论文标题
在polytope $ \ mathcal {u} _ {i，\ bar {j}} $的分区代数上
On the subdivision algebra for the polytope $\mathcal{U}_{I,\bar{J}}$
论文作者
论文摘要
Ceballos，Padrol和Sarmiento引入了Polytopes $ \ Mathcal {U} _ {i，\ bar {J}} $，为研究提供了几何方法，以研究$（i，\ bar {j}）$ - 塔玛里lattices。他们观察到某些$ \ Mathcal {u} _ {i，\ bar {j}} $与无循环的root polytopes之间的连接，并想知道是否可以使用Mészáros的subdivision代数来细分所有$ \ \ \ \ \ \ \} _ _ {i，我们从两个角度肯定地回答了这一点，一种是使用流多型的，另一个使用根多型。我们表明，$ \ Mathcal {u} _ {i，\ bar {j}} $完全等同于可以使用细分代数可以细分的流polytope。另外，我们发现$ \ Mathcal {u} _ {i，\ bar {j}} $的合适投影到一个无环的根多层，允许将root polytope的细分抬高到$ \ Mathcal {u} _} _ {i，\ bar {j}} $。结果，这意味着$ \ mathcal {u} _ {i，\ bar {j}} $的细分可以通过代数的代数解释，即使用Subdivision代数中的简化形式的单一形式的解释来获得。此外，我们表明$（i，\ bar {j}）$ - tamari复合物可以作为三角流层polytope获得。
The polytopes $\mathcal{U}_{I,\bar{J}}$ were introduced by Ceballos, Padrol, and Sarmiento to provide a geometric approach to the study of $(I,\bar{J})$-Tamari lattices. They observed a connection between certain $\mathcal{U}_{I,\bar{J}}$ and acyclic root polytopes, and wondered if Mészáros' subdivision algebra can be used to subdivide all $\mathcal{U}_{I,\bar{J}}$. We answer this in the affirmative from two perspectives, one using flow polytopes and the other using root polytopes. We show that $\mathcal{U}_{I,\bar{J}}$ is integrally equivalent to a flow polytope that can be subdivided using the subdivision algebra. Alternatively, we find a suitable projection of $\mathcal{U}_{I,\bar{J}}$ to an acyclic root polytope which allows subdivisions of the root polytope to be lifted back to $\mathcal{U}_{I,\bar{J}}$. As a consequence, this implies that subdivisions of $\mathcal{U}_{I,\bar{J}}$ can be obtained with the algebraic interpretation of using reduced forms of monomials in the subdivision algebra. In addition, we show that the $(I,\bar{J})$-Tamari complex can be obtained as a triangulated flow polytope.
