Baur-Marsh行列式的变形
Variations on Baur--Marsh's determinant
摘要：
Baur和Marsh计算了由A型簇代数中的簇变量组成的矩阵的行列式。在这篇文章中，我们希望描述两种变化。一方面，我们计算由Baur-Marsh矩阵中的簇变量的平方组成的矩阵的行列式。一个这样的行列式承认解释为凯莱-门格尔行列式。另一方面，我们希望给出D型簇代数中簇变量矩阵的行列式公式。这个簇代数与一个有标记的定向曲面相关联。正如在鲍尔-马什的设置矩阵索引的标记点的表面和一个条目是由集群变量对应于一个弧之间的两个标记点。我们的公式断言行列式可以再次被写成簇变量的乘积。
英文摘要：
Baur and Marsh computed the determinant of a matrix assembled from the cluster variables in a cluster algebra of type A. In this article we wish to describe two variations. On the one hand, we compute determinants of matrices assembled from the squares of the cluster variables in Baur--Marsh's matrix. One such determinant admits an interpretation as a Cayley--Menger determinant. On the other hand, we wish to present a formula for the determinant of a matrix of cluster variables in a cluster algebra of type D. This cluster algebra is associated with a marked oriented surface. As in Baur--Marsh's setup the matrix is indexed by the marked points of the surface and an entry is given by the cluster variable corresponding to an arc between two marked points. Our formula asserts that the determinant may again be written as a product of cluster variables.
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