Numerology of square equations
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N. A. Vavilov
Translated by: the author - St. Petersburg Math. J. 20 (2009), 687-707
- DOI: https://doi.org/10.1090/S1061-0022-09-01068-1
- Published electronically: July 21, 2009
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Abstract:
In the present work, which is a sequel of the paper “Can one see the signs of structure constants?”, we describe how one can see the form and the signs of the senior Weyl orbit of equations on the highest weight orbit directly in the weight diagram of microweight representations and adjoint representations for the simply-laced case. As special cases, the square equations we consider include the vanishing of second order minors, Plücker equations in polyvector and adjoint representations of classical groups, Cartan equations in spin and half-spin representations, Borel–Freudenthal equations defining the projective octave plane $\operatorname {E}_6/P_1$, and most of the equations defining Freudenthal’s variety $\operatorname {E}_7/P_7$. In view of forthcoming applications to the construction of decomposition of unipotents in the adjoint case, special emphasis is placed on the senior Weyl orbit of equations for the adjoint representations of groups of types $\operatorname {E}_6$, $\operatorname {E}_7$, and $\operatorname {E}_8$. This orbit consists of 270, 756, or 2160 equations, respectively, and we minutely discuss their form and signs. This generalizes Theorem 3 of the preceding paper “A third look at weight diagrams”, where we considered microweight representations of $\operatorname {E}_6$ and $\operatorname {E}_7$.References
- Armand Borel, Properties and linear representations of Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 1–55. MR 0258838
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629, DOI 10.1007/978-3-540-89394-3
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. MR 2109105
- N. A. Vavilov, How is one to view the signs of structure constants?, Algebra i Analiz 19 (2007), no. 4, 34–68 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 4, 519–543. MR 2381932, DOI 10.1090/S1061-0022-08-01008-X
- —, Decomposition of unipotents in adjoint representation of the Chevalley group of type $\mathrm {E}_6$, Algebra i Analiz (to appear). (Russian)
- N. A. Vavilov and M. R. Gavrilovich, $A_2$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$, Algebra i Analiz 16 (2004), no. 4, 54–87 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 4, 649–672. MR 2090851, DOI 10.1090/S1061-0022-05-00871-X
- N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, The structure of Chevalley groups: a proof from The Book, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 330 (2006), no. Vopr. Teor. Predst. Algebr. i Grupp. 13, 36–76, 271–272 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 140 (2007), no. 5, 626–645. MR 2253566, DOI 10.1007/s10958-007-0003-y
- N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, A Chevalley group of type $E_6$ in the 27-dimensional representation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 338 (2006), no. Vopr. Teor. Predst. Algebr. i Grupp. 14, 5–68, 261 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 145 (2007), no. 1, 4697–4736. MR 2354606, DOI 10.1007/s10958-007-0304-1
- N. A. Vavilov and S. I. Nikolenko, $A_2$-proof of structure theorems for a Chevalley group of type $F_4$, Algebra i Analiz 20 (2008), no. 4, 27–63 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 20 (2009), no. 4, 527–551. MR 2473743, DOI 10.1090/S1061-0022-09-01060-7
- N. A. Vavilov and E. Ya. Perel′man, Polyvector representations of $\textrm {GL}_n$, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 338 (2006), no. Vopr. Teor. Predst. Algebr. i Grupp. 14, 69–97, 261 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 145 (2007), no. 1, 4737–4750. MR 2354607, DOI 10.1007/s10958-007-0305-0
- N. A. Vavilov, E. B. Plotkin, and A. V. Stepanov, Calculations in Chevalley groups over commutative rings, Dokl. Akad. Nauk SSSR 307 (1989), no. 4, 788–791 (Russian); English transl., Soviet Math. Dokl. 40 (1990), no. 1, 145–147. MR 1020667
- N. A. Vavilov and N. P. Kharchev, Orbits of the stabilizer of subsystems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 338 (2006), no. Vopr. Teor. Predst. Algebr. i Grupp. 14, 98–124, 261–262 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 145 (2007), no. 1, 4751–4764. MR 2354608, DOI 10.1007/s10958-007-0306-z
- Yu. I. Manin, Kubicheskie formy: algebra, geometriya, arifmetika, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0360592
- T. A. Springer, Linear algebraic groups, Algebraic geometry, 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 5–136, 310–314, 315 (Russian). Translated from the English. MR 1100484
- Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
- Michael Aschbacher, The $27$-dimensional module for $E_6$. I, Invent. Math. 89 (1987), no. 1, 159–195. MR 892190, DOI 10.1007/BF01404676
- Michael Aschbacher, Some multilinear forms with large isometry groups, Geom. Dedicata 25 (1988), no. 1-3, 417–465. Geometries and groups (Noordwijkerhout, 1986). MR 925846, DOI 10.1007/BF00191936
- Roger W. Carter, Simple groups of Lie type, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Reprint of the 1972 original; A Wiley-Interscience Publication. MR 1013112
- Arjeh M. Cohen and Bruce N. Cooperstein, A characterization of some geometries of Lie type, Geom. Dedicata 15 (1983), no. 1, 73–105. MR 732544, DOI 10.1007/BF00146968
- Arjeh M. Cohen and Bruce N. Cooperstein, The $2$-spaces of the standard $E_6(q)$-module, Geom. Dedicata 25 (1988), no. 1-3, 467–480. Geometries and groups (Noordwijkerhout, 1986). MR 925847, DOI 10.1007/BF00191937
- A. M. Cohen and R. H. Cushman, Gröbner bases and standard monomial theory, Computational algebraic geometry (Nice, 1992) Progr. Math., vol. 109, Birkhäuser Boston, Boston, MA, 1993, pp. 41–60. MR 1230857, DOI 10.1007/978-1-4612-2752-6_{4}
- Bruce N. Cooperstein, A characterization of some Lie incidence structures, Geometriae Dedicata 6 (1977), no. 2, 205–258. MR 461279, DOI 10.1007/BF00181461
- Bruce N. Cooperstein, The geometry of root subgroups in exceptional groups. I, Geom. Dedicata 8 (1979), no. 3, 317–381. MR 550374, DOI 10.1007/BF00151515
- Bruce N. Cooperstein, The fifty-six-dimensional module for $E_7$. I. A four form for $E_7$, J. Algebra 173 (1995), no. 2, 361–389. MR 1325780, DOI 10.1006/jabr.1995.1092
- J. R. Faulkner and J. C. Ferrar, Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc. 9 (1977), no. 1, 1–35. MR 444729, DOI 10.1112/blms/9.1.1
- Alan Harebov and Nikolai Vavilov, On the lattice of subgroups of Chevalley groups containing a split maximal torus, Comm. Algebra 24 (1996), no. 1, 109–133. MR 1370526, DOI 10.1080/00927879608825557
- Stephen J. Haris, Some irreducible representations of exceptional algebraic groups, Amer. J. Math. 93 (1971), 75–106. MR 279103, DOI 10.2307/2373449
- Atanas Iliev and Laurent Manivel, The Chow ring of the Cayley plane, Compos. Math. 141 (2005), no. 1, 146–160. MR 2099773, DOI 10.1112/S0010437X04000788
- Hajime Kaji and Osami Yasukura, Projective geometry of Freudenthal’s varieties of certain type, Michigan Math. J. 52 (2004), no. 3, 515–542. MR 2097396, DOI 10.1307/mmj/1100623411
- V. Lakshmibai, C. Musili, and C. S. Seshadri, Geometry of $G/P$. III. Standard monomial theory for a quasi-minuscule $P$, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), no. 3, 93–177. MR 561813
- V. Lakshmibai, C. Musili, and C. S. Seshadri, Geometry of $G/P$. IV. Standard monomial theory for classical types, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), no. 4, 279–362. MR 553746
- V. Lakshmibai and C. S. Seshadri, Geometry of $G/P$. II. The work of de Concini and Procesi and the basic conjectures, Proc. Indian Acad. Sci. Sect. A 87 (1978), no. 2, 1–54. MR 490244
- V. Lakshmibai and C. S. Seshadri, Geometry of $G/P$. V, J. Algebra 100 (1986), no. 2, 462–557. MR 840589, DOI 10.1016/0021-8693(86)90089-X
- V. Lakshmibai and C. S. Seshadri, Standard monomial theory, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) Manoj Prakashan, Madras, 1991, pp. 279–322. MR 1131317
- V. Lakshmibai and J. Weyman, Multiplicities of points on a Schubert variety in a minuscule $G/P$, Adv. Math. 84 (1990), no. 2, 179–208. MR 1080976, DOI 10.1016/0001-8708(90)90044-N
- Woody Lichtenstein, A system of quadrics describing the orbit of the highest weight vector, Proc. Amer. Math. Soc. 84 (1982), no. 4, 605–608. MR 643758, DOI 10.1090/S0002-9939-1982-0643758-8
- Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1–62 (French). MR 240214
- S. Nikolenko and N. Semenov, Chow ring structure made simple, arXiv:math.AG/0606335 (2006), 1–17.
- Dmitri I. Panyushev, On actions of a maximal torus on orbits of highest weight vectors, J. Algebra 212 (1999), no. 2, 683–702. MR 1676860, DOI 10.1006/jabr.1998.7632
- V. Petrov, N. Semenov, and K. Zainoulline, Zero cycles on a twisted Cayley plane, Canad. Math. Bull. 51 (2008), no. 1, 114–124. MR 2384744, DOI 10.4153/CMB-2008-013-2
- Eugene Plotkin, On the stability of the $K_1$-functor for Chevalley groups of type $E_7$, J. Algebra 210 (1998), no. 1, 67–85. MR 1656415, DOI 10.1006/jabr.1998.7535
- Eugene Plotkin, Andrei Semenov, and Nikolai Vavilov, Visual basic representations: an atlas, Internat. J. Algebra Comput. 8 (1998), no. 1, 61–95. MR 1492062, DOI 10.1142/S0218196798000053
- C. S. Seshadri, Geometry of $G/P$. I. Theory of standard monomials for minuscule representations, C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 207–239. MR 541023
- Michael R. Stein, Stability theorems for $K_{1}$, $K_{2}$ and related functors modeled on Chevalley groups, Japan. J. Math. (N.S.) 4 (1978), no. 1, 77–108. MR 528869, DOI 10.4099/math1924.4.77
- Alexei Stepanov and Nikolai Vavilov, Decomposition of transvections: a theme with variations, $K$-Theory 19 (2000), no. 2, 109–153. MR 1740757, DOI 10.1023/A:1007853629389
- Nikolai A. Vavilov, Structure of Chevalley groups over commutative rings, Nonassociative algebras and related topics (Hiroshima, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 219–335. MR 1150262
- Nikolai Vavilov, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova 104 (2000), 201–250. MR 1809357
- N. A. Vavilov, Do it yourself structure constants for Lie algebras of types $E_l$, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), no. Vopr. Teor. Predst. Algebr. i Grupp. 8, 60–104, 281 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 120 (2004), no. 4, 1513–1548. MR 1875718, DOI 10.1023/B:JOTH.0000017882.04464.97
- Nikolai Vavilov, An $A_3$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$, Internat. J. Algebra Comput. 17 (2007), no. 5-6, 1283–1298. MR 2355697, DOI 10.1142/S0218196707003998
- Nikolai Vavilov and Eugene Plotkin, Chevalley groups over commutative rings. I. Elementary calculations, Acta Appl. Math. 45 (1996), no. 1, 73–113. MR 1409655, DOI 10.1007/BF00047884
Bibliographic Information
- N. A. Vavilov
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, 198504 St. Petersburg, Russia
- Email: nikolai-vavilov@yandex.ru
- Received by editor(s): April 1, 2007
- Published electronically: July 21, 2009
- Additional Notes: The basic ideas, which eventually led to the present paper, as also to \cite{47, 4} — a third look — were developed by the author in 1995 at the Universität Bielefeld (with the support of AvH-Stiftung, SFB-343 and INTAS 93-436). Preliminary versions of this text were taking shape since 1997, at the Universitá Milano I (with the support of the Cariplo Foundation for Fundamental Research), at the Newton Institute for Mathematical Sciences at Cambridge University, and at Bar Ilan University. At the final stage of this work the author was supported by RFBR 03-01-00349 (POMI RAN), by INTAS 00-566, and INTAS 03-51-3251, and by an express grant of the Russian Ministry of Education ‘Overgroups of semi-simple groups’ E02-1.0-61.
- © Copyright 2009 American Mathematical Society
- Journal: St. Petersburg Math. J. 20 (2009), 687-707
- MSC (2000): Primary 20G15, 20G35
- DOI: https://doi.org/10.1090/S1061-0022-09-01068-1
- MathSciNet review: 2492358