Generic bases for cluster algebras and the Chamber Ansatz
Generic bases for cluster algebras and the Chamber Ansatz
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DOI：
10.1090/s0894-0347-2011-00715-7
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发表时间：
2010-04
影响因子：
3.9
通讯作者：
C. Geiss;B. Leclerc;J. Schroer
中科院分区：
文献类型：
--
作者：
C. Geiss;B. Leclerc;J. Schroer
设Q是一个没有定向圈的有限代数，$\Lambda$是相应的预投射代数.设g是具有由Q给出的Cartan基准的Kac-Moody李代数，W是它的Weyl群. W中的w对应于Kac-Moody群的单幂单元N^w和李代数g。在以前的工作中，我们以自然的方式证明了N^w的坐标环\C[N^w]是一个簇代数。一个中心的作用是由生成函数\vphi_X的欧拉特征的某些品种的部分组成系列的X，其中X贯穿所有模块的Frobenius子范畴C_w的范畴的幂零$\Lambda$-模块。我们证明了对于C_w中的每个X，\vphi_X在适当改变变量后与Fu和Keller关于C_w中的任何团倾斜模T的团特征标一致.作为应用，我们得到了一个新的描述的一般基础的集群代数从\C[N^w]通过特殊化的系数为1。对于系数自由的非循环簇代数的特殊情况，证明了杜邦的一个猜想。
Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be its Weyl group. With w in W is associated a unipotent cell N^w of the Kac-Moody group with Lie algebra g. In previous work we proved that the coordinate ring \C[N^w] of N^w is a cluster algebra in a natural way. A central role is played by generating functions \vphi_X of Euler characteristics of certain varieties of partial composition series of X, where X runs through all modules in a Frobenius subcategory C_w of the category of nilpotent $\Lambda$-modules. We show that for every X in C_w, \vphi_X coincides after appropriate changes of variables with the cluster characters of Fu and Keller associated with any cluster-tilting module T of C_w. As an application, we get a new description of a generic basis of the cluster algebra obtained from \C[N^w] via specialization of coefficients to 1. For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.