Optimal Descriptions of Orbit Spaces and Strata of finite Groups

摘要

Let G is contained in GL_n(R) be a finite matrix group and X is contained in R~n be a G-variety. We propose a new approach for computing a stratification of X with respect to the orbit type of R~n respectively of the quotient X/G and we present new algorithms for this task. For X = R~n these algorithms yield an optimal description of each stratum and of the orbit space in terms of polynomial equations and inequalities (optimal with respect to the number of inequalities). Moreover we show that the dimension d of a stratum Σ_d of R~n/G is an upper and lower bound for the number of inequalities needed for a description of Σ_d and its closure, which improves the upper bound d(d+1)/2, which holds for general basic closed semialgebraic sets of dimension d. Additionally, our algorithms allow to compute strata of particular interest of X/G, which demands less computational resources. By performing computations as long as possible in R~n (and not in R~n/G) and by refining results of Procesi and Schwarz, it seems that our algorithms are more efficient than the present approach. We conclude by giving an application of our algorithms to the problem of constructing a potential for Nickel-Titanium alloys and compare the runtime with other algorithms.