Optimal descriptions of orbit spaces and strata of finite groups

ABSTRACT

Let G ⊆ GL;_n; (R be a finite matrix group and X⊆R ⁿ be a G-variety. We propose a new approach for computing a stratification of X with respect to the orbit type of Rⁿ respectively of the quotient X/G and we present new algorithms for this task. For X = Rⁿ 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 ∑ ;_d; of R;_n;/G is an upper and lower bound for the number of inequalities needed for a description of ∑ ;_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 Rⁿ (and not in Rⁿ/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.

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms

General Terms:
Algorithms

Keywords:
stratification, symbolic computation