论文标题
Lorentz和颗粒的置换不变II
Lorentz and permutation invariants of particles II
论文作者
论文摘要
Weyl的两个定理告诉我们，Lorentz-（和平等）的代数不变多项式在$ n $粒子的矩中是由点产品产生的，并且当$ n $超过spacetime d $ d $的冗余是由$（d+1）$ n $ n $ n $ matr的$ d $生成的。在这里，我们使用不变代数的Cohen-Macaulay结构来提供更直接的表征，以Hironaka分解。这种方法的好处之一是，它可以直接将置换组作用于颗粒上的情况（例如某些颗粒相同的情况下）进行直接推广。在第一个非平凡的情况下，$ n = D+1 $，我们给出了一个均匀的参数系统，该系统对于任意置换对称性的作用有效，并猜想了整个Hironaka分解的情况，而无需置换对称性。附录为相关希尔伯特系列的计算提供了公式，以$ d \ leq 4 $。
Two theorems of Weyl tell us that the algebra of Lorentz- (and parity-) invariant polynomials in the momenta of $n$ particles are generated by the dot products and that the redundancies which arise when $n$ exceeds the spacetime dimension $d$ are generated by the $(d+1)$-minors of the $n \times n$ matrix of dot products. Here, we use the Cohen-Macaulay structure of the invariant algebra to provide a more direct characterisation in terms of a Hironaka decomposition. Among the benefits of this approach is that it can be generalized straightforwardly to cases where a permutation group acts on the particles, such as when some of the particles are identical. In the first non-trivial case, $n=d+1$, we give a homogeneous system of parameters that is valid for the action of an arbitrary permutation symmetry and make a conjecture for the full Hironaka decomposition in the case without permutation symmetry. An appendix gives formulæ for the computation of the relevant Hilbert series for $d \leq 4$.
