Stasheff高阶的新解释--Tamari阶
New interpretations of the higher Stasheff--Tamari orders
摘要：
1996年，Edelman和Reiner在循环多面体的三角剖分上定义了两个更高的Stasheft-Tamari阶，并证明了它们是重合的。我们打开了一个代数的角度接近这个猜想，显示这些订单自然产生的表示论的高级Auslander代数的$A$型，表示$A_{n}^{d}$。为此，我们给出了新的组合解释的订单，使它们具有可比性。然后，我们将这些组合的解释到代数框架。我们还展示了如何三角化的奇数维循环多面体出现在$A_{n}^{d}$的表示论，即作为等价类的最大绿色序列。此外，我们给奇数维对应的已知描述的$2d$维三角剖分的非相交的$d$ -单形的最大尺寸。这包括在一个定义的两个新的属性，这意味着一组$d$单形产生一个$(2d+1)$维三角剖分。
英文摘要：
In 1996, Edelman and Reiner defined the two higher Stasheff--Tamari orders on triangulations of cyclic polytopes and conjectured them to coincide. We open up an algebraic angle for approaching this conjecture by showing how these orders arise naturally in the representation theory of the higher Auslander algebras of type $A$, denoted $A_{n}^{d}$. For this we give new combinatorial interpretations of the orders, making them comparable. We then translate these combinatorial interpretations into the algebraic framework. We also show how triangulations of odd-dimensional cyclic polytopes arise in the representation theory of $A_{n}^{d}$, namely as equivalence classes of maximal green sequences. We furthermore give the odd-dimensional counterpart to the known description of $2d$-dimensional triangulations as sets of non-intersecting $d$-simplices of a maximal size. This consists in a definition of two new properties which imply that a set of $d$-simplices produces a $(2d+1)$-dimensional triangulation.
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