Bn noted by Allcock, of which we give a simple algebraic proof here. This allows one to give a characterization of the Mikado braids of type Dn in terms of those of type Bn and also to describe them topologically. Using this topological representation and Athanasiadis and Reiners model for noncrossing partitions of type Dn which can be used to represent the simple elements, we deduce the above-mentioned conjecture.

Polytopes associated to dihedral groups

In this note we investigate the convex hull of those n ×  n permutation matrices that correspond to symmetries of a regular n -gon. We give the complete facet description. As an application, we show that this yields a Gorenstein polytope, and we determine the Ehrhart h * -vector.

The Non-canonical Gluings of two Affine Spaces

In this paper we determine the flag-transitive non-canonical gluings of two isomorphic desarguesian affine spaces. It turns out that there are fifteen sporadic examples and two infinite series. Moreover, we determine the universal covers of the fifteen sporadic gluings and of the canonical gluing.

Aperiodic logarithmic signatures

Abstract. In this paper we propose a method to construct logarithmic signatures which are not amalgamated transversal and further do not even have a periodic block. The latter property was crucial for the successful attack on the system MST3 by Blackburn, Cid and Mullan (2009). The idea for our construction is based on the theory in Szabós book “Topics in Factorization of Abelian Groups”.

A characterization of the Petersen−type geometry of the McLaughlin group

The McLaughlin sporadic simple group McL is the flag-transitive automorphism group of a Petersen-type geometry [Gscr ] = [Gscr ](McL) with the diagram diagram here where the edge in the middle indicates the geometry of vertices and edges of the Petersen graph. The elements corresponding to the nodes from the left to the right on the diagram P 3 3 are called points, lines, triangles and planes, respectively. The residue in [Gscr ] of a point is the P 3 -geometry [Gscr ](Mat 22 ) of the Mathieu group of degree 22 and the residue of a plane is the P 3 -geometry [Gscr ](Alt 7 ) of the alternating group of degree 7. The geometries [Gscr ](Mat 22 ) and [Gscr ](Alt 7 ) possess 3-fold covers [Gscr ](3Mat 22 ) and [Gscr ](3Alt 7 ) which are known to be universal. In this paper we show that [Gscr ] is simply connected and construct a geometry [Gscr ]˜ which possesses a 2-covering onto [Gscr ]. The automorphism group of [Gscr ]˜ is of the form 3 23 McL; the residues of a point and a plane are isomorphic to [Gscr ](3Mat 22 ) and [Gscr ](3Alt 7 ), respectively. Moreover, we reduce the classification problem of all flag-transitive P m n -geometries with n , m [ges ] 3 to the calculation of the universal cover of [Gscr ]˜.