论文标题
倒置安排和弱序秩序
Inversion arrangements and the weak Bruhat order
论文作者
论文摘要
对于每个排列$ w $，我们可以根据$ w $的反转来构建超平面$ \ mathcal {a} _W $的集合，这称为倒置超平面布置与$ w $相关。它是由后尼科夫猜想的，并由霍尔特曼，林努斯，莎拉什安和萨斯特兰确认，$ \ mathcal {a} _W $的区域数量小于或等于或等于在$ w $以下的$ w $低于$ w $的订单的数量，并且仅在$ w $ w $ wh $ whods and the＆w $ whopts and and plocy and and thefters off $ w $ whotess and theft $ w $ hyters and theft $ w $ hyters and theft $ w $ hyters and theft $ w。 351624。在本文中，我们表明$ \ MATHCAL {a} _W $的区域数量大于或等于弱的bruhat顺序中$ w $以下的排列数量，并且仅当$ w $避免使用模式231和312时，且仅在$ w $时就保持平等。
For each permutation $w$, we can construct a collection of hyperplanes $\mathcal{A}_w$ according to the inversions of $w$, which is called the inversion hyperplane arrangement associated to $w$. It was conjectured by Postnikov and confirmed by Hultman, Linusson, Shareshian and Sjöstrand that the number of regions of $\mathcal{A}_w$ is less than or equal to the number of permutations below $w$ in the Bruhat order, with the equality holds if and only if $w$ avoids the four patterns 4231, 35142, 42513 and 351624. In this paper, we show that the number of regions of $\mathcal{A}_w$ is greater than or equal to the number of permutations below $w$ in the weak Bruhat order, with the equality holds if and only if $w$ avoids the patterns 231 and 312.
