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N. A. Vavilov Department of Mathematics and Mechanics, State University of Saint-Petersburg St. Petersburg, Russia nikolai-vavilov@yandex.ru

Calculations in exceptional groups
Let be a reduced irreducible root system, R be a commutative ring with 1. We study the following three closely related groups, associated to (, R). · The (simply-connected) Chevalley group G(, R). · The (simply-connected) elementary Chevalley group E (, R). · The Steinberg group St(, R). We set K1 (, R) = G(, R)/E (, R) and denote by K2 (, R) the kernel of the natural pro jection St(, R) - E (, R). We are mainly interested in the four large exceptional groups of types E6 , E7 , E8 and F4 . There are two approaches towards the proof of structure theorems for G = G(, R): induction on dimension of R (localisation proofs) and induction on rank of (geometric proofs). Using Bak's method of localisation-completion, A.Bak, R.Hazrat and the author [1], [2] succeeded in finalising results on the nilpotent structure of K1 (, R). For example, (apart from the known exceptions of rank 2) we characterised E (, R) as the unique largest perfect subgroup of G(, R) and, among other things, proved the following results. Recall, that (R) denotes the Bass--Serre dimension of R. Theorem 1. Assume, that rk() 2, and = B2 , G2 . Then the subgroup E (, R, A, B ) is normal in G(, R), while C (, R, A, B ) coincides with the transporter Tran(G(, R), E (, R, A, B )). Theorem 2. Assume, that rk() 2, and (R) < . Then the group K1 (, R, I ) is nilpotent. However, our main emphasis is on calculations in exceptional groups, based on geometry and combinatorics of minimal modules. We wish to calculate with elements of E6 , E7 , E8 as 27 в 27, 56 в 56 or 248 в 248 matrices, respectively. Straightforward calculations with such matrices, using equations of degree 3 immediately run into formidable difficulties. Developing pivotal ideas of H.Matsumoto and M.Stein, the author, A.Stepanov and E.Plotkin [3] ­ [5], [11] proposed the first working method of such calculations, decomposition of unipotents, which only referred to quadratic equations on one column or row of a matrix from G. Recently the author and his students M.Gavrilovich, S.Nikolenko and A.Luzgarev, see, in particular, [6] ­ [10] developed another amazing geometric approach to the proof of the main structure theorems for G(, R), proof from the Book aka the A2 -proof, which only refers to linear equations on the Lie algebra of G. In the A2 -proof we look at the stabiliser of a column of a matrix of the form e + x, where x is an element of the corresponding Lie algebra L. In essense what we prove in [7], [8], [10] is the following result. Theorem 3. Let z G and g = [z , x (1)] is non-central. Then there exists a root type unipotent x = x1 ( )x2 ( ) such that (xg ) = g and [x, g ] = e. This easy observation immediately implies many known structure results about Chevalley groups, including description of their normal subgroups. We observe that this trick works for Chevalley groups of types E6 and E7 in minimal representations [6], not for two arbitrary columns, of course, not even for any two singular columns at distance 1, but for two columns of a root type inipotent at distance 1. 1


Theorem 4. Let z G and g = [z , x (1)] is non-central. Then there exists a root type unipotent x = x1 ( )x2 ( )x3 ( ) such, that (xg ) = g , (xg )µ = gµ for some weights , µ at distance 1 and [x, g ] = e. We state also some other closely related results, such as the main lemma of the A3 proof, proposed by the author to assault centrality of K2 (, R) for exceptional groups [6]; a characterisation of the extended Chevalley group of type E6 over an arbitrary commutative ring, found recently by the author and A.Luzgarev [9]; and finiteness of commutators in elementary generators, established by A.Stepanov and the author. References [1] A. Bak, R. Hazrat, N. Vavilov, Localisation-completion: application to classical-like groups, J. Pure Appl. Algebra (2008) (to appear). [2] R. Hazrat, N. Vavilov, K1 of Cheval ley groups are nilpotent, J. Pure Appl. Algebra 179 (2003), 99­116. [3] A. Stepanov, N. Vavilov, Decomposition of transvections: a theme with variations, K Theory 19 (2000), 109­153. [4] N. Vavilov, Structure of Cheval ley groups over commutative rings, Proc. Conf. Nonassociative algebras and related topics. Hiroshima ­ 1990, World Scientific, London et al., 1991, 219­335. [5] N. Vavilov, A third look at weight diagrams, Rendiconti del Seminario Matem. dell'Univ. di Padova 204 (2000), 1­45. [6] N. Vavilov, An A3 -proof of structure theorems for Cheval ley groups of types E6 and E7 , Int. J. Algebra Comput. 17 (2007), no. 5&6, 1283­1298. [7] N. A. Vavilov, M. R. Gavrilovich, An A2 -proof of the structure theorems for Cheval ley groups of types E6 and E7 , St.-Petersburg Math. J. 16 (2005), no. 4, 649­672. [8] N. A. Vavilov, M. R. Gavrilovich, S. I. Nikolenko, Structure of Cheval ley groups: the Proof from the Book, J. Math. Sci. 330 (2006), pages 36­76. [9] N. A. Vavilov, A. Yu. Luzgarev Normaliser of Cheval ley groups of type E6 , St.-Petersburg Math. J. 19 (2007), no. 5, 35­62. [10] N. A. Vavilov, S. I. Nikolenko, An A2 -proof of the structure theorems for Cheval ley groups of type F4 , St.-Petersburg Math. J. 20 (2008), no. 2. [11] N. Vavilov, E. Plotkin Cheval ley groups over commutative rings. I. Elementary calculations, Acta Applicandae Math. 45 (1996), no. 1, 73­113.

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