Coxeter Arrangement Research Articles

Hyperfactord of Shi arrangement Sh(A2) and Sh(A3)

In this paper, we introduce the region and the faces poset of shi arrangement that J. Y. Shi firstly introduced it. This is an affine arrangement, each of whose hyperplane is parallel to some"hyperplane of Coxeter arrangement"(Braid arrangement), the degrees and the exponents of this arrangement were found and we prove the shi arrangement is ahyperfactored arrangement when n=3 and not hyperfactored arrangement when n=4 arrangement.

A generalization of combinatorial identities for stable discrete series constants

This article is concerned with the constants that appear in Harish-Chandra’s character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra’s work the only information we have about these constants is that they are uniquely determined by an inductive property. Later, Goresky–Kottwitz–MacPherson (1997) and Herb (2000) gave different formulas for these constants. In this article, we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying the discrete series character identities of Morel (2011). We deduce this identity from a general abstract theorem giving a way to calculate the alternating sum of the values of a valuation on the chambers of a Coxeter arrangement. We also introduce a ring structure on the set of valuations on polyhedral cones in Euclidean space with values in a fixed ring. This gives a theoretical framework for the valuation appearing in Goresky– Kottwitz–MacPherson’s 1997 paper. In an appendix, we extend Herb’s notion of 2-structures to pseudo-root systems.

Equivariant Log Concavity and Representation Stability

AbstractWe expand upon the notion of equivariant log concavity and make equivariant log concavity conjectures for Orlik–Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik–Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{s}\mathfrak{l}_n$, we exploit the theory of representation stability to give computer-assisted proofs of these conjectures in low degree.

Accurate arrangements

Let A be a Coxeter arrangement of rank ℓ. In 1987 Orlik, Solomon and Terao conjectured that for every 1≤d≤ℓ, the first d exponents of A – when listed in increasing order – are realized as the exponents of a free restriction of A to some intersection of reflecting hyperplanes of 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 arrangement with this natural property involving their free restrictions accurate. In this paper we initialize their systematic study.Our principal result shows that MAT-free arrangements, a notion recently introduced by Cuntz and Mücksch, are accurate.This theorem in turn directly implies this special property for all ideal subarrangements of Weyl arrangements. In particular, this gives a new, simpler and uniform proof of the aforementioned conjecture of Orlik, Solomon and Terao for Weyl arrangements which is free of any case-by-case considerations.Another application of a slightly more general formulation of our main theorem shows that extended Catalan arrangements, extended Shi arrangements, and ideal-Shi arrangements share this property as well.We also study arrangements that satisfy a slightly weaker condition, called almost accurate arrangements, where we simply disregard the ordering of the exponents involved. This property in turn is implied by many well established concepts of freeness such as supersolvability and divisional freeness.

Counting chambers in restricted Coxeter arrangements

Solomon showed that the Poincare 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 $$E_8$$.

Stirling numbers in braid matroid Kazhdan–Lusztig polynomials

Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan–Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan–Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling numbers of the first kind and Stirling numbers of the second kind, as well as a non-recursive formula for the braid matroid Kazhdan–Lusztig polynomials.

The Varchenko determinant of a Coxeter arrangement

Abstract 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 for 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 associated determinants.

Limit directions for Lorentzian Coxeter systems

Every Coxeter group admits a geometric representation as a group generated by reflections in a real vector space. In the projective representation space of a Coxeter group, limit directions arising from a point are accumulation points of the orbit of this point. In particular, limit directions arising from roots are called limit roots. Recent studies show that limit roots lie on the isotropic cone of the representation space. In this paper, we study limit directions of Coxeter groups arising from any point when the representation space is a Lorentz space. We prove that the limit roots are the only light-like limit directions, and characterize the limit roots using eigenvectors of infinite-order elements. Then we describe the structure of space-like limit directions in terms of the projective Coxeter arrangement. Some non-Lorentzian cases are also discussed.

The Orlik-Terao Algebra and the Cohomology of Configuration Space

ABSTRACTWe give a recursive algorithm for computing the Orlik-Terao algebra of the Coxeter arrangement of type An − 1 as a graded representation of Sn, and we give a conjectural description of this representation in terms of the cohomology of the configuration space of n points in SU(2) modulo translation. We also give a version of this conjecture for more general graphical arrangements.

Coxeter arrangements in three dimensions

Let $$\mathcal {A}$$ be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of $$\mathcal {A}$$ are isometric. We prove that $$\mathcal {A}$$ is necessarily a Coxeter arrangement. As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open question whether this suffices to characterize Coxeter arrangements in higher dimensions. We also present the three families of affine arrangements in the plane which are not reflection arrangements, but in which all the regions are isometric.

The shape of a random affine Weyl group element and random core partitions

Let $W$ be a finite Weyl group and ${\hat{W}}$ be the corresponding affine Weyl group. We show that a large element in ${\hat{W}}$, randomly generated by (reduced) multiplication by simple generators, almost surely has one of $|W|$-specific shapes. Equivalently, a reduced random walk in the regions of the affine Coxeter arrangement asymptotically approaches one of $|W|$-many directions. The coordinates of this direction, together with the probabilities of each direction can be calculated via a Markov chain on $W$. Our results, applied to type $\tilde{A}_{n-1}$, show that a large random $n$-core obtained from the natural growth process has a limiting shape which is a piecewise-linear graph. In this case, our random process is a periodic analogue of TASEP, and our limiting shapes can be compared with Rost’s theorem on the limiting shape of TASEP.

Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the $n$-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement.

Splines on the Alfeld split of a simplex and type A root systems

Alfeld (1984) introduced a tetrahedral analog AS(Δ3) of the Clough–Tocher split of a triangle. A formula for the dimension of the spline space Ckr(AS(Δn)) is conjectured in Foucart and Sorokina (2013). We prove that the graded module of Cr-splines on the cone over AS(Δn) is isomorphic to the module Dr+1(An) of multiderivations on the type An Coxeter arrangement. A theorem of Terao shows that the module of multiderivations of a Coxeter arrangement is free and gives an explicit basis. As a consequence the conjectured formula holds.

The Orlik-Solomon algebra and the supersolvable class of arrangements

According to the powerful geometric properties of the hypersolvable order on the hyperplanes of a supersolvable arrangement, we introduced a sufficient condition on the Orlik-Solomon algebra for any central arrangement to have supersolvable analogue and we showed this condition as a necessary condition (not sufficient) on the Orlik-Solomon algebra for any central arrangement to be K(π; 1). Finally as an illustration of our result, we produce to the Orlik-Solomon algebra for the complexification of the Coxeter arrangement of type Ar and Br, for r ≥ 3, a structure by their supersolvable partitions analogues.

Ad-nilpotent ideals and the Shi arrangement

We extend the Shi bijection from the Borel subalgebra case to parabolic subalgebras. In the process, the I-deleted Shi arrangement Shi(I) naturally emerges. This arrangement interpolates between the Coxeter arrangement Cox and the Shi arrangement Shi, and breaks the symmetry of Shi in a certain symmetrical way. Among other things, we determine the characteristic polynomial χ(Shi(I),t) of Shi(I) explicitly for An−1 and Cn. More generally, let Shi(G) be an arbitrary arrangement between Cox and Shi. Armstrong and Rhoades recently gave a formula for χ(Shi(G),t) for An−1. Inspired by their result, we obtain formulae for χ(Shi(G),t) for Bn, Cn and Dn.

Modules of Differential Operators of Order 2 on Coxeter Arrangements

The collection of reflection hyperplanes of a finite reflection group is called a Coxeter arrangement. A Coxeter arrangement is known to be free. K. Saito has constructed a basis consisting of invariant elements for the module of derivations on a Coxeter arrangement. We study the module of \(\mathcal{A}\)-differential operators as a generalization of the study of the module of \(\mathcal{A}\)-derivations. In this article, we prove that the modules of differential operators of order 2 on Coxeter arrangements of types A, B and D are free, by exhibiting their bases. We also prove that the modules cannot have bases consisting of only invariant elements. Two keys for the proof of freeness are the “Cauchy-Sylvester theorem on compound determinants” and the “Saito-Holm criterion for freeness.”

On inductively free reflection arrangements

Abstract Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let 𝒜 = 𝒜(W) be the associated hyperplane arrangement of W. Terao [J. Fac. Sci. Univ. Tokyo 27 (1980), 293–320] has shown that each such reflection arrangement 𝒜 is free. There is the stronger notion of an inductively free arrangement. In 1992, Orlik and Terao [Arrangements of hyperplanes, Springer-Verlag, Berlin 1992, Conjecture 6.91] conjectured that each reflection arrangement is inductively free. It has been known for quite some time that the braid arrangement as well as the Coxeter arrangements of type B ℓ and type D ℓ are inductively free. Barakat and Cuntz [Adv. Math. 229 (2012), 691–709] completed this list only recently by showing that every Coxeter arrangement is inductively free. Nevertheless, Orlik and Terao's conjecture is false in general. In a paper which will appear in Tôhoku Math. J., we already gave two counterexamples to this conjecture among the exceptional complex reflection groups. In this paper we classify all inductively free reflection arrangements. In addition, we show that the notions of inductive freeness and that of hereditary inductive freeness coincide for reflection arrangements. As a consequence of our classification, we get an easy, purely combinatorial characterization of inductively free reflection arrangements 𝒜 in terms of exponents of the restrictions to any hyperplane of 𝒜.

Equivariant multiplicities of Coxeter arrangements and invariant bases

Let A be an irreducible Coxeter arrangement and W be its Coxeter group. Then W naturally acts on A. A multiplicity m:A→Z is said to be equivariant when m is constant on each W-orbit of A. In this article, we prove that the multi-derivation module D(A,m) is a free module whenever m is equivariant by explicitly constructing a basis, which generalizes the main theorem of Terao (2002) [12]. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the W-invariant part D(A,m)W for any multiplicity m is a free module over the W-invariant subring.

The Shi arrangements and the Bernoulli polynomials

The braid arrangement is the Coxeter arrangement of the type $A_\ell$. The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the braid arrangement and their parallel translations. In this paper, we give an explicit basis construction for the derivation module of the cone over the Shi arrangement. The essential ingredient of our recipe is the Bernoulli polynomials.

Bases for the derivation modules of two-dimensional ~multi-Coxeter arrangements and universal derivations

Let ¥cal{A} be an irreducible Coxeter arrangement and k be a multiplicity of ¥cal{A}. We study the derivation module D(¥cal{A}, k). Any two-dimensional irreducible Coxeter arrangement with even number of lines is decomposed into two orbits under the action of the Coxeter group. In this paper, we will explicitly construct a basis for D(¥cal{A}, k) assuming k is constant on each orbit. Consequently we will determine the exponents of (¥cal{A}, k) under this assumption. For this purpose we develop a theory of universal derivations and introduce a map to deal with our exceptional cases.