Between Shi and Ish

We introduce a new family of hyperplane arrangements in dimension n≥3 that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions – the connected components of the complement of the union of the hyperplanes – whic...

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Detalhes bibliográficos
Autor principal: Duarte, Rui (author)
Outros Autores: Guedes de Oliveira, António (author)
Formato: article
Idioma:eng
Publicado em: 2018
Assuntos:
Texto completo:http://hdl.handle.net/10773/21736
País:Portugal
Oai:oai:ria.ua.pt:10773/21736
Descrição
Resumo:We introduce a new family of hyperplane arrangements in dimension n≥3 that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions – the connected components of the complement of the union of the hyperplanes – which can be bijectively labeled with the Pak–Stanley labeling. In addition, we show that, in the cases of the Shi and the Ish arrangements, the number of labels with reverse centers of a given length is equal, and conjecture that the same happens with all of the members of the family.