Abstract
This paper continues the general survey in [30]. We consider the problem of finding in explicit form the basic invariants of symmetry groups of regular polyhedra and Gosset polyhedra. We study the structure of the invariants of infinite groups generated by oblique reflections.
Similar content being viewed by others
Literature cited
T. L. Agafonova and L. B. Medvedeva, “On the representation of equiaffine transformations of an affine space as a composition of reflections in planes,” Yarosl. Pedagog. Inst. (1982). (Dep. at VINITI 1 September 1982.)
J. F. Adams, Lectures on Lie Groups, W. A. Benjamin, New York-Amsterdam (1969).
A. D. Aleksandrov, “On filling a space by polyhedra,” Vestn. Leningr. Univ., No. 2, 33–43 (1954).
D. V. Alekseevskii, “Lie groups,” Itogi Nauki i Tekh. VINITI. Algebra. Topol. Geometriya,20, 153–192 (1982).
V. I. Arnol'd, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps [in Russian], Nauka, Moscow (1982).
I. A. Baltag and V. V. Banar', “Decomposition of equiaffine transformations of the Euclidean space R3 into a product of reflections in planes,” Kishinev. Univ., 1981. (Dep. at VINITI 10 April 1981.)
F. Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer-Verlag, Berlin-Göttingen-Heidelberg (1959).
Ya. P. Blank, “Conjugate nets of conical curves,” Dokl. Akad. Nauk SSSR,64, No. 6, 755–758 (1949).
W. Blaschke, Kreis und Kugel, 2nd ed., Walter de Gruyter, Berlin (1956).
V. G. Boltyanskii, “Combinatorial geometry,” Itogi Nauki i Tekh. VINITI. Algebra. Topol. Geometriya,19, 209–274 (1981).
N. Bourbaki, Lie Groups and Lie Algebras, Hermann, Paris (1975).
V. I. Vedernikov, “Symmetric spaces. Conjugate connections as a normalized connection,” Trudy Geometr. Sem. Inst. Informatsii AN SSSR (1966), pp. 63–88.
É. B. Vinberg, “The effective theory of invariants,” Algebra. Coll. of papers dedicated to the 90th anniversary of the birth of O. Yu. Shmidt [in Russian], Moscow (1982), pp. 27–33.
A. B. Givental', “Convolution of invariants of groups generated by reflections that are connected with simple singularities of functions,” Funkts. Anal. Prilozhen.,14, No. 2, 4–14 (1980).
S. Ya. Golubev, “On the inscription of regular simplexes in a multidimensional cube,” Ivanov. Univ., Ivanov. Energ. Inst. (1980). (Dep. at VINITI 2 June 1980.)
N. A. Gregor'ev, “Regular simplexes inscribed in a cube and Hadamard matrices,” Tr. Mat. Inst. Akad. Nauk SSSR,152, 87–88 (1980).
D. A. Gudkov, N. A. Kirsanova, and G. F. Nebukina, “Points of inflection and double tangents of curves of the fourth order. I,” Gor'k. Univ. (1982). (Dep. at VINITI 3 August 1982.)
D. A. Gudkov, N. A. Kirsanova, and G. F. Nebukina, “Points of inflection and double tangents of curves of the fourth order. II,” Gor'k. Univ. (1982). (Dep. at VINITI 3 January 1983.)
A. D. Deryugin, “The construction of Euclidean geometry and Lobachevsky geometry on the basis of the concept of symmetry,” Simferop. Univ. (1983). (Dep. at UkrNIINTI 15 April 1983.)
N. P. Dolbilin, “On three-dimensional and four-dimensional simple forms,” in: Problems of Crystallography [in Russian], Moscow State Univ. (1971), pp. 315–324.
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1979).
J. Dieudonne, La Geometrie des Groupes Classiques, 2nd ed., Springer-Verlag, Berlin-Göttingen-Heidelberg (1963).
J. Dieudonne, J. B. Carrell, and D. Mumford, Invariant Theory, Old and New, Academic Press, New York—London (1971).
K. D. Erastov, “Diametral hyperplanes,” Sb. Nauch. Tr. Tashkent. Univ., No. 623, 32–40 (1980).
M. A. Ermolin, “Kinematic properties of certain curves,” Prikl. Vopr. Differents. Geometrii, No. 1, Moscow (1982), pp. 117–127. (Dep. at VINITI 7 April 1982.)
D. P. Zhelobenko and A. I. Shtern, Representations of Lie Groups [in Russian], Nauka, Moscow (1983).
V. A. Zalgaller, Theory of Envelopes [in Russian], Nauka, Moscow (1975).
A. M. Zamorzaev, É. I. Galyarskii, and A. F. Palistrant, Chromatic Symmetry, Its Generalizations and Applications [in Russian], Shtiintsa, Kishinev (1978).
V. F. Ignatenko, “On plane algebraic curves with axes of symmetry,” Ukr. Geometrich. Sb., No. 21, 31–33 (1978).
V. F. Ignatenko, “The geometry of algebraic surfaces with symmetries,” Itogi Nauki i Tekh. VINITI. Problemy Geometrii,11, 203–240 (1980).
V. F. Ignatenko, “Algebraic surfaces with the symmetry group of the polyhedron 321,” Ukr. Geometrich. Sb., No. 23, 50–56 (1980).
V. F. Ignatenko, “On the problem of finding complete bases of algebras of polynomials invariant with respect to finite symmetry groups of the space En,” Proc. All-Union Symp. on Symmetry Theory and Its Generalizations [in Russian], Kishinev (1980), pp. 50–51.
V. F. Ignatenko, “On algebraic surfaces with symmetry groups An, Bn, Dn,” Ukr. Geometrich. Sb., No. 24, 33–39 (1981).
V. F. Ignatenko, “The general equation of an algebraic surface with the symmetry group of the polyhedron 221,” Ukr. Geometrich. Sb., No. 25, 56–60 (1982).
V. F. Ignatenko, “On invariants of finite groups generated by reflections,” Mat. Sb.,120, No. 4, 556–568 (1983).
V. F. Ignatenko, “On algebraic surfaces with the symmetry group of the polyhedron 421,” Ukr. Geometrich. Sb., No. 26, 48–55 (1983).
V. F. Ignatenko, “On a method of finding the basic invariants of finite groups generated by reflections,” Proc. All-Union School of Function Theory Dedicated to the Centenary of the Birth of N. N. Luzin [in Russian], Kemerovo (1983), p. 50.
V. F. Ignatenko, “On some applications of the diametral theory of an algebraic surface in the space Em,” Ukr. Geometrich. Sb., No. 27, 49–53 (1984).
V. F. Ignatenko and A. S. Leibin, “On the theory of planes of orthogonal symmetry of surfaces in En,” Ukr. Geometrich. Sb., No. 7, 39–54 (1970).
V. F. Ignatenko and A. S. Leibin, “On algebraic surfaces in E4 with the symmetry of regular four-dimensional simplexes and the 600-hedron,” Ukr. Geometrich. Sb., No. 11, 26–31 (1971).
V. F. Ignatenko and A. S. Leibin, “The general equation of an algebraic surface with the symmetry of a regular 600-hedron in the space E4,” Ukr. Geometrich. Sb., No. 13, 71–74 (1973).
V. F. Ignatenko and A. S. Leibin, “On the case of degeneracy of the equation of a diametral plane of a symmetric surface in Em,” Ukr. Geometrich. Sb., No. 17, 58–62 (1975).
J. W. S. Cassells, Rational Quadratic Forms, Academic Press, London—New York (1978).
M. D. Kovalev, “On a characteristic property of a disk and an n-dimensional ball,” Usp. Mat. Nauk,36, No. 6, 217–218 (1981).
H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, McGraw-Hill, New York (1967).
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, Berlin-New York (1980).
O. N. Kolesnikov, “On a rational transformation,” Izv. Krymsk. Gos. Pedagog. Inst.,35, 259–267 (1961).
A. I. Kostrikin, Introduction to Algebra [in Russian], Nauka, Moscow (1977).
A. I. Krivoruchko, “The structure of compact subsets of an affine plane that have infinite sets of axes of oblique symmetry,” Simferop. Univ. (1983). (Dep. at UkrNIINTI 11 July 1983.)
V. A. Kutin, “The area of a surface of a three-axis ellipsoid,” Perm. Univ. (1982). (Dep. at VINITI 6 June 1982.)
S. I. Kucherenko and N. A. Nikulin, “On some curves of the fourth order,” Prikl. Geometr. Inzh. Grafika, No. 30, 69–71 (1980).
É. A. Laudynya, “A characteristic circle of an affine transformation of the Euclidean plane,” Sb. Nauchn. Tr. Yarosl. Gos. Pedagog. Inst., No. 200, 56–67 (1982).
V. S. Makarov, “Geometric methods of constructing discrete groups of motions of Lobachevsky space,” Itogi Nauki i Tekhn. VINITI, Problemy Geometrii,15, 3–59 (1983).
A. I. Mandzyuk, “A generalization of a theorem of Prof. Mordukhai-Boltovskoi on quadratic diameters of a curve of the third order,” Tr. Mosk. Zootekhn. Inst.,4, 133–135 (1936).
Yu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic [in Russian], Nauka, Moscow (1972).
V. S. Martinenko, “On a property of the polyconic of a straight line with respect to a curve of the third order,” Dopovidi AN UkrSSR, No. 2, 150–152 (1961).
L. B. Medvedeva and Z. A. Skopets, “Representation of equiaffine transformations of a plane as a product of oblique symmetries,” Uch. Zap. Yaroslav. Pedagog. Inst., No. 92, 297–303 (1971).
A. I. Medyanik, “A regular simplex inscribed in a cube,” Ukr. Geometrich. Sb., No. 13, 109–112 (1973).
A. D. Milka, “Indecomposability of a convex surface,” Ukr. Geometrich. Sb., No. 13, 112–129 (1973).
A. S. Mishchenko and A. T. Fomenko, A Course of Differential Geometry and Topology [in Russian], Moscow State Univ. (1980).
D. D. Mordukhai-Boltovskoi, “Quadratic diameters and polars of curves of the third order,” Tr. Sev.-Kavk. Ass. Nauchno-Issled. Inst., No. 1, Issue 2, 31–39 (1926).
D. D. Mordukhai-Boltovskoi, “On a hyperplanar section of hypercones,” Tr. Sev.-Kavk. Ass. Nauchno-Issled. Inst., No. 1, Issue 2, 17–28 (1926).
A. F. Navrotskii and N. A. Nikulin, “A vector-cissoidal transformation of space,” Prikl. Geometr. Inzh. Grafika, No. 31, 64–66 (1981).
D. B. Persits, “Geometries over the algebra of antioctaves,” Izv. Akad. Nauk SSSR, Ser. Mat.,31, No. 6, 1263–1270 (1967).
A. T. Petrova, “Some singularities of one of the transcendental transformations of the plane,” Prikl. Geometr. Inzh. Grafika, No. 30, 64–67 (1980).
A. V. Pogorelov, Differential Geometry [in Russian], Nauka, Moscow (1969).
A. V. Pogorelov, Geometry [in Russian], Nauka, Moscow (1983).
É. G. Poznyak, “A relation between nonrigidity of the first and second order for surfaces of rotation,” Usp. Mat. Nauk,14, No. 6, 179–184 (1959).
V. L. Popov, “Constructive theory of invariants,” Izv. Akad. Nauk SSSR, Ser. Mat.,45, No. 5, 1100–1120 (1981).
V. L. Popov, “Syzygies in the theory of invariants,” Izv. Akad. Nauk SSSR, Ser. Mat.,47, No. 3, 544–622 (1983).
M. M. Postnikov, Fermat's Theorem [in Russian], Nauka, Moscow (1977).
L. M. Rogalya, “On some properties of ‘pearl-shaped’ curves,” Drogobych. Gos. Pedagog. Inst. (1982). (Dep. at VINITI 6 April 1982.)
B. A. Roxenfel'd, Multidimensional Spaces [in Russian], Nauka, Moscow (1966).
O. I. Rudnitskii, “On invariant finite groups generated by reflections in a three-dimensional unitary space,” Simferop. Univ. (1983). (Dep. at UkrNIINTI 1 June 1983.).
O. I. Rudnitskii, “Basic invariants of the unitary group EW (N4),” Proc. All-Union School on Function Theory, dedicated to the centenary of the birth of N. N. Luzin [in Russian], Kemerovo (1983), p. 94.
S. S. Ryshkov, “The geometry of positive quadratic forms,” Proc. Int. Congr. Math., Vancouver, 1974, Vol. 1 (1975), pp. 501–506.
S. S. Ryshkov and E. P. Baranovskii, “Classical methods of the theory of lattice packings,” Usp. Mat. Nauk,34, No. 4, 4–63 (1979).
S. S. Ryshkov and S. Sh. Shushbaev, “Positive forms of degree 2l > 2 and zeta-separating quadratic forms,” Dokl. Akad. Nauk SSSR,269, No. 6, 1316–1319 (1983).
S. S. Ryshkov and R. M. Érdal, “The geometry of integral roots of certain quadratic equations with many unknowns,” Dokl. Akad. Nauk SSSR,267, No. 3, 561–563 (1982).
E. Spenser, Theory of Invariants [Russian translation], Mir, Moscow (1974).
T. A. Springer, “Invariant theory,” Lect. Notes Math.,585 (1977).
R. B. Stekol'shchik, “A Coxeter transformation associated with an extended Dynkin diagram with multiple connections,” Tsentr. Avtomatiz. Nauchn. Issled. i Metrol. AN MSSR, Kishinev (1982). (Dep. at VINITI 2 November 1982.)
V. F. Subbotin and R. B. Stekol'shchik, “The Jordan form of a Coxeter transformation and applications to representations of finite graphs,” Funkts. Anal. Prilozhen.,12, No. 1, 84–85 (1978).
I. A. Sukhina and N. I. Shcheglova, “On the question of the construction of tangent planes and normals to surfaces of the second order,” Krasnodar. Politekh. Inst. (1981). (Dep. at VINITI 28 October 1981.)
V. A. Ternovskii, “On the invariants of the symmetry group of the polyhedron 221,” Simferop. Univ. (1983). (Dep. at UkrNIINTI 25 April 1983.)
V. A. Ternovskii, “Special invariants of the groups [N], F4, E6,” Proc. All-Union School on Function Theory, dedicated to the centenary of the birth of N. N. Luzin, Kemerovo (1983), p. 110.
V. A. Ternovskii, “On two classes of plane algebraic curves with axes of symmetry,” Ukr. Geometrich. Sb., No. 27, 116–118 (1984).
A. S. Fedenko, Spaces with Symmetries [in Russian], Belorussian Univ., Minsk (1977).
A. T. Fomenko, “On the geometry of the distribution of integer points in hyperdomains,” Tr. Seminara po Vektor. i Tenzor. Analizu, No. 21, 106–152 (1983).
A. T. Fomenko, Differential Geometry and Topology. Additional Chapters, Moscow State Univ. (1983).
M. L. Frank, “On the interpolation of certain closed curves,” Izv. Krym. Gos. Pedagog. Inst.,3, 217–227 (1930).
A. K. Tsikh, “Bezout's theorem in a space of function theory. On the solution of systems of algebraic equations,” in: Some Questions of Multidimensional Complex Analysis [in Russian], Krasnoyarsk (1980), pp. 185–196.
I. R. Shafarevich, Foundations of Algebraic Geometry [in Russian], Nauka, Moscow (1972).
G. Choquet, Geometry [Russian translation], Mir, Moscow (1970).
M. I. Shtogrin, “The Bravais type of a lattice and the full group of motions that match a lattice with itself,” Problems of Crystallography [in Russian], Moscow State Univ. (1971), pp. 299–314.
S. Sh. Shushbaev, “Computation of mixed classes in groups of integral automorphisms of certain perfect forms,” in: Questions of Computation and Applied Math. [in Russian], Vol. 66, Tashkent (1981), pp. 21–31.
P. Asisoff, “Kordemidtpunktskurve og tangentskaeringskurve,” Normat,28, No. 3, 116–118 (1980).
K. Atsuyama, “Another construction of real simple Lie algebras,” Kodai Math. J.,6, No. 1, 122–133 (1983).
P. Berard, “Spectres et groupes cristallographiques,” C. R. Acad. Sci.,288, No. 23, A1059–A1060 (1979).
S. Bilinski, “Die Invarianten einer diskreten Transformationsgruppe endlicher Ordnung,” RAD Jugosl. Acad. Znan. i Umjethn., No. 386, 89–93 (1980).
E. Bompiani, “Un teorema di Moutard sulle polari di una superficie algebra,” Atti Accad. Naz. Lincei. Rend. C1. Sci. Fis. Mat. Natur.,43, Nos. 1–2, 9–12 (1967).
N. Bourbaki, Elements de Mathematique. Groupes et Algebres de Lie, Ch. 9, Groupes de Lie Reels Compacts, Masson, Paris (1982).
R. Brauer, “Symmetrische Funktionen. Invarianten von linear Gruppen endlicher Ordnung,” Mat. J. Okayama Univ.,21, No. 2, 91–113 (1979).
F. Catanese and G. Ceresa, “Constructing sextic surfaces with a given number d of nodes,” J. Pure Appl. Algebra,23, No. 1, 1–12 (1982).
B. Chang, “Elements of order Coxeter number +1 in Chevalley groups,” Can. J. Math.,34, No. 4, 945–951 (1982).
J. P. Class, “Bitangents of plane quartics,” Bull. Austral. Math. Soc.,20, No. 2, 207–210 (1979).
H. S. M. Coxeter, “Discrete groups generated by reflections,” Ann. Math.,35, 588–621 (1934).
H. S. M. Coxeter, “Regular and semiregular polytopes. I,” Math. Z.,46, 380–407 (1940).
H. S. M. Coxeter, Regular Polytopes, Methuen, London (1948); Pitman, New York (1949, 1963).
H. S. M. Coxeter, “The product of the generators of a finite group generated by reflections,” Duke Math. J.,18, 765–782 (1951).
H. S. M. Coxeter, Regular Complex Polytopes, Cambridge Univ. Press, London-New York (1974).
H. S. M. Coxeter, “Ten toroids and 57 hemidodecahedra,” Geom. Dedic.,13, No. 1, 87–99 (1982).
H. S. M. Coxeter, P. Du Val, H. T. Flather, and J. F. Petrie, The 59 Icosahedra, Springer-Verlag, New York (1982).
H. T. Croft, “On maximal regular polyhedra inscribed in a regular polyhedron,” Proc. London Math. Soc.,41, No. 2, 279–296 (1980).
J. P. d'Angelo, “A Bezout type theorem for points of finite type on real hypersurfaces,” Duke Math. J.,50, No. 1, 197–201 (1983).
W. Degen, “Eine Kennzeichnung der Quadriken unter den Kegelschnittflächen von hyperbolischen Typ,” Arch. Math.,38, No. 1, 75–80 (1982).
V. V. Deodhar, “On the root system of a Coxeter group,” Commun. Algebra,10, No. 6, 611–630 (1982).
M. L. Eaton and M. Perlman, “Generating O(n) with reflections,” Pac. J. Math.,33, No. 1, 73–80 (1977).
E. W. Ellers, “Decomposition of equiaffinities into reflections,” Geom. Dedic.,6, No. 3, 297–304 (1977).
N. G. El-Sharkaway, H. A. Jahn, and R. C. King, “New integral bases for symmetric functions,” Math. Rept. Acad. Sci. Can.,2, No. 5, 243–246 (1980).
T. H. Fay and E. O'Neal, “Counting the petals of rose curves,” Math. Comput. Educ.,17, No. 1, 24–29 (1973).
L. Fejes Toth, “Some researches inspired by H. S. M. Coxeter,” in: The Geometric Vein (Coxeter Festschrift), Springer-Verlag, New York-Berlin (1981), pp. 271–277.
L. Flatto, “Basis sets of invariants for finite reflection groups,” Bull. Am. Math. Soc.74, No. 4, 730–734 (1968).
L. Flatto, “Regular polytopes and harmonic polynomials,” Can. J. Math.,22, 7–21 (1970).
L. Flatto, “Invariants of finite reflection groups,” Enseign. Math.,24, Nos. 3–4, 237–292 (1978).
L. Flatto and M. M. Wiener, “Invariants of finite reflection groups and mean value problems,” Am. J. Math.,91, No. 3, 591–598 (1969).
L. Flatto, “Invariants of finite reflection groups and mean value problems. II,” Am. J. Math.,92, No. 3, 552–561 (1970).
J. S. Frame, “The classes and representations of the groups of 27 lines and 28 bitangents,” Ann. Math.,32, 83–119 (1951).
G. Geise, “Eine die Kegelschnitte charakterisierende Eigenschaft,” Beitr. Algebra Geom.,14, 29–31 (1983).
E. Goursat, “Etude des surfaces qui admettent tous les plans de symetrie d'un polyedre regulier,” Ann. d'Ecole Norm. (3),4, 159–260 (1887).
J. Graham, “Three basic geometrical transformations: some applicable linear algebra,” Pentagon,39, No. 1, 22–29 (1979).
R. L. Greiss, “Quotients of infinite reflection groups,” Math. Ann.,263, 267–278 (1983).
R. Gulliver and F. Morgan, “The symmetry group of a curve preserves a plane,” Proc. Am. Math. Soc.,84, 408–411 (1982).
G. K. Haeuslein, “On the algebraic independence of symmetric functions,” Proc. Am. Math. Soc.,25, No. 1, 179–182 (1970).
Harish Chandra, “On some applications of the universal enveloping algebra of a semisimple Lie algebra,” Trans. Am. Math. Soc.,70, 28–96 (1951).
H. L. Hiller, “Coinvariant theory of a Coxeter group,” Geom. Vein: Coxeter Festschrift, New York (1981), pp. 555–559.
R. Hotta, “On Springer's representations,” J. Fac. Sci. Univ. Tokyo, Sec. 1A,28, No. 3, 863–876 (1981).
V. F. Ignatenko, “Geometry of algebraic surfaces with symmetries,” Zentralblatt Math.,473, 272–273 (1982).
F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen fünften Grades, Leipzig (1984).
M. Koca, “Explicit realization of E8,” Lect. Notes Phys.,180, 356–359 (1983).
T. Kondo, “The characters of the Weyl group of type F4,” J. Fac. Sci. Univ. Tokyo, Sec. 1,11, No. 2, 145–153 (1965).
L. Lecournu, “Sur les surfaces possedant les memes plans de symetrie que 1'un des polyedres reguliers,” Acta Math.,10, 201–280 (1887).
I. G. Macdonald, “Some conjectures for root systems and finite Coxeter groups,” Lect. Notes Math.,867, 90–97 (1981).
I. G. Macdonald, “Lie groups and combinatorics,” Contemp. Math.,9, 73–83 (1982).
H. Maürer, “Symmetries of quadrics,” in: Geometry — von Staudt's Point of View, Proc. NATO Adv. Study Inst. Bad Windsheim (1980), Dordrecht (1981), pp. 197–229.
B. R. Monson, “The Schläflian of a crystallographic Coxeter group,” Math. Repts. Acad. Sci. Can.,4, No. 3, 145–147 (1982).
W. Moser, Problems in Discrete Geometry, 2nd version, October 1977, Oberwalfach, 1977, July 10–16.
J. Naruki, “Über die Kleinsche Ikosaeder-Kurve sechsten Grades,” Math. Ann.,231, No. 3, 205–216 (1978).
P. E. Newstead, “Real classification of complex conies,” Mathematika,28, No. 1, 36–53 (1981).
W. Nolte, “Darstellung spezieller projektiver Spiegelungsgruppen durch ortogonale Gruppen,” J. Geom.,18, No. 2, 109–112 (1982).
A. Ostrowski, “Über die Darstellung von symmetrischen Funktionen durch Potenz summen,” Math. Ann.,132, No. 4, 362–372 (1956).
P. Petek, “A generalization of the theorem of symmetric functions,” Boll. Unione Mat. Ital.,A1, No. 2, 289–295 (1982).
H. Pfeuffer, “Über die reelle Spiegelungsgruppe ħ und die Klassenzahl der sechs dimensionalen Einheitsform,” Arch. Math.,31, No. 2, 126–132 (1978).
M. Piazzola-Beloch, “Proprieta diametrali delle superficie algebriche,” Atti IV Congr. dell'Unione Mat. Ital.,2, 425–430 (1953).
L. Pontrjagin, “Sur les nombres de Betti des groupes de Lie,” C. R. Acad. Sci. Paris,200, 1277–1280 (1935).
C. Pretki, “Uogolnione ogniska stozkowej,” Zesz. Nauk. Pozn. Geom., No. 10, 103–118 (1978).
M. Rais, “Le theoreme fondamentale des invariants pour les groupes finis,” Ann. Inst. Fourier,27, No. 4, 247–256 (1977).
B. Renschuch and W. Vogel, “Über den Bezoutschen Satz seit den Untersuchungen B. L. van der Waerden,” Wiss. Beitr. M.-Luther-Univ. Halle-Wittenberg, No. 27, 95–109 (1982).
C. A. Rogers, “Some problems in the geometry of convex bodies,” in: The Geometric Vein (Coxeter Festschrift), Springer-Verlag, New York-Berlin (1981), pp. 279–284.
B. A. Rosina, “Sulle curve algebriche piane con infiniti assi di simmetria obliqua (in particolare ortogonale), ” Ann. Univ. Ferrera, Sez. VII,15, No. 10, 153–159 (1970).
B. A. Rosina, “Classificazione delle curve algebriche piane di ordine qualunque secondo la teoria delle curve algebriche piane,” Atti Accad. Sci. Lett. Arti Palermo, Ser. IV, Parte 1,34, No. 2, 101–121 (1976).
K. Saito, T. Yano, and J. Sekiguchi, “On a certain generator system on the ring of invariants of a finite reflection group,” Commun. Algebra,8, No. 4, 373–408 (1980).
G. Sansone, “I punti di coordinate rationali e, in particolare, di coordinate intere della cubica ellittica y2=x3−x+1,” Ann. Mat. Pura Appl.,125, 1–11 (1980).
W. Schmid, “Poincare and Lie groups,” Bull. Am. Math. Soc.,6, No. 2, 175–186 (1982).
J. Sekiguchi and T. Yano, “The algebra of invariants of the Weyl group W(F4),” Sci. Repts. Saitama Univ.,A9, No. 2, 21–32 (1979).
J. Sekiguchi and T. Yano, “A note on the Coxeter group of type H3,” Sci. Repts. Saitama Univ.,A9, No. 2, 33–44 (1979).
J. Sekiguchi and T. Yano, “The microlocal structure of weighted homogeneous polynomials associated with Coxeter systems. I,” Tokyo J. Math.,2, No. 2, 193–219 (1979).
K. Sindelar, “Rozlozitelnost polar a jeji geometricky vyznam,” Acta Polytech., Ser. 4,7, No. 3, 89–93 (1978).
T. A. Springer, “Some remarks on involutions in Coxeter groups,” Commun. Algebra,10, No. 6, 631–636 (1982).
R. P. Stanley, “Invariants of finite groups and their applications to combinatorics,” Bull. Am. Math. Soc. (N.S.),1, No. 3, 475–511 (1979).
R. P. Stanley, “Interactions between commutative algebra and combinatorics,” Repts. Den. Math. Univ. Stockholm, No. 4 (1982).
R. Steinberg, “Finite reflection groups,” Trans. Am. Math. Soc.,91, No. 3, 493–504 (1959).
R. Steinberg, “Invariants of finite reflection groups,” Can. J. Math.,12, No. 4, 616–618 (1960).
H. Terao, “Arrangements of hyperplanes and their freeness. II. The Coxeter equality,” J. Fac. Sci. Univ. Tokyo, Sec. 1A,27, No. 2, 313–320 (1980).
J. A. Todd, “Polytopes associated with the general cubic surface,” J. London Math. Soc.,7, No. 27, 200–205 (1932).
J. Van der Jeugt, G. Van den Berghe, and H. De Heyer, “Boson realisation of the Lie algebra F4 and nontrivial zeros of Gj symbols,” J. Phys. A: Math. Gen.,16, No. 7, 1377–1382 (1983).
J. Van Ijzeren, “Parabolen van Feuerbach,” Nieuw Tijdschr. Wisk.,69, No. 3, 95–101 (1982).
W. O. Vogel, “Bewegflächen einer Ellipse als Quadriken,” Arch. Math.,38, No. 2, 124–130 (1982).
J. L. Walsh, “A mean value theorem for polynomials and harmonic polynomials,” Bull. Am. Math. Soc.,42, No. 12, 923–930 (1936).
W. Whiteley, “Logic and invariant theory. I. Invariant theory of projective properties,” Trans. Am. Math. Soc.,177, 121–139 (1973).
W. Whiteley, “Logic and invariant theory. II. Homogeneous coordinates, the introduction of higher quantities, and structural geometry,” J. Algebra,50, No. 2, 380–394 (1978).
D. J. Winter, “A combinatorial theory of symmetry and applications to Lie algebras,” Lect. Notes Math.,848 (1981).
I. M. Yaglom, “On the circular transformations of Möbius, Laguerre, and Lie,” in: The Geometric Vein (Coxeter Festschrift), New York (1981), pp. 345–353.
O. Yasukura and I. Yokota, “Subgroup SU(2) × spin(12)/Z2 of compact simple Lie group E7 and non-compact simple Lie group E7, σ of type E7 (−5),” Hiroshima Math. J.,12, No. 1, 59–76 (1982).
I. Yokota, “Subgroups of type A2-series in the exceptional Lie groups G2, F4, and E6,” J. Fac. Sci. Shinshu Univ.,14, No. 2, 87–94 (1979).
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 16, pp. 195–229 1984.
Rights and permissions
About this article
Cite this article
Ignatenko, V.F. Some questions in the geometric theory of invariants of groups generated by orthogonal and oblique reflections. J Math Sci 33, 933–953 (1986). https://doi.org/10.1007/BF01091738
Issue Date:
DOI: https://doi.org/10.1007/BF01091738