论文标题
代表理论中较高的stasheff- tamari命令
The higher Stasheff--Tamari orders in representation theory
论文作者
论文摘要
我们表明，Oppermann和Thomas在循环多面体的三角形和较高的Auslander代数之间发现的关系，表示$ a_ {n}^{d} $，是一个非常富裕的。 \ emph {较高的stasheff--tamari订单}是循环多面体三角构造的两个顺序，猜想是等效的，由卡普拉诺夫和沃沃沃斯基，弗沃德斯基，埃德曼和莱恩定义为1990年代。我们首先表明，这些订单均匀地与Riedtmann和Schofield定义的倾斜模块的自然订单相对应，并由Happel and Unger进行了研究。该结果使我们能够证明奇数循环多面体的三角剖分是在培养的，其等效类别为$ d $ - 毫米 - 毫米绿色序列的$ a_ {n}^{d} $，我们将其作为凯勒（Keller）原始最大绿色序列的更高维度泛化。我们进一步解释了较高的stasheff- tasheff-tamari订单，在奇数维度上，它们对应于$ d $ - 毫米绿色序列的等价类别的自然订单。 $ d $毫米绿色序列的这两个部分订单的猜想是与Brüstle，Dupont和Perotin的``no-Gap''猜想的定向版本相等的。我们的结果的推论是，此猜想适用于$ a_ {n} $，并且在这里（1-）最大绿色序列的等价类集合是一个晶格。
We show that the relationship discovered by Oppermann and Thomas between triangulations of cyclic polytopes and the higher Auslander algebras of type $A$, denoted $A_{n}^{d}$, is an incredibly rich one. The \emph{higher Stasheff--Tamari orders} are two orders on triangulations of cyclic polytopes, conjectured to be equivalent, defined in the 1990s by Kapranov and Voevodsky, and Edelman and Reiner. We first show that these orders correspond in even dimensions to natural orders on tilting modules defined by Riedtmann and Schofield and studied by Happel and Unger. This result allows us to show that triangulations of odd-dimensional cyclic polytopes are in bijection with equivalence classes of $d$-maximal green sequences of $A_{n}^{d}$, which we introduce as a higher-dimensional generalisation of the original maximal green sequences of Keller. We further interpret the higher Stasheff--Tamari orders in odd dimensions, where they correspond to natural orders on equivalences classes of $d$-maximal green sequences. The conjecture that these two partial orders on equivalence classes of $d$-maximal green sequences are equal amounts to an oriented version of the ``no-gap'' conjecture of Brüstle, Dupont, and Perotin. A corollary of our results is that this conjecture holds for $A_{n}$, and that here the set of equivalence classes of (1-)maximal green sequences is a lattice.
