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Register a new userThe freeness of ideal subarrangements of Weyl arrangements
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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.
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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}.
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.
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.
Let $q$ be a positive integer. In our recent paper, we proved that the cardinality of the complement of an integral arrangement, after the modulo $q$ reduction, is a quasi-polynomial of $q$, which we call the characteristic quasi-polynomial. In this paper, we study general properties of the characteristic quasi-polynomial as well as discuss two important examples: the arrangements of reflecting hyperplanes arising from irreducible root systems and the mid-hyperplane arrangements. In the root system case, we present a beautiful formula for the generating function of the characteristic quasi-polynomial which has been essentially obtained by Ch. Athanasiadis and by A. Blass and B. Sagan. On the other hand, it is hard to find the generating function of the characteristic quasi-polynomial in the mid-hyperplane arrangement case. We determine them when the dimension is less than six.
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.
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