The freeness of Ish arrangements

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• 作者：Takuro Abe^a ; ^{abe@imi.kyushu-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Daisuke Suyama^b ; ^{dsuyama@math.sci.hokudai.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Shuhei Tsujie^b ; ^{tsujie@math.sci.hokudai.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Hyperplane arrangement ; Ish arrangement ; Shi arrangement ; Coxeter arrangement ; Free arrangement ; Supersolvable arrangement ; Fiber-type arrangement
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：146
• 期：Complete
• 页码：169-183
• 全文大小：327 K

文摘

The Ish arrangement was introduced by Armstrong to give a new interpretation of the q,t$q,t$-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.