不变多项式的平坦基与E7和E8的P-矩阵
Flat Bases of Invariant Polynomials and P-matrices of E7 and E8
摘要：
设$G$是欧氏空间$V$的线性变换的紧群。$G$不变$C^\infty$函数可以表示为$G$不变齐次多项式的有限基本集的$C^\infty$函数，称为完整性基.轨道空间$V/G$的数学描述也依赖于完整性基：它是通过多项式方程和表示$P$ -矩阵（由完整性基确定的实对称矩阵）的秩和半正定性条件的不等式来实现的。构成完整基的$G$ -不变齐次多项式的基集的选择不唯一，因此轨道空间的数学描述也不唯一。如果$G$是一个不可约有限反射群，Saito等人在1980年刻画了$G$ -不变齐次多项式的一些特殊的基本集合，他们称之为{ \em flat}。他们还明确地发现了平坦的基本套不变齐次多项式的所有不可约有限反射群除了两个最大的群体$E_7$和$E_8$。本文明确地确定了$E_7$和$E_8$的不变齐次多项式的平坦基集及相应的$P$ -矩阵。使用本文报道的结果，能够容易地确定对应于$E_7$或$E_8$的任何其他完整性基础的$P$ -矩阵。从$P$矩阵中，人们可以写出定义相对于平坦基或任何其他完整基的$E_7$和$E_8$的轨道空间的方程和不等式。本文所得结果可具体用于分析研究对称群为有限反射群$E_7$和$E_8$之一或李群$E_7$和$E_8$之一的所有理论中的对称破缺。
英文摘要：
Let $G$ be a compact group of linear transformations of an Euclidean space $V$. The $G$-invariant $C^\infty$ functions can be expressed as $C^\infty$ functions of a finite basic set of $G$-invariant homogeneous polynomials, called an integrity basis. The mathematical description of the orbit space $V/G$ depends on the integrity basis too: it is realized through polynomial equations and inequalities expressing rank and positive semi-definiteness conditions of the $P$-matrix, a real symmetric matrix determined by the integrity basis. The choice of the basic set of $G$-invariant homogeneous polynomials forming an integrity basis is not unique, so it is not unique the mathematical description of the orbit space too. If $G$ is an irreducible finite reflection group, Saito et al. in 1980 characterized some special basic sets of $G$-invariant homogeneous polynomials that they called {\em flat}. They also found explicitly the flat basic sets of invariant homogeneous polynomials of all the irreducible finite reflection groups except of the two largest groups $E_7$ and $E_8$. In this paper the flat basic sets of invariant homogeneous polynomials of $E_7$ and $E_8$ and the corresponding $P$-matrices are determined explicitly. Using the results here reported one is able to determine easily the $P$-matrices corresponding to any other integrity basis of $E_7$ or $E_8$. From the $P$-matrices one may then write down the equations and inequalities defining the orbit spaces of $E_7$ and $E_8$ relatively to a flat basis or to any other integrity basis. The results here obtained may be employed concretely to study analytically the symmetry breaking in all theories where the symmetry group is one of the finite reflection groups $E_7$ and $E_8$ or one of the Lie groups $E_7$ and $E_8$ in their adjoint representations.
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。
