论文标题
布鲁哈特订单中的布尔元素
Boolean elements in the Bruhat order
论文作者
论文摘要
我们表明，只有当它避免了一组Billey-Postnikov模式时，我们就表明$ w \ in W $是布尔的，我们会明确描述。我们的证明是基于对反转集的分析，并且非常均匀。我们还介绍了避免线性图案的概念，并表明布尔元素的特征是避免了$ 3 $线性图案$ s_1 s_2 s_2 s_1 \ in W（a_2）$，$ s_2 s_1 s_1 s_1 s_3 s_2 \ in W（a_3）$和$ s_2 s_2 s_1 s_1 s_1 s_3 s_3 s_4 s_4 s_4 s_4 $ s_4 $ s_4 $ s_4 $ s_4 $ s_4 $ s_4 $。我们还考虑了$ K $ -Boolean Weyl Group Elements更一般的情况。我们说，如果每种发电机的$ w $最多包含$ w $的每个减少表达式，则$ w \是$ k $ -Boolean。我们表明，对称群体$ s_n $的$ 2 $ -Boolean元素的特征是避免了$ 3421,4312,4321，$和$ 456123 $的$ 3421,4312,4321，$和$ 456123 $，并为$ 2 $ $ s_n $的$ 2 $ boolean Elements提供理性生成功能。
We show that $w\in W$ is boolean if and only if it avoids a set of Billey-Postnikov patterns, which we describe explicitly. Our proof is based on an analysis of inversion sets, and it is in large part type-uniform. We also introduce the notion of linear pattern avoidance, and show that boolean elements are characterized by avoiding just the $3$ linear patterns $s_1 s_2 s_1 \in W(A_2)$, $s_2 s_1 s_3 s_2 \in W(A_3)$, and $s_2 s_1 s_3 s_4 s_2 \in W(D_4)$. We also consider the more general case of $k$-boolean Weyl group elements. We say that $w\in W$ is $k$-boolean if every reduced expression for $w$ contains at most $k$ copies of each generator. We show that the $2$-boolean elements of the symmetric group $S_n$ are characterized by avoiding the patterns $3421,4312,4321,$ and $456123$, and give a rational generating function for the number of $2$-boolean elements of $S_n$.
