绝对移动空间与任意Coxeter群中的非交叉划分偏序集
Absolute moved spaces and noncrossing partition posets in arbitrary Coxeter groups
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中文总结 AI 辅助
本文提出绝对移动空间概念，替代任意Coxeter群中非交叉划分偏序集元素的移动空间，保持维数且具单射性，并证明反射子群映射单射，给出秩三格性质新证明及秩四非格例子。
中文摘要 AI 辅助
在Coxeter群$W$的绝对序中，单位元与Coxeter元素$c$之间的区间$[1,c]_T$是当$W$为对称群时出现的非交叉划分偏序集的一般化。当$W$有限时，该偏序集总是格，自然地将每个元素$w\in [1,c]_T$与其在$W$的几何表示$V$中的\textit{移动空间}$\mathsf{Mov}(w)=\mathrm{Im}(w - \mathrm{Id}_V)$相关联。该空间的维数等于$w$的反射长度$\ell_T(w)$，并在$V$的子空间格中给出了$[1,c]_T$的实现。它是研究$[1,c]_T$的重要工具。当$W$无限时，元素$w\in [1,c]_T$的移动空间一般不再具有维数$\ell_T(w)$，且不同元素可能具有相同的移动空间。我们提出在任意Coxeter群中元素$w\in [1,c]_T$的移动空间的替代品，称为$w$的\textit{绝对移动空间}。$V$的这个子空间$\mathsf{AM}(w)$总是包含$\mathsf{Mov}(w)$且维数等于$\ell_T(w)$，并且不同元素具有不同的绝对移动空间。这使我们能够推导出在完全一般性下成立的非交叉划分偏序集的若干性质，并证明从$[1,c]_T$到$W$的反射子群的自然映射（该映射将$w\in [1,c]_T$关联到由绝对序中位于$w$下方的反射生成的子群$P(w)$）总是单射。此外，我们还推导出当$W$秩为三时$[1,c]_T$的格性质的新证明，并展示了秩为四的无限Coxeter群及Coxeter元素选择的无穷多个新例子，使得$[1,c]_T$不是格。
英文摘要
The interval $[1,c]_T$ between the identity element and a Coxeter element $c$ in the absolute order on a Coxeter group $W$ is a generalization of the poset of noncrossing partitions arising when $W$ is the symmetric group. When $W$ is finite, this poset is always a lattice, and it is natural to associate to every element $w\in [1,c]_T$ its \textit{moved space} $\mathsf{Mov}(w)=\mathrm{Im}(w - \mathrm{Id}_V)$ in the geometric representation $V$ of $W$. It has dimension equal to the reflection length $\ell_T(w)$ of $w$, and gives a realization of $[1,c]_T$ inside the lattice of subspaces of $V$. It is an important tool in the study of $[1,c]_T$. When $W$ is infinite, the moved space of an element $w\in [1,c]_T$ no longer has dimension $\ell_T(w)$ in general, and distinct elements may have the same moved space. We propose a replacement for the moved space of an element $w\in [1,c]_T$ in an arbitrary Coxeter group, that we call \textit{absolute moved space} of $w$. This subspace $\mathsf{AM}(w)$ of $V$ always contains $\mathsf{Mov}(w)$ and has dimension equal to $\ell_T(w)$, and distinct elements have distinct absolute moved spaces. This allows us to derive several properties of noncrossing partition posets that hold in full generality, and to show that the natural map from $[1,c]_T$ to reflection subgroups of $W$, which to $w\in [1,c]_T$ associates the subgroup $P(w)$ generated by reflections lying below $w$ in the absolute order, is always injective. Among others, we also derive a new proof of the lattice property of $[1,c]_T$ when $W$ has rank three, and exhibit infinitely many new examples of infinite Coxeter groups of rank four and choices of Coxeter elements for which $[1,c]_T$ fails to be a lattice.
发表机构
- Université Clermont Auvergne（克莱蒙费朗大学）
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