A geometrical construction for the polynomial invariants of some reflection groups

In these notes we investigate the ring of real polynomials in four variables, which are invariant under the action of the reflection groups [3, 4, 3] and [3, 3, 5]. It is well known that they are rationally generated in degree 2,6,8,12 and 2,12,20,30. We give a different proof of this fact by giving explicit equations for the generating polynomials. 0 Introduction There are four groups generated by reflections which operate on the fourdimensional Euclidian space. These are the symmetry groups of some regular four dimensional polytopes and are described in [Co2] p. 145 and table I p. 292-295. With the notation there the groups and their orders are Group [3, 3, 3] [3, 3, 4] [3, 4, 3] [3, 3, 5] Order 120 384 1152 14400 They operate in a natural way on the ring of polynomialsR = R[x0, x1, x2, x3] and it is well known that the ring of invariants R (G one of the groups above) is algebraically generated by a set of four independent polynomials (cf. [Bu] p. 357). Coxeter shows in [Co1] that the rings R, G = [3, 3, 3] or [3, 3, 4] are generated in degree 2, 3, 4, 5 resp. 2, 4, 6, 8 and since the product of the degrees is equal to the order of the group, any other invariant polynomial is a combination with real coefficients of products of these invariants (i.e., in the terminology of [Co1], the ring R is rationally generated by the polynomials). Coxeter also gives equations for the generators. In the case of the groups [3, 4, 3] and [3, 3, 5] he recalls a result of Racah, who shows with 1 the help of the theory of Lie groups that the rings R are rationally generated in degree 2, 6, 8, 12 resp. 2, 12, 20, 30 (cf. [Ra]). Neither Coxeter nor Racah give equations for the polynomials. In these notes we construct the generators and give a different proof of the result of Racah. The invariant of degree two is well known ( cf. [Co1] ) and can be given as q = x0 + x 2 1 + x 2 2 + x 2 3. We construct the other invariants in a completely geometrical way. For proving that our polynomials together with the quadric generate the ring R, we show some relations between them and the invariant forms of the binary tetrahedral group and of the binary icosahedral group. It is a pleasure to thank W. Barth of the University of Erlangen for many helpful discussions. 1 Notations and preliminaries Denote by R the ring of polynomials in four variables with real coefficients R[x0, x1, x2, x3], by G a finite group of homogeneous linear substitutions, and by R the ring of invariant polynomials. 1. A set of polynomials F1, . . . , Fn in R is called algebraically dependent if there is a non trivial relation ∑ αI(F i1 1 · . . . · F in n ) = 0, where I = (i1, . . . , in) ∈ N, αI ∈ R. 2. The polynomials are called algebraically independent if they are not dependent. For the ring R, there always exists a set of four algebraically independent polynomials (cf. [Bu], thm. I, p. 357). 3. We say thatR is algebraically generated by a set of polynomials F1, . . . , F4, if for any other polynomial P ∈ R we have an algebraic relation ∑ αI(P i0 · F i1 1 · . . . · F i4 4 ) = 0. 4. We say that the ring R is rationally generated by a set of polynomials F1, . . . , F4, if for any other polynomial P ∈ R we have a relation ∑ αI(F i1 1 · . . . · F i4 4 ) = P, αI ∈ R 5. The four polynomials of 3 are called a basic set if they have the smallest possible degree (cf. [Co1]). 6. There are two classical 2 : 1 coverings ρ : SU(2) → SO(3) and σ : SU(2)× SU(2) → SO(4),