Zeros of covariant vector fields for the point groups : invariant formulation

All finite as well as infinite (matrix) point subgroups of full orthogonal groups in two and three dimensions are considered. For each point group a polynomial integrity basis for invariants and the basic polynomial vector fields are first given. Then, the strata are defined via equations and inequalities involving the integrity basis. Finally, equations for zeros of a covariant vector field are given on each stratum in terms of the integrity basis, which appears via coefficients in the expansion of the vector field on the vector-field basis. All the results are tabulated and an illustration using the cubic group is presented. Mathematical background sufficient for extensions of the results is also given.

[1]  M. Jarić Orbit space and minima of Landau–Higgs potentials , 1983 .

[2]  M. Jarić Extrema of Landau and Higgs Polynomials and Fixed Points of Renormalization-Group Equations , 1982 .

[3]  Jaisam Kim General Method for Analyzing Higgs Potentials , 1982 .

[4]  M. Jarić Spontaneous symmetry breaking and the chain criterion , 1981 .

[5]  Louis Michel,et al.  SYMMETRY DEFECTS AND BROKEN SYMMETRY. CONFIGURATIONS - HIDDEN SYMMETRY , 1980 .

[6]  Richard P. Stanley,et al.  Invariants of finite groups and their applications to combinatorics , 1979 .

[7]  V. Kopský Tensorial covariants for the 32 crystal point groups , 1979 .

[8]  R. T. Sharp,et al.  Polynomial irreducible tensors for point groups , 1978 .

[9]  Richard P. Stanley,et al.  Relative invariants of finite groups generated by pseudoreflections , 1977 .

[10]  J. Birman,et al.  New algorithms for the Molien function , 1977 .

[11]  Gerald W. Schwarz SMOOTH FUNCTIONS INVARIANT UNDER THE ACTION OF A COMPACT LIE GROUP , 1975 .

[12]  L. Flatto,et al.  Invariants of Finite Reflection Groups and Mean Value Problems II , 1969 .

[13]  J. J. Hopfdeld Fine structure in the optical absorption edge of anisotropic crystals , 1960 .

[14]  D. Montgomery,et al.  THE EXISTENCE OF A SLICE , 1957 .

[15]  Burnett Meyer,et al.  On the Symmetries of Spherical Harmonics , 1954, Canadian Journal of Mathematics.

[16]  G. C. Shephard,et al.  Finite Unitary Reflection Groups , 1954, Canadian Journal of Mathematics.

[17]  H. S. M. Coxeter,et al.  The product of the generators of a finite group generated by reflections , 1951 .