On the Periodicity Conjecture for Y-systems
On the Periodicity Conjecture for Y-systems
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关于Y系统的周期性猜想
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
A. Volkov
中科院分区：
文献类型：
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作者：
A. Volkov
这个猜想（最终由阿列克谢·扎莫洛奇科夫提出）断言了所谓的Y系统的所有解的周期性。这些系统自然地与有限型的不可分解的Cartan矩阵对相关联，并且约束周期等于相应Coxeter数之和的两倍。这个猜想到目前为止只在其中一个秩等于1时才被证明，在这种情况下，Y系统与Fomin-Zelevinsky的簇代数有内在的联系。本文利用初等射影几何证明了两个Cartan矩阵都是A型矩阵且两个秩都是任意的情形。
The conjecture in question (due ultimately to Alexei Zamolodchikov) asserts the periodicity of all the solutions to the so-called Y-systems. Those systems are naturally associated to pairs of indecomposable Cartan matrices of finite type, and the conjectured period is equal to twice the sum of the respective Coxeter numbers. This conjecture has so far been proven only if one of the ranks equals one, in which case the Y-systems are intrinsically related to Fomin-Zelevinsky’s cluster algebras. In this paper, I use elementary projective geometry to prove the case when the two Cartan matrices involved are of type A with both ranks arbitrary.