Shi Threshold Arrangement

نویسنده

Seunghyun Seo

چکیده

Richard Stanley suggested the problem of finding the characteristic polynomial of a certain hyperplane arrangement defined by xi + xj = 0, 1, which is called the Shi threshold arrangement. We present the answer of the problem, using the finite field method.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

On Free Deformations of the Braid Arrangement

There has been considerable interest in the past in analyzing specific families of hyperplane arrangements from the perspective of freeness. Examples of such families have primarily included classes of subarrangements of Coxeter arrangements. The subarrangements of the braid arrangement An , the Weyl arrangement of type An−1, are known as the graphical arrangements. They correspond naturally to...

متن کامل

A simple bijection for the regions of the Shi arrangement of hyperplanes

The Shi arrangement Sn is the arrangement of affine hyperplanes in R n of the form xi−xj = 0 or 1, for 1 ≤ i < j ≤ n. It dissects R n into (n+1) regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of Sn containing the hyperplanes xi − xj = 0 and to the extended Shi arrangements.

متن کامل

A bijection between (bounded) dominant Shi regions and core partitions

It is well-known that Catalan numbers Cn = 1 n+1 ( 2n n ) count the number of dominant regions in the Shi arrangement of type A, and that they also count partitions which are both n-cores as well as (n + 1)-cores. These concepts have natural extensions, which we call here the m-Catalan numbers and m-Shi arrangement. In this paper, we construct a bijection between dominant regions of the m-Shi a...

متن کامل

The Shi arrangement and the Ish arrangement

This paper is about two arrangements of hyperplanes. The first — the Shi arrangement — was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type A. The second — the Ish arrangement — was recently defined by the first author who used the two arrangements together to give a new interpretation of the q, t-Catalan numbers of Garsia and Haiman. In the prese...

متن کامل