Edmonds问题与双表示轨道半群的成员问题
Edmonds' problem and the membership problem for orbit semigroups of quiver representations
摘要：
由J. Edmonds提出的代数复杂性中的一个中心问题要求判定$N \times N$复矩阵的给定$l$-元组$\V=(\V_1, \ldots, \V_l)$的跨度是否包含非奇异矩阵。在本文中，我们提供了一个不变量的理论方法来解决这个问题。将$\V$看作是$l$-Kronecker矩阵$\K_l$的表示，埃德蒙兹的问题可以被重新表述为要求决定在权重$(1,-1)$的表示空间$(\CC^{N\times N})^l$上是否存在在$\V$处不消失的半不变式。换句话说，Edmonds的问题是要求决定权$(1,-1)$是否属于$\V$的轨道半群。设$Q$是任意的非循环数，$\V$是$Q$的表示.我们研究了$\V$轨道半群的隶属度问题，主要研究了所谓的$\V$-饱和权。我们首先表明，对于任何给定的$\V$-饱和重量$σ$，检查如果$σ$属于轨道半群的$\V$可以在确定性多项式时间。其次，设$(Q, \R)$是一个无圈有界代数，其上有界代数为$A=KQ/\langle \R \rangle$，并假定$\V$满足$\R$中的关系。证明了如果$A/\Ann_A(\V)$是驯服代数，则$\V$的权半群中的任何权$σ$都是$\V$-饱和的.我们的研究结果提供了一个系统的方法来产生家庭的矩阵元组，埃德蒙兹的问题可以有效地解决。
英文摘要：
A central problem in algebraic complexity, posed by J. Edmonds, asks to decide if the span of a given $l$-tuple $\V=(\V_1, \ldots, \V_l)$ of $N \times N$ complex matrices contains a non-singular matrix. In this paper, we provide a quiver invariant theoretic approach to this problem. Viewing $\V$ as a representation of the $l$-Kronecker quiver $\K_l$, Edmonds' problem can be rephrased as asking to decide if there exists a semi-invariant on the representation space $(\CC^{N\times N})^l$ of weight $(1,-1)$ that does not vanish at $\V$. In other words, Edmonds' problem is asking to decide if the weight $(1,-1)$ belongs to the orbit semigroup of $\V$. Let $Q$ be an arbitrary acyclic quiver and $\V$ a representation of $Q$. We study the membership problem for the orbit semi-group of $\V$ by focusing on the so-called $\V$-saturated weights. We first show that for any given $\V$-saturated weight $σ$, checking if $σ$ belongs to the orbit semigroup of $\V$ can be done in deterministic polynomial time. Next, let $(Q, \R)$ be an acyclic bound quiver with bound quiver algebra $A=KQ/\langle \R \rangle$ and assume that $\V$ satisfies the relations in $\R$. We show that if $A/\Ann_A(\V)$ is a tame algebra then any weight $σ$ in the weight semigroup of $\V$ is $\V$-saturated. Our results provide a systematic way of producing families of tuples of matrices for which Edmonds' problem can be solved effectively.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
