Skip to main content
Log in

Some questions in the geometric theory of invariants of groups generated by orthogonal and oblique reflections

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Literature cited

  1. 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.)

  2. J. F. Adams, Lectures on Lie Groups, W. A. Benjamin, New York-Amsterdam (1969).

    Google Scholar 

  3. A. D. Aleksandrov, “On filling a space by polyhedra,” Vestn. Leningr. Univ., No. 2, 33–43 (1954).

    Google Scholar 

  4. D. V. Alekseevskii, “Lie groups,” Itogi Nauki i Tekh. VINITI. Algebra. Topol. Geometriya,20, 153–192 (1982).

    Google Scholar 

  5. V. I. Arnol'd, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  6. 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.)

  7. F. Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer-Verlag, Berlin-Göttingen-Heidelberg (1959).

    Google Scholar 

  8. Ya. P. Blank, “Conjugate nets of conical curves,” Dokl. Akad. Nauk SSSR,64, No. 6, 755–758 (1949).

    Google Scholar 

  9. W. Blaschke, Kreis und Kugel, 2nd ed., Walter de Gruyter, Berlin (1956).

    Google Scholar 

  10. V. G. Boltyanskii, “Combinatorial geometry,” Itogi Nauki i Tekh. VINITI. Algebra. Topol. Geometriya,19, 209–274 (1981).

    Google Scholar 

  11. N. Bourbaki, Lie Groups and Lie Algebras, Hermann, Paris (1975).

    Google Scholar 

  12. V. I. Vedernikov, “Symmetric spaces. Conjugate connections as a normalized connection,” Trudy Geometr. Sem. Inst. Informatsii AN SSSR (1966), pp. 63–88.

  13. 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.

  14. 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).

    Google Scholar 

  15. 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.)

  16. N. A. Gregor'ev, “Regular simplexes inscribed in a cube and Hadamard matrices,” Tr. Mat. Inst. Akad. Nauk SSSR,152, 87–88 (1980).

    Google Scholar 

  17. 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.)

  18. 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.)

  19. 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.)

  20. N. P. Dolbilin, “On three-dimensional and four-dimensional simple forms,” in: Problems of Crystallography [in Russian], Moscow State Univ. (1971), pp. 315–324.

  21. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  22. J. Dieudonne, La Geometrie des Groupes Classiques, 2nd ed., Springer-Verlag, Berlin-Göttingen-Heidelberg (1963).

    Google Scholar 

  23. J. Dieudonne, J. B. Carrell, and D. Mumford, Invariant Theory, Old and New, Academic Press, New York—London (1971).

    Google Scholar 

  24. K. D. Erastov, “Diametral hyperplanes,” Sb. Nauch. Tr. Tashkent. Univ., No. 623, 32–40 (1980).

    Google Scholar 

  25. 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.)

    Google Scholar 

  26. D. P. Zhelobenko and A. I. Shtern, Representations of Lie Groups [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  27. V. A. Zalgaller, Theory of Envelopes [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  28. A. M. Zamorzaev, É. I. Galyarskii, and A. F. Palistrant, Chromatic Symmetry, Its Generalizations and Applications [in Russian], Shtiintsa, Kishinev (1978).

    Google Scholar 

  29. V. F. Ignatenko, “On plane algebraic curves with axes of symmetry,” Ukr. Geometrich. Sb., No. 21, 31–33 (1978).

    Google Scholar 

  30. V. F. Ignatenko, “The geometry of algebraic surfaces with symmetries,” Itogi Nauki i Tekh. VINITI. Problemy Geometrii,11, 203–240 (1980).

    Google Scholar 

  31. V. F. Ignatenko, “Algebraic surfaces with the symmetry group of the polyhedron 321,” Ukr. Geometrich. Sb., No. 23, 50–56 (1980).

    Google Scholar 

  32. 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.

  33. V. F. Ignatenko, “On algebraic surfaces with symmetry groups An, Bn, Dn,” Ukr. Geometrich. Sb., No. 24, 33–39 (1981).

    Google Scholar 

  34. 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).

    Google Scholar 

  35. V. F. Ignatenko, “On invariants of finite groups generated by reflections,” Mat. Sb.,120, No. 4, 556–568 (1983).

    Google Scholar 

  36. V. F. Ignatenko, “On algebraic surfaces with the symmetry group of the polyhedron 421,” Ukr. Geometrich. Sb., No. 26, 48–55 (1983).

    Google Scholar 

  37. 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.

  38. 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).

    Google Scholar 

  39. 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).

    Google Scholar 

  40. 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).

    Google Scholar 

  41. 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).

    Google Scholar 

  42. 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).

    Google Scholar 

  43. J. W. S. Cassells, Rational Quadratic Forms, Academic Press, London—New York (1978).

    Google Scholar 

  44. M. D. Kovalev, “On a characteristic property of a disk and an n-dimensional ball,” Usp. Mat. Nauk,36, No. 6, 217–218 (1981).

    Google Scholar 

  45. H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, McGraw-Hill, New York (1967).

    Google Scholar 

  46. H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, Berlin-New York (1980).

    Google Scholar 

  47. O. N. Kolesnikov, “On a rational transformation,” Izv. Krymsk. Gos. Pedagog. Inst.,35, 259–267 (1961).

    Google Scholar 

  48. A. I. Kostrikin, Introduction to Algebra [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  49. 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.)

  50. V. A. Kutin, “The area of a surface of a three-axis ellipsoid,” Perm. Univ. (1982). (Dep. at VINITI 6 June 1982.)

  51. S. I. Kucherenko and N. A. Nikulin, “On some curves of the fourth order,” Prikl. Geometr. Inzh. Grafika, No. 30, 69–71 (1980).

    Google Scholar 

  52. 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).

    Google Scholar 

  53. 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).

    Google Scholar 

  54. 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).

    Google Scholar 

  55. Yu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  56. 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).

    Google Scholar 

  57. 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).

    Google Scholar 

  58. A. I. Medyanik, “A regular simplex inscribed in a cube,” Ukr. Geometrich. Sb., No. 13, 109–112 (1973).

    Google Scholar 

  59. A. D. Milka, “Indecomposability of a convex surface,” Ukr. Geometrich. Sb., No. 13, 112–129 (1973).

    Google Scholar 

  60. A. S. Mishchenko and A. T. Fomenko, A Course of Differential Geometry and Topology [in Russian], Moscow State Univ. (1980).

  61. 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).

    Google Scholar 

  62. D. D. Mordukhai-Boltovskoi, “On a hyperplanar section of hypercones,” Tr. Sev.-Kavk. Ass. Nauchno-Issled. Inst., No. 1, Issue 2, 17–28 (1926).

    Google Scholar 

  63. A. F. Navrotskii and N. A. Nikulin, “A vector-cissoidal transformation of space,” Prikl. Geometr. Inzh. Grafika, No. 31, 64–66 (1981).

    Google Scholar 

  64. D. B. Persits, “Geometries over the algebra of antioctaves,” Izv. Akad. Nauk SSSR, Ser. Mat.,31, No. 6, 1263–1270 (1967).

    Google Scholar 

  65. A. T. Petrova, “Some singularities of one of the transcendental transformations of the plane,” Prikl. Geometr. Inzh. Grafika, No. 30, 64–67 (1980).

    Google Scholar 

  66. A. V. Pogorelov, Differential Geometry [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  67. A. V. Pogorelov, Geometry [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  68. 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).

    Google Scholar 

  69. V. L. Popov, “Constructive theory of invariants,” Izv. Akad. Nauk SSSR, Ser. Mat.,45, No. 5, 1100–1120 (1981).

    Google Scholar 

  70. V. L. Popov, “Syzygies in the theory of invariants,” Izv. Akad. Nauk SSSR, Ser. Mat.,47, No. 3, 544–622 (1983).

    Google Scholar 

  71. M. M. Postnikov, Fermat's Theorem [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  72. L. M. Rogalya, “On some properties of ‘pearl-shaped’ curves,” Drogobych. Gos. Pedagog. Inst. (1982). (Dep. at VINITI 6 April 1982.)

  73. B. A. Roxenfel'd, Multidimensional Spaces [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  74. 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.).

  75. 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.

  76. S. S. Ryshkov, “The geometry of positive quadratic forms,” Proc. Int. Congr. Math., Vancouver, 1974, Vol. 1 (1975), pp. 501–506.

    Google Scholar 

  77. S. S. Ryshkov and E. P. Baranovskii, “Classical methods of the theory of lattice packings,” Usp. Mat. Nauk,34, No. 4, 4–63 (1979).

    Google Scholar 

  78. 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).

    Google Scholar 

  79. 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).

    Google Scholar 

  80. E. Spenser, Theory of Invariants [Russian translation], Mir, Moscow (1974).

    Google Scholar 

  81. T. A. Springer, “Invariant theory,” Lect. Notes Math.,585 (1977).

  82. 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.)

  83. 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).

    Google Scholar 

  84. 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.)

  85. V. A. Ternovskii, “On the invariants of the symmetry group of the polyhedron 221,” Simferop. Univ. (1983). (Dep. at UkrNIINTI 25 April 1983.)

  86. 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.

  87. V. A. Ternovskii, “On two classes of plane algebraic curves with axes of symmetry,” Ukr. Geometrich. Sb., No. 27, 116–118 (1984).

    Google Scholar 

  88. A. S. Fedenko, Spaces with Symmetries [in Russian], Belorussian Univ., Minsk (1977).

    Google Scholar 

  89. 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).

    Google Scholar 

  90. A. T. Fomenko, Differential Geometry and Topology. Additional Chapters, Moscow State Univ. (1983).

  91. M. L. Frank, “On the interpolation of certain closed curves,” Izv. Krym. Gos. Pedagog. Inst.,3, 217–227 (1930).

    Google Scholar 

  92. 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.

  93. I. R. Shafarevich, Foundations of Algebraic Geometry [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  94. G. Choquet, Geometry [Russian translation], Mir, Moscow (1970).

    Google Scholar 

  95. 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.

  96. 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.

    Google Scholar 

  97. P. Asisoff, “Kordemidtpunktskurve og tangentskaeringskurve,” Normat,28, No. 3, 116–118 (1980).

    Google Scholar 

  98. K. Atsuyama, “Another construction of real simple Lie algebras,” Kodai Math. J.,6, No. 1, 122–133 (1983).

    Google Scholar 

  99. P. Berard, “Spectres et groupes cristallographiques,” C. R. Acad. Sci.,288, No. 23, A1059–A1060 (1979).

    Google Scholar 

  100. S. Bilinski, “Die Invarianten einer diskreten Transformationsgruppe endlicher Ordnung,” RAD Jugosl. Acad. Znan. i Umjethn., No. 386, 89–93 (1980).

    Google Scholar 

  101. 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).

    Google Scholar 

  102. N. Bourbaki, Elements de Mathematique. Groupes et Algebres de Lie, Ch. 9, Groupes de Lie Reels Compacts, Masson, Paris (1982).

    Google Scholar 

  103. R. Brauer, “Symmetrische Funktionen. Invarianten von linear Gruppen endlicher Ordnung,” Mat. J. Okayama Univ.,21, No. 2, 91–113 (1979).

    Google Scholar 

  104. 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).

    Google Scholar 

  105. B. Chang, “Elements of order Coxeter number +1 in Chevalley groups,” Can. J. Math.,34, No. 4, 945–951 (1982).

    Google Scholar 

  106. J. P. Class, “Bitangents of plane quartics,” Bull. Austral. Math. Soc.,20, No. 2, 207–210 (1979).

    Google Scholar 

  107. H. S. M. Coxeter, “Discrete groups generated by reflections,” Ann. Math.,35, 588–621 (1934).

    Google Scholar 

  108. H. S. M. Coxeter, “Regular and semiregular polytopes. I,” Math. Z.,46, 380–407 (1940).

    Google Scholar 

  109. H. S. M. Coxeter, Regular Polytopes, Methuen, London (1948); Pitman, New York (1949, 1963).

    Google Scholar 

  110. H. S. M. Coxeter, “The product of the generators of a finite group generated by reflections,” Duke Math. J.,18, 765–782 (1951).

    Google Scholar 

  111. H. S. M. Coxeter, Regular Complex Polytopes, Cambridge Univ. Press, London-New York (1974).

    Google Scholar 

  112. H. S. M. Coxeter, “Ten toroids and 57 hemidodecahedra,” Geom. Dedic.,13, No. 1, 87–99 (1982).

    Google Scholar 

  113. H. S. M. Coxeter, P. Du Val, H. T. Flather, and J. F. Petrie, The 59 Icosahedra, Springer-Verlag, New York (1982).

    Google Scholar 

  114. H. T. Croft, “On maximal regular polyhedra inscribed in a regular polyhedron,” Proc. London Math. Soc.,41, No. 2, 279–296 (1980).

    Google Scholar 

  115. 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).

    Google Scholar 

  116. W. Degen, “Eine Kennzeichnung der Quadriken unter den Kegelschnittflächen von hyperbolischen Typ,” Arch. Math.,38, No. 1, 75–80 (1982).

    Google Scholar 

  117. V. V. Deodhar, “On the root system of a Coxeter group,” Commun. Algebra,10, No. 6, 611–630 (1982).

    Google Scholar 

  118. M. L. Eaton and M. Perlman, “Generating O(n) with reflections,” Pac. J. Math.,33, No. 1, 73–80 (1977).

    Google Scholar 

  119. E. W. Ellers, “Decomposition of equiaffinities into reflections,” Geom. Dedic.,6, No. 3, 297–304 (1977).

    Google Scholar 

  120. 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).

    Google Scholar 

  121. T. H. Fay and E. O'Neal, “Counting the petals of rose curves,” Math. Comput. Educ.,17, No. 1, 24–29 (1973).

    Google Scholar 

  122. 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.

    Google Scholar 

  123. L. Flatto, “Basis sets of invariants for finite reflection groups,” Bull. Am. Math. Soc.74, No. 4, 730–734 (1968).

    Google Scholar 

  124. L. Flatto, “Regular polytopes and harmonic polynomials,” Can. J. Math.,22, 7–21 (1970).

    Google Scholar 

  125. L. Flatto, “Invariants of finite reflection groups,” Enseign. Math.,24, Nos. 3–4, 237–292 (1978).

    Google Scholar 

  126. L. Flatto and M. M. Wiener, “Invariants of finite reflection groups and mean value problems,” Am. J. Math.,91, No. 3, 591–598 (1969).

    Google Scholar 

  127. L. Flatto, “Invariants of finite reflection groups and mean value problems. II,” Am. J. Math.,92, No. 3, 552–561 (1970).

    Google Scholar 

  128. J. S. Frame, “The classes and representations of the groups of 27 lines and 28 bitangents,” Ann. Math.,32, 83–119 (1951).

    Google Scholar 

  129. G. Geise, “Eine die Kegelschnitte charakterisierende Eigenschaft,” Beitr. Algebra Geom.,14, 29–31 (1983).

    Google Scholar 

  130. 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).

    Google Scholar 

  131. J. Graham, “Three basic geometrical transformations: some applicable linear algebra,” Pentagon,39, No. 1, 22–29 (1979).

    Google Scholar 

  132. R. L. Greiss, “Quotients of infinite reflection groups,” Math. Ann.,263, 267–278 (1983).

    Google Scholar 

  133. R. Gulliver and F. Morgan, “The symmetry group of a curve preserves a plane,” Proc. Am. Math. Soc.,84, 408–411 (1982).

    Google Scholar 

  134. G. K. Haeuslein, “On the algebraic independence of symmetric functions,” Proc. Am. Math. Soc.,25, No. 1, 179–182 (1970).

    Google Scholar 

  135. Harish Chandra, “On some applications of the universal enveloping algebra of a semisimple Lie algebra,” Trans. Am. Math. Soc.,70, 28–96 (1951).

    Google Scholar 

  136. H. L. Hiller, “Coinvariant theory of a Coxeter group,” Geom. Vein: Coxeter Festschrift, New York (1981), pp. 555–559.

  137. R. Hotta, “On Springer's representations,” J. Fac. Sci. Univ. Tokyo, Sec. 1A,28, No. 3, 863–876 (1981).

    Google Scholar 

  138. V. F. Ignatenko, “Geometry of algebraic surfaces with symmetries,” Zentralblatt Math.,473, 272–273 (1982).

    Google Scholar 

  139. F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen fünften Grades, Leipzig (1984).

  140. M. Koca, “Explicit realization of E8,” Lect. Notes Phys.,180, 356–359 (1983).

    Google Scholar 

  141. T. Kondo, “The characters of the Weyl group of type F4,” J. Fac. Sci. Univ. Tokyo, Sec. 1,11, No. 2, 145–153 (1965).

    Google Scholar 

  142. L. Lecournu, “Sur les surfaces possedant les memes plans de symetrie que 1'un des polyedres reguliers,” Acta Math.,10, 201–280 (1887).

    Google Scholar 

  143. I. G. Macdonald, “Some conjectures for root systems and finite Coxeter groups,” Lect. Notes Math.,867, 90–97 (1981).

    Google Scholar 

  144. I. G. Macdonald, “Lie groups and combinatorics,” Contemp. Math.,9, 73–83 (1982).

    Google Scholar 

  145. 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.

  146. B. R. Monson, “The Schläflian of a crystallographic Coxeter group,” Math. Repts. Acad. Sci. Can.,4, No. 3, 145–147 (1982).

    Google Scholar 

  147. W. Moser, Problems in Discrete Geometry, 2nd version, October 1977, Oberwalfach, 1977, July 10–16.

  148. J. Naruki, “Über die Kleinsche Ikosaeder-Kurve sechsten Grades,” Math. Ann.,231, No. 3, 205–216 (1978).

    Google Scholar 

  149. P. E. Newstead, “Real classification of complex conies,” Mathematika,28, No. 1, 36–53 (1981).

    Google Scholar 

  150. W. Nolte, “Darstellung spezieller projektiver Spiegelungsgruppen durch ortogonale Gruppen,” J. Geom.,18, No. 2, 109–112 (1982).

    Google Scholar 

  151. A. Ostrowski, “Über die Darstellung von symmetrischen Funktionen durch Potenz summen,” Math. Ann.,132, No. 4, 362–372 (1956).

    Google Scholar 

  152. P. Petek, “A generalization of the theorem of symmetric functions,” Boll. Unione Mat. Ital.,A1, No. 2, 289–295 (1982).

    Google Scholar 

  153. H. Pfeuffer, “Über die reelle Spiegelungsgruppe ħ und die Klassenzahl der sechs dimensionalen Einheitsform,” Arch. Math.,31, No. 2, 126–132 (1978).

    Google Scholar 

  154. M. Piazzola-Beloch, “Proprieta diametrali delle superficie algebriche,” Atti IV Congr. dell'Unione Mat. Ital.,2, 425–430 (1953).

    Google Scholar 

  155. L. Pontrjagin, “Sur les nombres de Betti des groupes de Lie,” C. R. Acad. Sci. Paris,200, 1277–1280 (1935).

    Google Scholar 

  156. C. Pretki, “Uogolnione ogniska stozkowej,” Zesz. Nauk. Pozn. Geom., No. 10, 103–118 (1978).

    Google Scholar 

  157. M. Rais, “Le theoreme fondamentale des invariants pour les groupes finis,” Ann. Inst. Fourier,27, No. 4, 247–256 (1977).

    Google Scholar 

  158. 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).

    Google Scholar 

  159. 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.

    Google Scholar 

  160. 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).

    Google Scholar 

  161. 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).

    Google Scholar 

  162. 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).

    Google Scholar 

  163. 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).

    Google Scholar 

  164. W. Schmid, “Poincare and Lie groups,” Bull. Am. Math. Soc.,6, No. 2, 175–186 (1982).

    Google Scholar 

  165. 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).

    Google Scholar 

  166. J. Sekiguchi and T. Yano, “A note on the Coxeter group of type H3,” Sci. Repts. Saitama Univ.,A9, No. 2, 33–44 (1979).

    Google Scholar 

  167. 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).

    Google Scholar 

  168. K. Sindelar, “Rozlozitelnost polar a jeji geometricky vyznam,” Acta Polytech., Ser. 4,7, No. 3, 89–93 (1978).

    Google Scholar 

  169. T. A. Springer, “Some remarks on involutions in Coxeter groups,” Commun. Algebra,10, No. 6, 631–636 (1982).

    Google Scholar 

  170. R. P. Stanley, “Invariants of finite groups and their applications to combinatorics,” Bull. Am. Math. Soc. (N.S.),1, No. 3, 475–511 (1979).

    Google Scholar 

  171. R. P. Stanley, “Interactions between commutative algebra and combinatorics,” Repts. Den. Math. Univ. Stockholm, No. 4 (1982).

  172. R. Steinberg, “Finite reflection groups,” Trans. Am. Math. Soc.,91, No. 3, 493–504 (1959).

    Google Scholar 

  173. R. Steinberg, “Invariants of finite reflection groups,” Can. J. Math.,12, No. 4, 616–618 (1960).

    Google Scholar 

  174. 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).

    Google Scholar 

  175. J. A. Todd, “Polytopes associated with the general cubic surface,” J. London Math. Soc.,7, No. 27, 200–205 (1932).

    Google Scholar 

  176. 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).

    Google Scholar 

  177. J. Van Ijzeren, “Parabolen van Feuerbach,” Nieuw Tijdschr. Wisk.,69, No. 3, 95–101 (1982).

    Google Scholar 

  178. W. O. Vogel, “Bewegflächen einer Ellipse als Quadriken,” Arch. Math.,38, No. 2, 124–130 (1982).

    Google Scholar 

  179. J. L. Walsh, “A mean value theorem for polynomials and harmonic polynomials,” Bull. Am. Math. Soc.,42, No. 12, 923–930 (1936).

    Google Scholar 

  180. W. Whiteley, “Logic and invariant theory. I. Invariant theory of projective properties,” Trans. Am. Math. Soc.,177, 121–139 (1973).

    Google Scholar 

  181. 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).

    Google Scholar 

  182. D. J. Winter, “A combinatorial theory of symmetry and applications to Lie algebras,” Lect. Notes Math.,848 (1981).

  183. I. M. Yaglom, “On the circular transformations of Möbius, Laguerre, and Lie,” in: The Geometric Vein (Coxeter Festschrift), New York (1981), pp. 345–353.

  184. 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).

    Google Scholar 

  185. 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).

    Google Scholar 

Download references

Authors

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 16, pp. 195–229 1984.

Rights and permissions

Reprints 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

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01091738

Keywords

Navigation