63 research outputs found
The Shi arrangements and the Bernoulli polynomials
The braid arrangement is the Coxeter arrangement of the type . 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.Comment: We fixed a typ
The Primitive Derivation and Discrete Integrals
The 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 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
The freeness of Ish arrangements
International audienceThe Ish arrangement was introduced by Armstrong to give a new interpretation of the -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 freeL’arrangement Ish a été introduit par Armstrong pour donner une nouvelle interprétation des nombres -Catalan de Garsia et Haiman. Armstrong et Rhoades ont montré qu’il y avait des ressemblances frappantes entre l’arrangement Shi et l’arrangement Ish et ont posé des conjectures. L’une d’elles est de savoir si l’arrangement Ish est un arrangement libre ou pas. Dans cet article, nous vérifions que l’arrangement Ish est supersoluble et donc libre. De plus, on donne une condition nécessaire et suffisante pour que l’arrangement Ish réduit soit libre
The freeness of Ish arrangements
The Ish arrangement was introduced by Armstrong to give a new interpretation of the -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 fre
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