Counting faces in the extended Shi arrangement

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We define the level of a subset X of Euclidean space to be the dimension of the smallest subspace such that the distance between each element of X and the subspace is bounded. We prove that the number of faces in the n-dimensional extended Shi arrangement Aˆn r having codimension k and level m is given by m⋅(nk)⋅Δr kΔm−1xn−1|x=rn−1, where Δ is the difference operator and Δr is the difference operator of step r, that is, Δrp(x)=p(x)−p(x−r). This generalizes a result of Athanasiadis which counts the number of faces of different dimensions in the extended Shi arrangement Aˆn r. The proof relies on a multi-variated Abel identity due to Françon.

Original languageEnglish
Pages (from-to)55-64
Number of pages10
JournalAdvances in Applied Mathematics
StatePublished - Aug 2019

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Publisher Copyright:
© 2019 Elsevier Inc.

ASJC Scopus subject areas

  • Applied Mathematics


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