SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 964-993, January 2020.
A synchrony subspace of $\mathbb{R}^{n}$ is defined by setting certain components of the vectors equal according to an equivalence relation. Synchrony subspaces invariant under a given set of square matrices ordered by inclusion form a lattice. Applications of these invariant synchrony subspaces include equitable and almost equitable partitions of the vertices of a graph used in many areas of graph theory, balanced and exo-balanced partitions of coupled cell networks, and coset partitions of Cayley graphs. We study the basic properties of invariant synchrony subspaces and provide many examples of the applications. We also present what we call the split and cir algorithm for finding the lattice of invariant synchrony subspaces. Our theory and algorithm is further generalized for nonsquare matrices. This leads to the notion of tactical decompositions studied for its application in design theory.

中文翻译：

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矩阵集的不变同步子空间

SIAM应用动力系统杂志，第19卷，第2期，第964-993页，2020年1月。
通过根据等价关系将向量的某些分量设置为相等来定义$ \ mathbb {R} ^ {n} $的同步子空间。在给定的一组平方矩阵下，包含子排序的同步子空间不变，形成一个晶格。这些不变同步子空间的应用包括在图论的许多领域中使用的图的顶点的相等和几乎相等的划分，耦合单元网络的平衡和exo平衡划分以及Cayley图的同集划分。我们研究不变同步子空间的基本性质，并提供许多应用实例。我们还介绍了所谓的分割和cir算法，用于查找不变同步子空间的晶格。我们的理论和算法被进一步推广到非平方矩阵。