Shi Arrangement Research Articles

Shi Arrangements Restricted to Weyl Cones

Shi Arrangements Restricted to Weyl Cones

Shi arrangements and low elements in Coxeter groups

AbstractGiven an arbitrary Coxeter system and a non‐negative integer , the ‐Shi arrangement of is a subarrangement of the Coxeter hyperplane arrangement of . The classical Shi arrangement () was introduced in the case of affine Weyl groups by Shi to study Kazhdan–Lusztig cells for . As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in and that the union of their inverses form a convex subset of the Coxeter complex. The set of ‐low elements in were introduced to study the word problem of the corresponding Artin–Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in . In this article, we generalize and extend Shi's results to any Coxeter system for any : (1) the set of minimal length elements of the regions in a ‐Shi arrangement is precisely the set of ‐low elements, settling a conjecture of the first and third authors in this case; (2) the union of the inverses of the (0‐)low elements form a convex subset in the Coxeter complex, settling a conjecture by the third author, Nadeau and Williams.

Shi arrangements and low elements in affine Coxeter groups

Abstract Given an affine Coxeter group W, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan–Lusztig cells for W. Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in W. Low elements in W were introduced to study the word problem of the corresponding Artin–Tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in W. In this article, we show, in the case of an affine Coxeter group, that the set of minimal length elements of the regions in the Shi arrangement is precisely the set of low elements, settling a conjecture of Dyer and the second author in this case. As a by-product of our proof, we show that the descent walls – the walls that separate a region from the fundamental alcove – of any region in the Shi arrangement are precisely the descent walls of the alcove of its corresponding low element.

Vertex-weighted digraphs and freeness of arrangements between Shi and Ish

We introduce and study a digraph analogue of Stanley’s ψ-graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in the literature including the Catalan arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish, recently introduced by Duarte and Guedes de Oliveira. The arrangements between Shi and Ish all are proved to have the same characteristic polynomial with all integer roots, thus raising the natural question of their freeness. We define two operations on digraphs, which we shall call king and coking elimination operations and prove that subject to certain conditions on the weight ψ, the operations preserve the characteristic polynomials and freeness of the associated arrangements. As an application, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone.

Freeness for restriction arrangements of the extended Shi and Catalan arrangements

The extended Shi and Catalan arrangements are well investigated arrangements. In this paper, we prove that the cone of the extended Catalan arrangement of type A is always hereditarily free, while we determine the dimension in which the cone of the extended Shi arrangement of type A is hereditarily free. For this purpose, using digraphs, we define a class of arrangements which is closed under restriction, and which contains the extended Shi and Catalan arrangements. We also characterize the freeness for the cone of this arrangement by graphical conditions.

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.

Cone types, automata, and regular partitions in Coxeter groups

In this article we introduce the notion of a 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 minimal length element. We further develop the theory of cone types in Coxeter groups by identifying the minimal set of roots required to express a cone type as an intersection of half-spaces. This set of boundary roots is closely related to the elementary inversion sets of Brink and Howlett, and also to the notion of the base of an inversion set introduced by Dyer.

Bijections for Faces of the Shi and Catalan Arrangements

In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in ${R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection between the $k$-dimensional faces of the Shi arrangement for any $k$ and a set of decorated binary trees. Furthermore, we show how these trees can be converted to a simple set of functions of the form $f: [n-1] \to [n+1]$ together with a marked subset of $\text{Im}(f)$. This correspondence gives the first bijective proof of the formula of Athanasiadis. In the process, we also obtain a bijection and counting formula for the faces of the Catalan arrangement. All of our results generalize to both extended arrangements.

The Primitive Derivation and Discrete Integrals

he modules of logarithmic derivations for the (extended) Catalan and Shi arrangements associated with root systems are known to be free. However, except for a few cases, explicit bases for such modules are not known. In this paper, we construct explicit bases for type A root systems. Our construction is based on Bandlow-Musiker's integral formula for a basis of the space of quasiinvariants. The integral formula can be considered as an expression for the inverse of the primitive derivation introduced by K. Saito. We prove that the discrete analogues of the integral formulas provide bases for Catalan and Shi arrangements.

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 faces in the extended Shi arrangement

We define the level of a subset X of Euclidean space to be the dimension of the smallest subspace such that the distance between each element of X and the subspace is bounded. We prove that the number of faces in the n-dimensional extended Shi arrangement Aˆnr having codimension k and level m is given by m⋅(nk)⋅ΔrkΔm−1xn−1|x=rn−1, where Δ is the difference operator and Δr is the difference operator of step r, that is, Δrp(x)=p(x)−p(x−r). This generalizes a result of Athanasiadis which counts the number of faces of different dimensions in the extended Shi arrangement Aˆnr. The proof relies on a multi-variated Abel identity due to Françon.

Partial parking functions

We characterise the Pak–Stanley labels of the regions of a family of hyperplane arrangements that interpolate between the Shi arrangement and the Ish arrangement.

Hyperplane arrangements between Shi and Ish

We introduce a new family of hyperplane arrangements in dimension n≥3 that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of this family have the same number of regions — the connected components of the complement of the union of the hyperplanes — which can be bijectively labeled with the Pak-Stanley labelling. In addition, we characterise the Pak-Stanley labels of the regions of this family of hyperplane arrangements.

Worpitzky partitions for root systems and characteristic quasi-polynomials

We introduce a partition of (coweight) lattice points inside the dilated fundamental parallelepiped into those of partially closed simplices. This partition can be considered as a generalization and a lattice points interpretation of the classical formula of Worpitzky. This partition, and the generalized Eulerian polynomial, recently introduced by Lam and Postnikov, can be used to describe the characteristic (quasi)polynomials of Shi and Linial arrangements. As an application, we prove that the characteristic quasi-polynomial of the Shi arrangement turns out to be a polynomial. We also present several results on the location of zeros of characteristic polynomials, related to a conjecture of Postnikov and Stanley. In particular, we verify the "functional equation" of the characteristic polynomial of the Linial arrangement for any root system, and give partial affirmative results on "Riemann hypothesis" for the root systems of type $E_6, E_7, E_8$, and $F_4$.

A basis construction of the extended Catalan and Shi arrangements of the type A2

In [10], Terao proved the freeness of multi-Coxeter arrangements with constant multiplicities by giving an explicit construction of bases. Combining it with algebro-geometric method, Yoshinaga proved the freeness of the extended Catalan or Shi arrangements in [12]. However, there have been no explicit constructions of the bases for the logarithmic derivation modules of the extended Catalan and Shi arrangements. In this paper, we give the first explicit construction of them when the root system is of the type A2.

Between Shi and Ish

We introduce a new family of hyperplane arrangements in dimension n≥3 that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions – the connected components of the complement of the union of the hyperplanes – which can be bijectively labeled with the Pak–Stanley labeling. In addition, we show that, in the cases of the Shi and the Ish arrangements, the number of labels with reverse centers of a given length is equal, and conjecture that the same happens with all of the members of the family.

The freeness of Ish arrangements

The Ish arrangement was introduced by Armstrong to give a new interpretation of the q,t-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free.

Parking Functions, Shi Arrangements, and Mixed Graphs

The Shi arrangement is the set of all hyperplanes in ℝn of the form xj − xk = 0 or 1 for 1 ≤ j < k ≤ n. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this arrangement is (n + 1)n−1. An unrelated combinatorial concept is that of a parking function, i.e., a sequence (x1, x2, …, xn) of positive integers that, when rearranged from smallest to largest, satisfies xk ≤ k. (There is an illustrative reason for the term parking function.) It turns out that the number of parking functions of length n also equals (n + 1)n−1, a result given by Konheim and Weiss in 1966. A natural problem consists of finding a bijection between the n-dimensional Shi arrangement and the parking functions of length n. Pak and Stanley (1996) and Athanasiadis and Linusson (1999) gave such (quite different) bijections. We will shed new light on the former bijection by taking a scenic route through certain mixed graphs.

The braid and the Shi arrangements and the Pak–Stanley labelling

In this article we study a construction, due to Pak and Stanley, with which every region R of the Shi arrangement is (bijectively) labelled with a parking function λ(R). In particular, we construct an algorithm that returns R out of λ(R). This is done by relating λ to another bijection, that labels every region S of the braid arrangement with r(S), the unique central parking function f such that λ−1(f)⊆S. We also prove that λ maps the bounded regions of the Shi arrangement bijectively onto the prime parking functions. Finally, we introduce a variant (that we call “s-parking”) of the parking algorithm that is in the very origin of the term “parking function”. S-parking may be efficiently used in the context of our new algorithm, but we show that in some (well defined) cases it may even replace it.

Bigraphical arrangements

We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley in 1996. We also give bounds on the number of regions of a bigraphical arrangement.