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Ad-nilpotent ideals and The Shi arrangement

Abstract

We extend the Shi bijection from the Borel subalgebra case to parabolic subalgebras. In the process, the II-deleted Shi arrangement Shi(I)\texttt{Shi}(I) naturally emerges. This arrangement interpolates between the Coxeter arrangement Cox\texttt{Cox} and the Shi arrangement Shi\texttt{Shi}, and breaks the symmetry of Shi\texttt{Shi} in a certain symmetrical way. Among other things, we determine the characteristic polynomial Ο‡(Shi(I),t)\chi(\texttt{Shi}(I), t) of Shi(I)\texttt{Shi}(I) explicitly for Anβˆ’1A_{n-1} and CnC_n. More generally, let Shi(G)\texttt{Shi}(G) be an arbitrary arrangement between Cox\texttt{Cox} and Shi\texttt{Shi}. Armstrong and Rhoades recently gave a formula for Ο‡(Shi(G),t)\chi(\texttt{Shi}(G), t) for Anβˆ’1A_{n-1}. Inspired by their result, we obtain formulae for Ο‡(Shi(G),t)\chi(\texttt{Shi}(G), t) for BnB_n, CnC_n and DnD_n.Comment: The third version, quasi-antichains are shown to be in bijection with elements of L(Cox). arXiv admin note: text overlap with arXiv:1009.1655 by other author

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