AI 中文总结
研究行和为零的对称矩阵的循环全正性问题，通过在格拉斯曼流形坐标环上构造种子并与\(\mathcal{CM}_n\)比较，关联循环全正性与格拉斯曼流形正性，还证明\(\mathcal{LM}_n\)与电气网络非紧化空间坐标环同构。
AI 中文摘要
本文研究行和为零的对称矩阵的循环全正性问题，这类矩阵恰是电气网络的响应矩阵。Alman、Lian和Tran在两个相关框架（簇代数\(\mathcal{CM}_n\)和洛朗现象代数\(\mathcal{LM}_n\)）中描述了循环全正性测试。首先在格拉斯曼流形\(\mathrm{Gr}(n - 1, 2n)\)的坐标环上的斯科特簇代数结构中构造了一个完全由循环子式组成的种子，并将其诱导的簇结构与\(\mathcal{CM}_n\)比较，得出奇数\(n\)时二者同构，借此关联循环全正性与格拉斯曼流形中的正性。其次证明洛朗现象代数\(\mathcal{LM}_n\)与电气网络非紧化空间的坐标环同构，或等价于格罗夫代数的某个局部化。
英文摘要
The paper studies the problem of circular total positivity of the symmetric matrices with zero row sums. These matrices are exactly response matrices of the electrical networks. Alman, Lian and Tran described tests for circular total positivity in two related frameworks: the cluster algebra $\mathcal{CM}_n$ and the Laurent Phenomenon algebra $\mathcal{LM}_n$. Our first result is the construction of a seed in Scott's cluster algebra structure on the coordinate ring of the Grassmannian $\mathrm{Gr}(n-1,2n)$ that consists entirely of circular minors. We compare the cluster structure induced by this seed with $\mathcal{CM}_n$. In particular, for odd $n$ the cluster algebra structure $\mathcal{CM}_n$ is isomorphic to the cluster algebra structure on $\mathrm{Gr}(n-1,2n)$ subject to natural freezing and trivialization of certain cluster variables in their initial seeds. We use this isomorphism to relate circular total positivity to positivity in the Grassmannian. Our second result is that the Laurent Phenomenon algebra $\mathcal{LM}_n$ is isomorphic to the coordinate ring of the noncompactified space of electrical network, or equivalently, to a certain localization of the grove algebra.
