Modules of differential operators of order 2 on Coxeter arrangements
Norihiro Nakashima
备注：
The title has changed. The previous title is "Cauchy-Sylvester's theorem on compound determinants and modules of differential operators on Coxeter arrangements."
We prove that the modules of differential operators of order 2 on the classical Coxeter arrangements are free by exhibiting bases. For this purpose, we use Cauchy-Sylvester's theorem on compound determinants and Saito-Holm's criterion. In the case type $A$, we apply Cauchy-Sylvester's theorem on compound determinants to Vandermond determinant. By using the Schur polynomials, we define operators which form a part of a basis of modules of differential operators on the classical Coxeter arrangements of type $A$. In the cases of type $B$ and type $D$, the proofs go similarly to the case of type $A$ with some adjustments of operators and determinants.
The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relati…
Solomon showed that the Poincaré polynomial of a Coxeter group $W$ satisfies a product decomposition depending on the exponents of $W$. This polynomial coincides with the rank-generating function of the poset of regions of the underlying Coxeter arrangement. In this note we determine all instances when the analogous factorization property of the rank-generating function of the poset of regions holds for a restriction of a Coxeter arrangement. It turns out that this is always the case with the exception of some instances in type $E_8$.
The reflection arrangement of a Coxeter group is a well known instance of a free hyperplane arrangement. In 2002, Terao showed that equipped with a constant multiplicity each such reflection arrangement gives rise to a free multiarrangement. In this note we show that this multiarrangment satisfies the stronger property of inductive freeness in case the Coxeter group is of type $A$.
Mirror graphs were introduced by Brešar et al. in 2004 as an intriguing class of graphs: vertex-transitive, isometrically embeddable into hypercubes, having a strong connection with regular maps and polytope structure. In this article we settle the structure of mirror graphs by characterizing them as precisely the Cayley graphs of the finite Coxeter groups or equivalently the tope graphs of reflection arrangements - well understood and classified structures. We provide a polynomial algorithm for their recognition.
We refine Brieskorn's study of the cohomology of the complement of the reflection arrangement of a finite Coxeter group $W$. As a result we complete the verification of a conjecture by Felder and Veselov that gives an explicit basis of the space of $W$-invariants in this cohomology ring.
We introduce and prove the $n$-dimensional Pizza Theorem: Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{R}^{n}$. If $K$ is a measurable set of finite volume, the {pizza quantity} of $K$ is the alternating sum of the volumes of the regions obtained by intersecting $K$ with the arrangement $\mathcal{H}$. We prove that if $\mathcal{H}$ is a Coxeter arrangement different from $A_{1}^{n}$ such that the group of isometries $W$ generated by the reflections in the hyperplanes of $\mathcal{H}$ contains the map $-\mathrm{id}$, and if $K$ is a translate of a convex body that is stable under $…
We introduce the notion of a Tits arrangement on a convex open cone as a special case of (infinite) simplicial arrangements. Such an object carries a simplicial structure similar to the geometric representation of Coxeter groups. The standard constructions of subarrangements and restrictions, which are known in the case of finite hyperplane arrangements, work as well in this more general setting.
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 arrangements with particularly nice geometric, algebraic, topological, and combinatorial properties are the supersolvable arrangements. In this paper we give a complete classification of supersolvable simplicial arrangements (in all ranks). For each fixed rank, our classification already includes almost all known simplicial arrangements. Surprisingly, for irr…
Associated to each irreducible crystallographic root system $Φ$, there is a certain cell complex structure on the torus obtained as the quotient of the ambient space by the coroot lattice of $Φ$. This is the Steinberg torus. A main goal of this paper is to exhibit a module structure on (the set of faces of) this complex over the (set of faces of the) Coxeter complex of $Φ$. The latter is a monoid under the Tits product of faces. The module structure is obtained from geometric considerations involving affine hyperplane arrangements. As a consequence, a module structure is obtained on the space …
Let $\mathscr A$ be a Coxeter arrangement of rank $\ell$. In 1987 Orlik, Solomon and Terao conjectured that for every $1\leq d \leq \ell$, the first $d$ exponents of $\mathscr A$ -- when listed in increasing order -- are realized as the exponents of a free restriction of $\mathscr A$ to some intersection of reflecting hyperplanes of $\mathscr A$ of dimension $d$. This conjecture does follow from rather extensive case-by-case studies by Orlik and Terao from 1992 and 1993, where they show that all restrictions of Coxeter arrangements are free. We call a general free arrangements with this natura…
In this article we introduce the notion of a \textit{regular partition} of a Coxeter group. We develop the theory of these partitions, and show that the class of regular partitions is essentially equivalent to the class of automata (not necessarily finite state) recognising the language of reduced words in the Coxeter group. As an application of this theory we prove that each cone type in a Coxeter group has a unique minimal length representative. This result can be seen as an analogue of Shi's classical result that each component of the Shi arrangement of an affine Coxeter group has a unique …
In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of an abstract string module and the perfect matching lattice of the corresponding abstract snake graph. In particular, we make explicit the direct correspondence between a submodule of a string module and the perfect matching of the corresponding snake graph. For every string module, we define a Coxeter element in a symmetric group, and we establish a bijection between these lattices and the interval in the weak Bruhat order determined by the Coxeter element. Using the correspondenc…
A basis construction for the Shi arrangement of the type $B_{\ell}$ or $C_{\ell}$
Shi排列是由Weyl排列的超平面和它们的平行移动组成的超平面的仿射排列。它是由J. Y. Shi在仿射外尔群的Kazhdan-Lusztig表示的研究中取得了重要的成就。M.吉永表明，圆锥体在每个施安排是免费的。本文构造了$B_{\ell}$或$C_{\ell}$型Shi型安排上锥的导子模的一个显式基.
We define a type B analogue of the category of finite sets with surjections, and we study the representation theory of this category. We show that the opposite category is quasi-Grobner, which implies that submodules of finitely generated modules are again finitely generated. We prove that the generating functions of finitely generated modules have certain prescribed poles, and we obtain restrictions on the representations of type B Coxeter groups that can appear in such modules. Our main example is a module that categorifies the degree i Kazhdan-Lusztig coefficients of type B Coxeter arrangem…
In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all finite Coxeter groups. This in turn follows from Deligne's seminal work from 1972, where he showed that the complexification of every real simplicial arrangement is a $K(π,1)$-arrangement. In this paper we study the $K(π,1)$-property for a certain class of subarrangements of Weyl arrangements, the so called arrangements of ideal type ${\mathscr A}_I$. These stem from ideals $I$ in the set of positive …
