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Register a new userThe freeness of Ish arrangements
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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.
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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.
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.
A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. Our proof of the main theorem heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.
Let $ G $ be a simple graph of $ ell $ vertices $ {1, dots, ell } $ with edge set $ E_{G} $. The graphical arrangement $ mathcal{A}_{G} $ consists of hyperplanes $ {x_{i}-x_{j}=0} $, where $ {i, j } in E_{G} $. It is well known that three properties, chordality of $ G $, supersolvability of $ mathcal{A}_{G} $, and freeness of $ mathcal{A}_{G} $ are equivalent. Recently, Richard P. Stanley introduced $ psi $-graphical arrangement $ mathcal{A}_{G, psi} $ as a generalization of graphical arrangements. Lili Mu and Stanley characterized the supersolvability of the $ psi $-graphical arrangements and conjectured that the freeness and the supersolvability of $ psi $-graphical arrangements are equivalent. In this paper, we will prove the conjecture.
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.
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