Related papers: Ziegler's Multi-Reflection Arrangements are free
Let $\mathcal C :f=0$ be a curve arrangement in the complex projective plane. If $\mathcal C$ contains a curve subarrangement consisting of at least three members in a pencil, then one obtains an explicit syzygy among the partial…
In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is…
We re-cast in a more combinatorial and computational form the foldings approach of John Stallings and pursue a detailed study of the subgroup structure of free groups. In particular, we introduce the notions of an "algebraic" and a "free"…
We introduce the notion of multielement order separability and study this property for free groups and free products.
Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane…
We prove Anzis and Tohaneanu conjecture, that is the Dirac-Motzkin conjecture for supersolvable line arrangements in the projective plane over an arbitrary field of characteristic zero. Moreover, we show that a divisionally free…
We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.
We introduce the concept of hyperreflection groups, which are a generalization of Coxeter groups. We prove the Deletion and Exchange Conditions for hyperreflection groups, and we discuss special subgroups and fundamental sectors of…
A set of general physical principles is proposed as the structural basis for the theory of complex systems. First the concept of harmony is analyzed and its different aspects are uncovered. Then the concept of reflection is defined and…
Let $G$ be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of $G$. This resolution is used to define…
In this work we study line arrangements consisting in lines passing through three non-aligned points. We call them triangular arrangements. We prove that any combinatorics of a triangular arrangement is always realized by a…
This paper aims to systematically study mystic reflection groups that emerged independently in the paper [Selecta Math. (N.S.) 14 (2009), 325-372, arXiv:0806.0867] by the authors and in the paper [Algebr. Represent. Theory 13 (2010),…
We introduce the notion of reflexivity for combinatory algebras. Reflexivity can be thought of as an equational counterpart of the Meyer-Scott axiom of combinatory models, which indeed allows us to characterise an equationally definable…
Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This…
Let R be a local ring and A a connected differential graded algebra over R which is free as a graded R-module. Using homological perturbation theory techniques, we construct a minimal free multi model for A having properties similar to that…
The standard generators of tridiagonal algebras, recently introduced by Terwilliger, are shown to generate a new (in)finite family of mutually commuting operators which extends the Dolan-Grady construction. The involution property relies on…
It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying…
In 1963 [Ann. of Math. {\bf 78}, 267-288], Gerstenhaber invented a \emph{comp(osition)} calculus in the Hochschild complex of an associative algebra. In this paper, the first steps of the Gerstenhaber theory are exposed in an abstract (comp…
Dixon's multipoles for a system of N relativistic positive-energy scalar particles are evaluated in the rest-frame instant form of dynamics. The Wigner hyperplanes (intrinsic rest frame of the isolated system) turn out to be the natural…
In 1968, R. Steinberg proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise non-isomorphic. We extend this result to a more general…