论文标题
轨道闭合的最小长度作为半经典极限
Minimal length in an orbit closure as a semiclassical limit
论文作者
论文摘要
考虑连接的复合复合物还原基在有限维矢量空间上的作用。不变理论的基本结果指出，当且仅当某些均质不变多项式在V上是非零时，即V，即V不在零锥中时，向量V的轨道闭合与原点分开。因此，有效地找到轨道闭合和原点之间的最小距离可能会导致Null Cone成员的确定性算法，这是一个重要的多项式身份测试问题，包括非交通性Edmonds问题。最近与优化的联系导致了许多不变理论中许多问题的有效算法。 在这里，我们探讨了轨道闭合和不变多项式之间的著名二元性的完善，这认为以下两个数量是重合的：（1）轨道闭合和起源之间的欧几里得距离对数，以及（2）$ v^niffime of v^^feftiment offime of v^^^k}的指数增长速率。可以从S. Zhang的工作（Archimedean Ploce，1994年的几何还原性）的工作中推导出该结果，该作品在算术几何形状中使用了复杂的工具。我们提供了一个新的独立基本证明，其灵感来自局部中央限制定理的傅立叶分析证明。我们将结果推广到最高权重矢量和同种型组件上，并探讨了这种半经典限制与代表理论中多重性的渐近行为之间的联系，经典和量子统计中的大偏差理论以及由Mathieu重新制定的Jacobian猜想。我们的公式表明，在许多情况下，可以有效地计算它们以任意精确。
Consider the action of a connected complex reductive group on a finite-dimensional vector space. A fundamental result in invariant theory states that the orbit closure of a vector v is separated from the origin if and only if some homogeneous invariant polynomial is nonzero on v, i.e. v is not in the null cone. Thus, efficiently finding the minimum distance between the orbit closure and the origin can lead to deterministic algorithms for null cone membership, an important polynomial identity testing problem including the non-commutative Edmonds problem. This connection to optimization has recently led to efficient algorithms for many problems in invariant theory. Here we explore a refinement of the famous duality between orbit closures and invariant polynomials, which holds that the following two quantities coincide: (1) the logarithm of the Euclidean distance between the orbit closure and the origin and (2) the rate of exponential growth of the 'invariant part' of $v^{\otimes k}$ in the semiclassical limit as k tends to infinity. This result can be deduced from work of S. Zhang (Geometric reductivity at Archimedean places, 1994), which uses sophisticated tools in arithmetic geometry. We provide a new and independent elementary proof inspired by the Fourier-analytic proof of the local central limit theorem. We generalize the result to projections onto highest weight vectors and isotypical components, and explore connections between such semiclassical limits and the asymptotic behavior of multiplicities in representation theory, large deviations theory in classical and quantum statistics, and the Jacobian conjecture as reformulated by Mathieu. Our formulas imply that they can be computed, in many cases efficiently, to arbitrary precision.
