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A geometrical construction for the polynomial invariants of some reflection groups. (English) Zbl 1106.13005
There are four groups generated by reflections which operate on the four-dimensional Euclidean space. They are symmetry groups of regular polytopes. In the paper under review the author studies the algebra of invariants of two of them, \([3,4,3]\) and \([3,3,5]\) in the notation of H. S. M. Coxeter [Regular polytopes. 2nd ed. New York: The Macmillan Company; London: Collier-Macmillan Ltd. (1963; Zbl 0118.35902)] of order 1152 and 14400, respectively, often denoted in the literature as \(F_4\) and \(H_4\). An old result of G. Racah [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 8, 108–112 (1950; Zbl 0036.15601)] based on the theory of Lie groups gives that the algebras of invariants \({\mathbb R}[x_0,x_1,x_2,x_3]^G\), \(G=[3,4,3]\) and \([3,3,5]\), are generated by invariants of degree \(2,6,8,12\) and \(2,12,20,30\), respectively. In the literature, there are several constructions which give the explicit form of these invariants.
In the present paper the author presents a different construction. She considers special \([3,4,3]\)- and \([3,3,5]\)-orbits of lines on the quadric \({\mathbb P}_1\times{\mathbb P}_1\) in \({\mathbb P}_3\) and constructs the invariant polynomials with some additional geometric considerations. She uses very little computer algebra, only MAPLE for some computations in two of the propositions. The construction gives a simple proof of the result of Racah and establishes relations with the invariants of some binary subgroups of \(SU(2)\). It also may be helpful in the study of the geometry of algebraic surfaces defined by the zero sets of the algebraic invariants. The author points to families of surfaces with many symmetries. For example, it is possible to determine immediately the base locus of these families which consists of sets of lines on \({\mathbb P}_1\times{\mathbb P}_1\).

13A50 Actions of groups on commutative rings; invariant theory
20F55 Reflection and Coxeter groups (group-theoretic aspects)
14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
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