Mixed Eulerian Numbers and Peterson Schubert Calculus
Mixed Eulerian Numbers and Peterson Schubert Calculus
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DOI：
10.1093/imrn/rnad030
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发表时间：
2021-04
影响因子：
1
通讯作者：
T. Horiguchi
中科院分区：
文献类型：
--
作者：
T. Horiguchi
让$\Phi $是一个根系统。Postnikov引入并研究了混合$\Phi $-欧拉数。这些数字表示$\Phi $-超单形的混合体积。作为这些数的特殊化，可以得到通常的欧拉数、卡塔兰数和二项式系数。Berget-Spink-Tseng最近的工作给出了当$\Phi $为$A$型时混合$\Phi $-欧拉数的一个简单计算。本文将混合$\Phi $-欧拉数与PetersonSchubert微积分联系起来。通过使用连接，我们提供了一个组合模型的计算Berget-Spink-Tseng的左右图，介绍了Abe-Horiguchi-Kuwata-Zeng的彼得森舒伯特演算的目的。我们还得到了一个简单的计算混合$\Phi $-欧拉数在任意的李类型从彼得森舒伯特演算。
Let $\Phi $ be a root system. Postnikov introduced and studied the mixed $\Phi $-Eulerian numbers. These numbers indicate the mixed volumes of $\Phi $-hypersimplices. As specializations of these numbers, one can obtain the usual Eulerian numbers, the Catalan numbers, and the binomial coefficients. Recent work of Berget–Spink–Tseng gave a simple computation for the mixed $\Phi $-Eulerian numbers when $\Phi $ is of type $A$. In this paper, we connect a relation between mixed $\Phi $-Eulerian numbers and Peterson Schubert calculus. By using the connection, we provide a combinatorial model for the computation of Berget–Spink–Tseng in terms of left–right diagrams that were introduced by Abe–Horiguchi–Kuwata–Zeng for the purpose of Peterson Schubert calculus. We also derive a simple computation for the mixed $\Phi $-Eulerian numbers in arbitrary Lie types from Peterson Schubert calculus.