A third look at weight diagrams

Vavilov, Nikolai

Rendiconti del Seminario Matematico della Università di Padova, Tome 104 (2000), pp. 201-250.

[A] Abe E., Chevalley groups over local rings, Tôhoku Math. J., 21, no. 3 (1969), pp. 474-494. | MR | Zbl

[A1] Aschbacher M., The 27-dimensional module for E6, I, Inv. Math., 89, no. 1 (1987), pp. 159-195. | MR | Zbl

[A2] Aschbacher M., Some multilinear forms with large isometry groups, Geom. dedic., 25, no. 1-3 (1988), pp. 417-465. | MR | Zbl

[A2] Aschbacher M., The geometry of trilinear forms, Finite Geometries, Buildings and Related topics, Oxford Univ. Press, 1990, pp. 75-84. | MR | Zbl

[ABS] Azad H. - Barry M. - Seitz G.M., On the structure of parabolic subgroups, Commun. Algebra, 18 (1990), pp. 551-562. | MR | Zbl

[BV] Bak A. - Vavilov N.A., Structure of hyperbolic unitary groups I, Elementary subgroup, Algebra Colloquim, 7, no. 2 (2000), pp. 159-196. | MR | Zbl

[BE] Baston R.J. - Eastwood M.G., The Penrose transform, its interactions with representation theory, Clarendon Press, 1984. | MR | Zbl

[B] Borel A., Properties and linear representations of Chevalley groups, Lecture Notes Math., 131 (1970), pp. 1-55. | MR | Zbl

[B1] Bourbaki N., Groupes et algèbres de Lie. Ch. 4-6, Hermann, Paris, 1968. | MR | Zbl

[B2] Bourbaki N., Groupes et algèbres de Lie. Ch. 7, 8, Hermann, Paris, 1975. | MR

[BCN] Brouwer A.E. - Cohen A.M. - Neumaier A., Distance regular graphs, Springer-Verlag, N. Y. et al., 1989. | MR | Zbl

[Br] Brown R.B., Groups of type E7, J. reine angew. Math., 236, no. 1 (1969), 79-102. | MR | Zbl

[C] Carter R.W., Simple groups of Lie type, Wiley, London et al., 1972. | MR | Zbl

[CC1] Cohen A.M. - Cooperstein B.N., The 2-space of the standard E6(q)-mudule, Geom. dedic., 25, no. 1-3 (1988), pp. 467-480. | MR | Zbl

[CC2] Cohen A.M. - Cushman R.H., Gröbner bases and standard monomial theory, Computational algebraic geometry, Progess in Mathematics 109, Birkhäuser, 1993, pp. 41-60. | MR | Zbl

[CW] Cohen A.M. - Wales D.B., Finite subgroups of F4(C) and E6(C), Proc. London Math. Soc., 74, no. 1 (1997), pp. 105-150. | MR | Zbl

[C1] Cooperstein B.N., The geometry of root subgroups in exceptional groups, Geom. dedic., 8, no. 3 (1978), pp. 317-381; 15, no. 1 (1983), pp. 1-45. | MR | Zbl

[C2] Cooperstein B.N., The fifty-six-dimensional module for E 7. A four form for E7, J. Algebra, 173 (1995), pp. 361-389. | MR | Zbl

[C3] Cooperstein B N., Four forms I. Basic concepts and examples (to appear).

[CIK] Curtis C.W. - Iwahori N. - Kilmoyer R., Hecke algebras and characters of parabolic type of finite groups with (B, N)- pairs, Publ. Math. Inst. Hautes Et. Sci., no. 40 (1971), pp. 81-116. | Numdam | MR | Zbl

[DMV] Di Martino L. - Vavilov N.A., (2, 3)-generation of E6(q) (to appear).

[FF] Faulkner J.R. - Ferrar J.C., Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc., 9 (1977), pp. 1-35. | MR | Zbl

[GS] Gilkey P. - Seitz G., Some representations of exceptional Lie algebras, Geom. dedic., 25, no. 1-3 (1988), pp. 407-416. | MR | Zbl

[G] Griess R.L., A Moufang loop, the exceptional Jordan algebra and a cubic form in 27 variables, J. Algebra, 13 (1990), pp. 281-293. | MR | Zbl

[HOM] Hahn A. - O'Meara O.T., The classical groups and K-theory, Springer-Verlag, N. Y. et al., 1989. | MR | Zbl

[Ha] Haris S.J., Some irreducible representations of exceptional algebraic groups, Amer. J. Math., 93, no. 1 (1971), pp. 75-106. | MR | Zbl

[Hr] Hartshorn R., Algebraic Geometry, Springer-Verlag, N. Y. et al., 1977. | MR

[Hé] Hée J.-Y., Groupes de Chevalley et groupes classiques, Publ. Math. Univ. Paris VII, 17 (1984), pp. 1-54. | MR | Zbl

[Hi1] Hiller H., Combinatorics and intersections of Schubert varieties, Comment. Math. Helv., 57 (1982), pp. 41-59. | MR | Zbl

[Hi1] Hiller H., Geometry of Coxeter groups, Pitman, Boston and London, 1982. | MR | Zbl

[Ho] Howe R., Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, Israel Math. Conf. Proc. (to appear). | MR | Zbl

[H] Humphreys J.E., Introduction to Lie algebras and representation theory, Springer, Berlin et al., 1980. | MR | Zbl

[J] Joseph A., Quantum groups and their primitive ideals, Springer-Verlag, N.Y. et al., 1995. | MR | Zbl

[K1] Kashiwara M., Crystallizing the q-analogue of universal enveloping algebra, Comm. Math. Phys., 133 (1990), pp. 249-260. | MR | Zbl

[K2] Kashiwara M., On crystal base of q-analogue of universal envoloping algebras, Duke Math. J., 63 (1991), pp. 456-516. | MR | Zbl

[K3] Kashiwara M., On crystal bases, Canadian Math. Soc. Conf. Proc., 16 (1995), pp. 155-197. | MR | Zbl

[KN] Kashiwara M. - Nakashima T., Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra, 165 (1994), pp. 295-345. | MR | Zbl

[LS1] Lakshmibai V. - Seshadri C.S., Geometry of G/P. V, J. Algebra, 100 (1986), pp. 462-557. | MR | Zbl

[LS1] Lakshmibai V. - Seshadri C.S., Standard monomial theory, Hyderabad Conference on Algebraic Groups, Manoj Prakashan, Madras, 1991, pp. 279-323. | MR | Zbl

[LW] Lakshmibai V. - Weyman J., Multiplicities of points on a Schubert variety in a minuscule G/P, Adv. Math., 84, no. 2 (1990), pp. 179-208. | MR | Zbl

[Li] Lichtenstein W., A system of quadrics describing the orbit of the highest weight vector, Proc. Amer. Math. Soc., 84, no. 4 (1982), pp. 605-608. | MR | Zbl

[LS] Liebeck W.M. - Saxl J., On the orders of the maximal subgroups of the finite exceptional groupes of Lie type, Proc. London Math. Soc., 55 (1987), pp. 299-330. | MR | Zbl

[Li1] Littelmann P., A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math., 116 (1994), 229-246. | MR | Zbl

[Li2] Littelmann P., Crystal graphs and Young tableau, J. Algebra, 175, no. 1 (1995), pp. 65-87. | MR | Zbl

[Li3] Littelmann P., Path and root operators in representation theory, Ann. Math., 142 (1995), pp. 499-525. | MR | Zbl

[L1] Lusztig G., Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., 3 (1990), pp. 447-498. | MR | Zbl

[L2] Lusztig G., Canonical bases arising from quantized envoloping algebras. II, Progr. Theor. Physics, 102 (1990), pp. 175-202. | MR | Zbl

[L2] Lusztig G., Introduction to quantum groups, Birkhäuser, Boston et al., 1993. | MR | Zbl

[Mn] Manin Yu. I., Cubic forms: algebra, geometry, arithmetic, North-Holland, Amsterdam-London, 1974. | MR | Zbl

[Mr] Mars J.G.M., Les nombers de Tamagawa de certains groupes exceptionnels, Bull. Soc. Math. France, 94 (1966), pp. 97-140. | Numdam | MR | Zbl

[Ma] Marsh R.J., On the adjoint module of a quantum group, Preprint University Warwick, no. 79 (1994), pp. 1-12.

[M] Matsumoto H., Sur les sous-groupes arithmétiques des groupes semisimples deployés, Ann. Sci. Ecole Norm. Sup., 4ème sér., 2 (1969), pp. 1-62. | Numdam | MR | Zbl

[MPS] Michel L. - Patera J. - Sharp R., The Demazure-Tits subgroup of a simple Lie group, J. Math. Phys., 29 no. 4 (1988), pp. 777-796. | MR | Zbl

[PR] Parker C. - Röhrle G., Minuscule Representations, Preprint Universität Bielefeld, no. 72 (1993).

[Pl1] Plotkin E.B., Stability theorems for K1-functors for Chevalley groups, Proc. Conf. Nonassociative algebras and related topics. Hiroshima- 1990, World Scientific, London et al., 1991, pp. 203-217. | MR | Zbl

[Pl2] Plotkin E.B., Surjective stabilization for K1-functors for some exceptional Chevalley groups, J. Soviet Math., 64, no. 1 (1993), pp. 751-767. | MR | Zbl

[Pl3] Plotkin E.B., On a problem of H. Bass for Chevalley groups of type E7, Y. Algebra, 210, no. 1 (1998), pp. 67-85. | Zbl

[PSV] Plotkin E.B. - Semenov A.A. - Vavilov N.A., Visual basic representations: an atlas, Int. J. Algebra and Computations, 8, no. (1998), pp. 61-97. | MR | Zbl

[P1] Proctor R.A., Bruhat lattices, plane partition generating functions, and minuscule representations, Europ. J. Combinatorics, 5 (1984), pp. 331-350. | MR | Zbl

[P2] Proctor R.A., A Dynkin diagram classification theorem arising from a combinatorial problem, Adv. Math., 62, no. 2 (1986), pp. 103-117. | MR | Zbl

[Sch] Scharlau R., Buildings, Handbook of Incidence Geometry, North Jolland, Amsterdam, 1995, pp. 477-645. | MR | Zbl

[Se] Seshadri C.S., Geometry of G/P. I. Standard monomial theory for minuscule P, C. P. Ramanujan: a tribute, Tata Press, Bombay, 1978, pp. 207-239. | MR | Zbl

[S1] Springer T.A., On the geometric algebra of the projective octave plane, Proc. Nederl. Akad. Wetensch., 65 (1962), pp. 451-468. | MR | Zbl

[S2] Springer T.A., Jordan algebras and algebraic groups, Springer-Verlag, N. Y. et al., 1973. | MR | Zbl

[S3] Springer T.A., Linear algebraic groups, Fundam. trends in Math., 55 (1989), pp. 5-136 (in Russian, English translation in Springer-Verlag). | MR | Zbl

[St1] Stein M.R., Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math., 93, no. 4 (1971), pp. 965-1004. | MR | Zbl

[St2] Stein M.R., Stability theorems for K1, K 2 and related functors modeled on Chevalley groups, Japan J. Math., 4, no. 1 (1978), pp. 77-108. | MR | Zbl

[S] Steinberg R., Lectures on Chevalley groups, Yale University, 1968. | MR

[St] Stembridge J.R., On minuscule representations, plane partitions and involutions in complex Lie groups, Duke J. Math., 73, no. 2 (1994), pp. 469-490. | MR | Zbl

[SV] Stepanov A.V. - Vavilov N.A., Decomposition of transvections: a theme with variations, K-theory, 19 (2000), pp. 109-153. | MR | Zbl

[Te] Testerman D.M., A1-type overgroups of elements of order p in semisimple algebraic groups and associated finite groups, J. Algebra, 177, no. 1 (1995), pp. 34-76. | MR | Zbl

[T] Tits J., Normalisateurs de tores. I. Groupes de Coxeter étendus, J. Algebra, 4, no. 1 (1966), pp. 96-116. | MR | Zbl

[V1] Vavilov N.A., On the problem of normality of the elementary subgroup in a Chevalley group, Algebraic and Discrete Systems, Ivanovo Univ., 1988, pp. 7-25 (in Russian). | MR | Zbl

[V2] Vavilov N.A., Structure of Chevalley groups over commutative rings, Proc. Conf. Nonassociative algebras and related topics. Hiroshima- 1990, World Scientific, London et al., 1991, pp. 219-335. | MR | Zbl

[V3] Vavilov N.A., Do it yourself structure constants for Lie algebras of type El, Preprint Universität Bielefeld, no. 35 (1993), pp. 1-42. | Zbl

[V4] Vavilov N.A., Intermediate subgroups in Chevalley groups, Proc. Conf. Groups of Lie Type and their Geometries (Como - 1993), Cambridge Univ. Press, 1995, pp. 233-280. | MR | Zbl

[V5] Vavilov N A., The ubiquity of microweights (to appear).

[V6] Vavilov N.A., Can one see the signs of the structure constants ?, St. Petersburg J. Math. (In Russian, to appear).

[V7] Vavilov N A., Geometry of the minimal modules for Chevalley groups of types E6 and E7 (to appear).

[VPe] Vavilov N.A. - Perelman E. Ya, Transvections in polyvector representations, St. Petersburg J. Math. (In Russian, to appear).

[VP] Vavilov N.A. - Plotkin E.B., Chevalley groups over commutative rings. I. Elementary calculations, Acta Applicandae Math., 45, no. 1 (1996), pp. 73-113. | MR | Zbl

[VP] Vavilov N.A. - Plotkin E.B. - Stepanov A.V., Calculations in Chevalley groups over commutative rings, Soviet Math. Dokl., 40 no. 1 (1989), pp. 145-147. | MR | Zbl

[W] Waterhouse W.C., Automorphisms of det(xij): a group scheme approach, Adv. Math., 65, no. 2 (1987), pp. 171-203. | MR | Zbl