论文标题
舒伯特多项式的定居者品种，混合的欧拉数和主要专业
The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials
论文作者
论文摘要
我们计算在舒伯特类别的基础上，计算了定位型品种的共同体学类别的扩展。生成的结构常数$ a_w $表示为\ emph {normolized}混合欧拉数字的总和自然而然地索引了$ w $。该描述意味着$ a_w $对于所有排列的$ w \ in Length $ n-1 $ in s_n $均为阳性，从而回答了田园，Horiguchi，Masuda和Park的问题。我们使用相同的表达式来建立$ a_w $的不变性，并通过最长的单词进行倒置和共轭，然后为数字建立一个有趣的环状规则。然后，我们通过利用与Postnikov的分裂对称性相关的利用，朝着对$ a_w $的更深入的组合理解。最后，就某些Tableau的下降而言，当$ w $ vexillary时，我们能够对$ a_w $进行组合解释。它部分基于数字$ a_w $与舒伯特多项式的主要专业之间的关系。在此过程中，我们证明了结果并提出了有关排列，舒伯特多项式和相关对象的独立兴趣的问题。我们还绘制了如何将方法扩展到其他谎言类型的方法，特别是强调了Klyachko的身份。
We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants $a_w$ are expressed as a sum of \emph{normalized} mixed Eulerian numbers indexed naturally by reduced words of $w$. The description implies that the $a_w$ are positive for all permutations $w\in S_n$ of length $n-1$, thereby answering a question of Harada, Horiguchi, Masuda and Park. We use the same expression to establish the invariance of $a_w$ under taking inverses and conjugation by the longest word, and subsequently establish an intriguing cyclic sum rule for the numbers. We then move toward a deeper combinatorial understanding for the $a_w$ by exploiting in addition the relation to Postnikov's divided symmetrization. Finally, we are able to give a combinatorial interpretation for $a_w$ when $w$ is vexillary, in terms of certain tableau descents. It is based in part on a relation between the numbers $a_w$ and principal specializations of Schubert polynomials. Along the way, we prove results and raise questions of independent interest about the combinatorics of permutations, Schubert polynomials and related objects. We also sketch how to extend our approach to other Lie types, highlighting in particular an identity of Klyachko.
