论文标题
通过蒙特卡洛模拟的代数关系
Algebraic Relations Via a Monte Carlo Simulation
论文作者
论文摘要
复杂的正交组对多项式函数对$ n \ times n $矩阵的共轭作用产生了不变多项式的分级代数。该代数的一组跨度与一组未标记的循环图相对应有，该图在二面体对称性下等效的有向边缘。当不变性的程度为$ n+1 $时，我们表明不变空间之间的关系尺寸在$ n $中线性增长。此外，我们提出了两种获得关系空间基础的方法。首先，我们使用称为年轻对称器的组代数的基础构建基础，但是随着$ n $的增加，这很快在计算上变得昂贵。因此，我们通过使用Monte Carlo算法反复生成随机矩阵，为该问题提出了一种更有效的方法。
The conjugation action of the complex orthogonal group on the polynomial functions on $n \times n$ matrices gives rise to a graded algebra of invariant polynomials. A spanning set of this algebra is in bijective correspondence to a set of unlabeled, cyclic graphs with directed edges equivalent under dihedral symmetries. When the degree of the invariants is $n+1$, we show that the dimension of the space of relations between the invariants grows linearly in $n$. Furthermore, we present two methods to obtain a basis of the space of relations. First, we construct a basis using an idempotent of the group algebra referred to as Young symmetrizers, but this quickly becomes computationally expensive as $n$ increases. Thus, we propose a more computationally efficient method for this problem by repeatedly generating random matrices using a Monte Carlo algorithm.
