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arXiv 2307.024732023-07-05math.CO

pircon 自同构的不动元素

Fixed elements of pircon automorphisms

Mikael Hansson, Vincent Umutabazi

备注：: 10 pages

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AI总结：

本文证明 pircon 自同构的不动元素仍构成 pircon，并将其应用于对称群，证明无不动点带符号对合在超八面体群 Bruhat 序对偶下形成 EL 可壳的 pircon。

摘要：

我们证明，由 pircon 的任意自同构的不动元素所诱导的子偏序集也是一个 pircon。根据 Abdallah、Hansson 和 Hultman 的一个结果，pircon 中任意开区间的序复形是一个 PL 球或 PL 球面。我们将主要结果应用于形如 $S_{2n}$ 的对称群。一个推论是，在超八面体群上 Bruhat 序的对偶下，无不动点的带符号对合构成一个 pircon。最后，我们证明该偏序集实际上是 EL 可壳的，这是 Can、Cherniavsky 和 Twelbeck 的一个结果的 $B$ 型类似。

英文摘要：

We prove that the subposet induced by the fixed elements of any automorphism of a pircon is also a pircon. By a result of Abdallah, Hansson, and Hultman, the order complex of any open interval in a pircon is a PL ball or a PL sphere. We apply our main results to symmetric groups of the form $S_{2n}$. A consequence is that the fixed point free signed involutions form a pircon under the dual of the Bruhat order on the hyperoctahedral group. Finally, we prove that this poset is, in fact, EL-shellable, which is a type $B$ analogue of a result of Can, Cherniavsky, and Twelbeck.

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