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The emph{Shi arrangement} is the set of all hyperplanes in $mathbb R^n$ of the form $x_j - x_k = 0$ or $1$ for $1 le j < k le n$. 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 emph{parking function}, i.e., a sequence $(x_1, x_2, ..., x_n)$ of positive integers that, when rearranged from smallest to largest, satisfies $x_k le k$. (There is an illustrative reason for the term emph{parking function}.) It turns out that the number of parking functions of length $n$ also equals $(n+1)^{n-1}$, a result due to Konheim and Weiss from 1966. A natural problem consists of finding a bijection between the $n$-dimensional Shi arragnement and the parking functions of length $n$. Stanley and Pak (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.
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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.
In this article we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual-partition formula. Then it immediately follows that there exists a saturated free filtration of the cone of any affine Weyl arrangement such that each filter is a free subarrangement satisfying the dual-partition formula. This generalizes the main result in cite{ABCHT} which affirmatively settled a conjecture by Sommers and Tymoczko cite{SomTym}.
Let $W$ be a finite Weyl group and $A$ be the corresponding Weyl arrangement. A deformation of $A$ is an affine arrangement which is obtained by adding to each hyperplane $HinA$ several parallel translations of $H$ by the positive root (and its integer multiples) perpendicular to $H$. We say that a deformation is $W$-equivariant if the number of parallel hyperplanes of each hyperplane $Hin A$ depends only on the $W$-orbit of $H$. We prove that the conings of the $W$-equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers. This generalizes Yoshinagas theorem conjectured by Edelman-Reiner.
In his affirmative answer to the Edelman-Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules. However, all we know about the bases is their existence and degrees. In this article, we introduce two distinguished bases for the modules. More specifically, we will define and study the simple-root basis plus (SRB+) and the simple-root basis minus (SRB-) when a primitive derivation is fixed. They have remarkable properties relevant to the simple roots and those properties characterize the bases.
We introduce and study a digraph analogue of Stanleys $psi$-graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in 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 $psi$, 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.
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