Tilting Modules over the Path Algebra of Type A, Polytopes, and Catalan Numbers
Tilting Modules over the Path Algebra of Type A, Polytopes, and Catalan Numbers
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A 型路径代数、多面体和加泰罗尼亚数的倾斜模块
DOI：
10.1090/conm/652/12982
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
L. Hille
中科院分区：
文献类型：
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作者：
L. Hille
众所周知，A型路代数上倾斜模的个数与加泰罗尼亚数Cn重合。证明了An型根的所有根的凸壳C(An)是一个整体积为(n+1)Cn+1=2n n�的多面体.此外，我们将某些多面体与倾斜模集和支承倾斜模集关联起来，并证明它们的体积分别与这些模数重合。最后，我们证明了这些多面体可以只用根系统来定义，并将它们的体积联系起来，这样我们就可以以一种新的方式得到上述结果。
It is well known that the number of tilting modules over a path algebra of type An coincides with the Catalan number Cn. Moreover, the number of support tilting modules of type An is the Catalan number Cn+1. We show that the convex hull C(An) of all roots of a root system of type An is a polytope with integral volume (n + 1)Cn+1 = 2n n � . Moreover, we associate to the set of tilting modules and to the set of support tilting modules certain polytopes and show that their volumes coincide with the number of those modules, respectively. Finally, we show that these polytopes can be defined just using the root system and relate their volumes so that we can derive the above results in a new way.