Related papers: Freeness for multiarrangements of hyperplanes over…
Notions of freedom and independence for hypergraphs of models of a theory are defined. Properties of these notions and their applications to some natural classes of theories are studied.
We describe a structure of PRO on hypermatrices. This structure allows us to define multilinear representations of PROs and in particular of free Pros. As an example of applications, we investigate the relations of the representations of…
Athanasiadis studied arrangements obtained by adding shifted hyperplanes to the braid arrangement. Similarly, Bailey studied arrangements obtained by adding tilted hyperplanes to the braid arrangement. These two kinds of arrangements are…
We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient…
We extend the relation between random matrices and free probability theory from the level of expectations to the level of all correlation functions (which are classical cumulants of traces of products of the matrices). We introduce the…
The exhaustive study of the rigid symmetries of arbitrary free field theories is motivated, along several lines, as a preliminary step in the completion of the higher-spin interaction problem in full generality. Some results for the…
We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…
The addition-deletion theorems for hyperplane arrangements, which were originally shown in [H. Terao, Arrangements of hyperplanes and their freeness I, II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 293--320], provide useful ways to…
We introduce the notion of multielement order separability and study this property for free groups and free products.
A new description of free massless superfields of arbitrary superspin $Y$ ($Y>1/2$) is proposed. Following the first-order philosophy, we relax some of the properties (reality, gauge redundancy) of the unconstrained higher spin…
Hyperplane Arrangements of rank $3$ admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this…
We study the classes of free and plus-one generated hyperplane arrangements. Specifically, we describe how to compute the associated prime ideals of the Jacobian ideal of such an arrangement from its lattice of intersection. Moreover, we…
In this paper we study Cynk-Szemberg octic hyperplane arrangements from the perspective of homological properties of their derivation modules. In particular, we define the notion of the type of hyperplane arrangements that will be used in…
We consider the triple $(\mathcal{A},\mathcal{A}',\mathcal{A}^H)$ of hyperplane arrangements and the division of their characteristic polynomials. We show that the freeness of $\mathcal{A}^H$ and the division of $\chi(\mathcal{A};t)$ by…
We construct counterexamples to Yoshinaga's conjecture that every free arrangement is either inductively free or rigid in characteristic zero. The smallest example has $13$ hyperplanes, its intersection lattice has a one dimensional moduli…
In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated…
New features are described for models with multi-particle area-dependent potentials, in any number of dimensions. The corresponding many-body field theories are investigated for classical configurations. Some explicit solutions are given,…
By way of Ziegler restrictions we study the relation between nearly free plane arrangements and combinatorics and we give a Yoshinaga-type criterion for plus-one generated plane arrangements.
It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function, whenever the ground field…
Let ${\mathcal C}= \bigcup_{i=1}^n C_i \subseteq \mathbb{P}^2$ be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a…