线性拟阵积：一种综合方法

中文摘要 AI 辅助

最近建立的关于均匀拟阵的张量、对称和外积与抽象（双）刚性拟阵之间的对偶性，连接了此前被认为无关的两个领域的定理和开放问题。特别是，Mason在1981年提出的一个问题询问任意两个均匀拟阵是否允许一个最自由的积。通过对偶性，Cruickshank、Jackson、Jordán和Tanigawa对其的重新表述询问一般双刚性拟阵是否是其参数的最自由抽象双刚性拟阵。Jackson和Tanigawa的一个相关猜想询问一般$2$-超连通拟阵是否是最自由的$\{K_4,K_{3,3}\}$-拟阵。我们表明这些问题可以有效地解决，并且对于线性可表示拟阵类常常可以完全解决。为此，我们引入了星基正规形式，允许比较同一域上抽象（双）刚性拟阵的表示，从而细化了底层拟阵上的弱序关系。利用它，我们证明了一般双刚性拟阵是最自由的线性可表示抽象$(a,b)$-双刚性拟阵。这证实了Mason关于线性拟阵的猜想。我们证明每个可表示的抽象$2$-刚性拟阵都允许一个刚性矩阵表示。这是对一般$2$-刚性拟阵极大性性质的加强。我们还证明每个可表示的抽象$2$-刚性拟阵，其中每个$K_{3,3}$副本都是一个回路，都允许一个超连通矩阵表示。由此得出，一般刚性和超连通族$\mathcal{R}_2$和$\mathcal{H}_2$是仅有的线性可表示$2$-刚性族。

英文摘要

The recently established duality between tensor, symmetric and exterior products of uniform matroids on the one hand, and abstract (bi-)rigidity matroids on the other hand, connects theorems and open questions from both areas, previously thought unrelated. In particular, a 1981 question of Mason asks if any two uniform matroids admit a freest product. Via duality, its reformulation due to Cruickshank, Jackson, Jordán and Tanigawa asks if the generic birigidity matroid is the freest abstract birigidity matroid for its parameters. A related conjecture of Jackson and Tanigawa asks if the generic $2$-hyperconnectivity matroid is the freest $\{K_4,K_{3,3}\}$-matroid. We show that these problems can be addressed effectively, and often solved completely for the class of linearly representable matroids. To this end, we introduce star-basis normal forms that allow to compare representations of abstract (bi-)rigidity matroids over the same field, resulting in a refinement of the weak order relation on the underlying matroids. Utilizing it, we prove that the generic birigidity matroid is the freest linearly representable abstract $(a,b)$-birigidity matroid. This confirms Mason's conjecture for linear matroids. We prove that every representable abstract $2$-rigidity matroid admits a rigidity matrix representation. This is a strengthening of the maximality property of the generic $2$-rigidity matroid. We also prove that every representable abstract $2$-rigidity matroid in which every copy of $K_{3,3}$ is a circuit admits a hyperconnectivity matrix representation. It follows that the generic rigidity and hyperconnectivity families $\mathcal{R}_2$ and $\mathcal{H}_2$ are the only linearly representable $2$-rigidity families.

发表机构

• Charles University（查理大学）

机构由 AI 辅助整理，请以论文原文为准。