# Calculations in Exceptional Groups, an Update

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This paper is a slightly expanded text of our talk at the PCA-2014. There we announced two recent results concerning explicit polynomial equations defining exceptional Chevalley groups in microweight or adjoint representations. One of these results is an explicit characteristic-free description of equations on the entries of a matrix from the simply connected Chevalley group *G*(*E* _{7}, *R*) in the 56-dimensional representation *V*. Before, a similar description was known for the group *G*(*E* _{6}, *R*) in the 27-dimensional representation, whereas for the group of type E_{7} it was only known under the simplifying assumption that 2 ϵ *R* ^{*}. In particular, we compute the normalizer of *G*(*E* _{7}, *R*) in GL(56, *R*) and establish that, as also the normalizer of the elementary subgroup *E*(E_{7}, *R*), it coincides with the extended Chevalley group \( \overline{G}\left({\mathrm{E}}_7,R\right) \). The construction is based on the works of J. Lurie and the first author on the E_{7}-invariant quartic forms on *V*. Another major new result is a complete description of quadratic equations defining the highest weight orbit in the adjoint representations of Chevalley groups of types E_{6}, E_{7}, and E_{8}. Part of these equations not involving zero weights, the so-called square equations (or *π*/2-equations) were described by the second author. Recently, the first author succeeded in completing these results, explicitly listing also the equations involving zero weight coordinates linearly (the 2*π*/3-equations) and quadratically (the *π*-equations). Also, we briefly discuss recent results by S.Garibaldi and R.M.Guralnick on octic invariants for E_{8}.

## Keywords

Commutative Ring Adjoint Representation Chevalley Group High Weight Vector Exceptional Group## Preview

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## References

- 1.H. Apte and A. Stepanov, “Local-global principle for congruence subgroups of Chevalley groups,”
*Cent. Eur. J. Math.*,**12**, No. 6, 801–812 (2014).MATHMathSciNetGoogle Scholar - 2.M. Aschbacher, “Some multilinear forms with large isometry groups,”
*Geom. Dedicata*,**25**, No. 1–3, 417–465 (1988).MATHMathSciNetGoogle Scholar - 3.A. Bak, R. Hazrat, and N. Vavilov, “Localization-completion strikes again: relative K
_{1}is nilpotent,”*J. Pure Appl. Algebra*,**213**, 1075–1085 (2009).MATHMathSciNetCrossRefGoogle Scholar - 4.S. Berman and R. V. Moody, “Extensions of Chevalley groups,”
*Israel J. Math.*,**22**, No. 1, 42–51 (1975).MATHMathSciNetCrossRefGoogle Scholar - 5.A. Borel and J. Tits, “Groupes réductifs,”
*Inst. Hautes Études Sci Publ. Math.*, No. 27, 55–150 (1965).Google Scholar - 6.R. B. Brown, “Groups of type E
_{7},”*J. Reine Angew. Math.*,**236**, 79–102 (1969).MATHMathSciNetGoogle Scholar - 7.E. I. Bunina, “Automorphisms of Chevalley groups of different types over commutative rings,”
*J. Algebra*,**355**, No. 1, 154–170 (2012).MATHMathSciNetCrossRefGoogle Scholar - 8.M. Cederwall and J. Palmkvist, “The octic E
_{8}invariant,”*J. Math. Phys.*,**48**, No. 7, 073505 (2007).MathSciNetCrossRefGoogle Scholar - 9.B. N. Cooperstein, “The fifty-six-dimensional module for E
_{7}. I. A four form for E_{7},”*J. Algebra*,**173**, No. 2, 361–389 (1995).MATHMathSciNetCrossRefGoogle Scholar - 10.M. Demazure and A. Grothendieck (with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud, and J.-P. Serre),
*Schémas en groupes*(SGA 3). Séminaire de Géométrie Algébrique du Bois Marie 1962–64, revised and annotated edition of the 1970 French original, edited by Ph. Gille and P. Polo, tomes 1–3. Société Mathématique de France, Paris (2011).Google Scholar - 11.J. R. Faulkner and J. C. Ferrar, “Exceptional Lie algebras and related algebraic and geometric structures,”
*Bull. London Math. Soc.*,**9**, No. 1, 1–35 (1977).MATHMathSciNetCrossRefGoogle Scholar - 12.S. Garibaldi and R. M. Guralnick, “Simple algebraic groups are (usually) determined by an invariant,” arXiv:1309.6611v1.Google Scholar
- 13.S. Garibaldi and R. M. Guralnick, “Simple groups stabilizing polynomials,” arXiv:1309.6611v2.Google Scholar
- 14.R. Hazrat, V. Petrov, and N. Vavilov, “Relative subgroups in Chevalley groups,”
*J. K-Theory*,**5**, No. 3, 603–618 (2010).MATHMathSciNetCrossRefGoogle Scholar - 15.R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “The yoga of commutators,”
*J. Math. Sci. (N. Y.)*,**179**, No. 6, 662–678 (2011).MATHMathSciNetCrossRefGoogle Scholar - 16.R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “The yoga of commutators: further applications,”
*J. Math. Sci. (N. Y.)*,**200**, No. 6, 742–768 (2014).MATHMathSciNetCrossRefGoogle Scholar - 17.R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “Commutators width in Chevalley groups,”
*Note Mat.*,**33**, No. 1, 139–170 (2013).MATHMathSciNetGoogle Scholar - 18.R. Hazrat and N. Vavilov, “
*K*_{1}of Chevalley groups are nilpotent,”*J. Pure Appl. Algebra*,**179**, No. 1, 99–116 (2003).MATHMathSciNetCrossRefGoogle Scholar - 19.R. Hazrat, N. Vavilov, and Z. Zhang, “Relative commutator calculus in Chevalley groups,”
*J. Algebra*,**383**, No. 1, 262–293 (2013).MathSciNetCrossRefGoogle Scholar - 20.R. Hazrat, N. Vavilov, and Z. Zhang, “Generation of relative commutator subgroups in Chevalley groups,”
*Proc. Edinburgh Math. Soc.*, 1–19 (2014); arXiv:1212.5432v1.Google Scholar - 21.W. van der Kallen, “Another presentation for Steinberg groups,”
*Indag. Math.*,**39**, No. 4, 304–312 (1977).MathSciNetCrossRefGoogle Scholar - 22.E. A. Kulikova and A. K. Stavrova, “Centralizer of the elementary subgroup of an isotropic reductive group,”
*Vestnik St. Petersburg Univ. Math.*,**46**, No. 1, 22–28 (2013).MATHMathSciNetCrossRefGoogle Scholar - 23.A. Lavrenov, “Another presentation for symplectic Steinberg groups,” arXiv:1405.4296.Google Scholar
- 24.W. Lichtensein, “A system of quadrics describing the orbit of the highest weight vector,”
*Proc. Amer. Math. Soc.*,**84**, No. 4, 605–608 (1982).MathSciNetCrossRefGoogle Scholar - 25.M. W. Liebeck and G. M. Seitz, “On the subgroup structure of exceptional groups of Lie type,”
*Trans. Amer. Math. Soc.*,**350**, No. 9, 3409–3482 (1998).MATHMathSciNetCrossRefGoogle Scholar - 26.J. Lurie, “On simply laced Lie algebras and their minuscule representations,”
*Comment. Math. Helv.*,**76**, No. 3, 515–575 (2001).MATHMathSciNetCrossRefGoogle Scholar - 27.A. Yu. Luzgarev, “Fourth-degree invariants for
*G*(E_{7},*R*) not depending on the characteristic,”*Vestnik St. Petersburg Univ. Math.*,**46**, No. 1, 29–34 (2013).Google Scholar - 28.A. Luzgarev, “Equations determining the orbit of the highest weight vector in the adjoint representation,” arXiv:1401.0849.Google Scholar
- 29.A. Luzgarev, V. Petrov, and N. Vavilov, “Explicit equations on orbit of the highest weight vector,” to appear.Google Scholar
- 30.A. Yu. Luzgarev and A. K. Stavrova, “The elementary subgroup of an isotropic reductive group is perfect,”
*St. Petersburg Math. J.*,**23**, No. 5, 881–890 (2012).Google Scholar - 31.H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,”
*Ann. Sci. Ecole Norm. Sup. (4)*,**2**, 1–62 (1969).Google Scholar - 32.V. A. Petrov and A. K. Stavrova, “Elementary subgroups in isotropic reductive groups,”
*St. Petersburg Math. J.*,**20**, No. 4, 625–644 (2009).MATHMathSciNetCrossRefGoogle Scholar - 33.E. Plotkin, “Stability theorems of
*K*_{1}-functor for Chevalley groups,” in:*Nonassociative Algebras and Related Topics*(Hiroshima, 1990), World Sci. Publ., River Edge, New Jersey (1991), pp. 203–217.Google Scholar - 34.E. Plotkin, “Surjective stabilization of the K
_{1}-functor for some exceptional Chevalley groups,”*J. Soviet Math.*,**64**, No. 1, 751–766 (1993).MathSciNetCrossRefGoogle Scholar - 35.E. Plotkin, “On the stability of the K
_{1}-functor for Chevalley groups of type E_{7},”*J. Algebra*,**210**, No. 1, 67–85 (1998).MATHMathSciNetCrossRefGoogle Scholar - 36.E. Plotkin, A. Semenov, and N. Vavilov, “Visual basic representations: an atlas,”
*Internat. J. Algebra Comput.*,**8**, No. 1, 61–95 (1998).MATHMathSciNetCrossRefGoogle Scholar - 37.A. N. Rudakov, “Deformations of simple Lie algebras,”
*Izv. Akad. Nauk USSR Ser. Mat.*,**5**, No. 5, 1113–1119 (1971).Google Scholar - 38.G. M. Seitz,
*The Maximal Subgroups of Classical Algebraic Groups*, Mem. Amer. Math. Soc.,**67**, No. 365 (1987).Google Scholar - 39.S. S. Sinchuk, “Parabolic factorizations of reductive groups,” Ph. D. Thesis, St.Petersburg State University (2013).Google Scholar
- 40.S. S. Sinchuk, “Improved stability for the odd-dimensional orthogonal group,”
*J. Math. Sci. (N. Y.)*,**199**, No. 3, 343–349 (2014).MATHMathSciNetCrossRefGoogle Scholar - 41.A. Stavrova, “Homotopy invariance of non-stable K
_{1}-functors,”*J. K-Theory*,**13**, No. 2, 199–248 (2014).MATHMathSciNetCrossRefGoogle Scholar - 42.A. Stavrova, “On the congruence kernel of isotropic groups over rings,” arXiv:1305.0057.Google Scholar
- 43.A. Stavrova, “Non-stable K
_{1}-functors of multiloop groups,” arXiv:1404.7587.Google Scholar - 44.M. R. Stein, “Stability theorems for
*K*_{1},*K*_{2}and related functors modeled on Chevalley groups,”*Japan. J. Math. (N.S.)*,**4**, No. 1, 77–108 (1978).Google Scholar - 45.A. Stepanov, “Elementary calculus in Chevalley groups over rings,”
*J. Prime Res. Math.*,**9**, 79–95 (2013).MATHMathSciNetGoogle Scholar - 46.A. Stepanov, “Structure of Chevalley groups over rings via universal localization,” to appear in
*J. K-Theory*(2014); arXiv:1303.6082v3.Google Scholar - 47.A. Stepanov, “Non-abelian
*K*-theory of Chevalley groups over rings,” to appear in*J. Math. Sci. (N. Y.)*(2014).Google Scholar - 48.A. Stepanov, “Structure theory and subgroups of Chevalley groups over rings,” Dr. Sci. Thesis (Habilitation), St.Petersburg State University (2014).Google Scholar
- 49.A. Stepanov and N. Vavilov, “Decomposition of transvections: a theme with variations,”
*K-Theory*,**19**, No. 2, 109–153 (2000).Google Scholar - 50.A. Stepanov and N. Vavilov, “On the length of commutators in Chevalley groups,”
*Israel J. Math.*,**185**, No. 1, 253–276 (2011).MATHMathSciNetCrossRefGoogle Scholar - 51.A. A. Suslin, “On the structure of the special linear group over polynomial rings,”
*Izv. Akad. Nauk USSR Ser. Mat.*,**11**, 221–238 (1977).MATHCrossRefGoogle Scholar - 52.A. Suslin, “Quillen’s solution of Serre’s problem,”
*J. K-Theory*,**11**, No. 3, 549–552 (2013).MATHMathSciNetCrossRefGoogle Scholar - 53.N. A. Vavilov, “Structure of Chevalley groups over commutative rings,” in
*Nonassociative Algebras and Related Topics*(Hiroshima, 1990), World Sci. Publ., River Edge, New Jersey (1991), pp. 219–335.Google Scholar - 54.N. A. Vavilov, “A third look at weight diagrams,”
*Rend. Sem. Mat. Univ. Padova*,**104**, 201–250 (2000).MATHMathSciNetGoogle Scholar - 55.N. A. Vavilov, “An A
_{3}-proof of structure theorems for Chevalley groups of types E_{6}and E_{7},”*Internat. J. Algebra Comput.*,**17**, No. 5–6, 1283–1298 (2007).MATHMathSciNetCrossRefGoogle Scholar - 56.N. A. Vavilov, “Can one see the signs of structure constants?,”
*St. Petersburg Math. J.*,**19**, No. 4, 519–543 (2008).MATHMathSciNetCrossRefGoogle Scholar - 57.N. A. Vavilov, “Weight elements of Chevalley groups,”
*St. Petersburg Math. J.*,**20**, No. 1, 23–57 (2009).MATHMathSciNetCrossRefGoogle Scholar - 58.N. A. Vavilov, “Numerology of quadratic equations,”
*St. Petersburg Math. J.*,**20**, No. 5, 687–707 (2009).MATHMathSciNetCrossRefGoogle Scholar - 59.N. A. Vavilov, “Some more exceptional numerology,”
*J. Math. Sci. (N. Y.)*,**171**, No. 3, 317–321 (2010).MATHMathSciNetCrossRefGoogle Scholar - 60.N. A. Vavilov, “An A
_{3}-proof of structure theorems for Chevalley groups of types E_{6}and E_{7}. II. Fundamental lemma,”*St. Petersburg Math. J.*,**23**, No. 6, 921–942 (2012).MATHMathSciNetCrossRefGoogle Scholar - 61.N. A. Vavilov, “A closer look at weight diagrams of types (E6,
*ϖ*_{1}) and (E7,*ϖ*_{7}),” to appear (2014).Google Scholar - 62.N. A. Vavilov and M. R. Gavrilovich, “A
_{2}-proof of structure theorems for Chevalley groups of types E_{6}and E_{7},”*St. Petersburg Math. J.*,**16**, No. 4, 649–672 (2005).MATHMathSciNetCrossRefGoogle Scholar - 63.N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, “Structure of Chevalley groups: the proof from the Book,”
*J. Math. Sci. (N. Y.)*,**140**, No. 5, 626–645 (2007).MathSciNetCrossRefGoogle Scholar - 64.N. A. Vavilov and V. G. Kazakevich, “More variations on the decomposition of transvections,”
*J. Math. Sci. (N. Y.)*,**171**, No. 3, 322–330 (2010).MATHMathSciNetCrossRefGoogle Scholar - 65.N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of Chevalley groups of type E
_{6},”*St. Petersburg Math. J.*,**19**, No. 5 (2008), 699–718.Google Scholar - 66.N. A. Vavilov and A. Yu. Luzgarev, “Chevalley group of type E
_{7}in the 56-dimensional representation,”*J. Math. Sci. (N. Y.)*,**180**, No. 3, 197–251 (2012).Google Scholar - 67.N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of Chevalley groups of type E
_{7},” to appear in*St. Petersburg Math. J*.Google Scholar - 68.N. A. Vavilov and A. Yu. Luzgarev, “A
_{2}-proof of structure theorems for the Chevalley group of type E_{8},” to appear in*St. Petersburg Math. J*.Google Scholar - 69.N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, “Chevalley group of type E
_{6}in the 27-dimensional representation,”*J. Math. Sci. (N. Y.)*,**145**, No. 1 (2007), 4697–4736.Google Scholar - 70.N. Vavilov, A. Luzgarev, and A. Stepanov, “Calculations in exceptional groups over rings,”
*J. Math. Sci. (N. Y.)*,**373**, 48–72 (2009).MathSciNetGoogle Scholar - 71.N. A. Vavilov and S. I. Nikolenko, “A
_{2}-proof of structure theorems for the Chevalley group of type F_{4},”*St. Petersburg Math. J.*,**20**, No. 4, 527–551 (2009).MATHMathSciNetCrossRefGoogle Scholar - 72.N. A. Vavilov and E. B. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,”
*Acta Appl. Math.*,**45**, No. 1, 73–113 (1996).MATHMathSciNetCrossRefGoogle Scholar - 73.N. A. Vavilov, E. B. Plotkin, and A. V. Stepanov, “Calculations in Chevalley groups over commutative rings,”
*Soviet Math. Dokl.*,**40**, No. 1, 145–147 (1990).MathSciNetGoogle Scholar - 74.W. C.Waterhouse, “Automorphisms of det(
*X*_{ij}): the group scheme approach,”*Adv. Math.*,**65**, No. 2, 171–203 (1987).MATHMathSciNetCrossRefGoogle Scholar