Fourth-degree invariants for G(E7,R) not depending on the characteristic

The Chevalley group of type E₇ over a field of characteristic different from 2 coincides with the stabilizer of a fourth-degree form on a 56-dimensional vector space. Removing the constraint on the characteristic requires considering asymmetric forms. The space of four-linear forms stabilized by the Chevalley group of type E₇ in the minimal representation over an arbitrary commutative ring is described.     

A New Functor from E 6-mod to E 7-mod

We find a new representation of the simple Lie algebra of type E ₇ on the polynomial algebra in 27 variables. Using this representation and Shen's idea of mixed product, we construct a new functor from E ₆-Mod to E ₇-Mod. A condition for the functor to map a finite-dimensional irreducible E ₆-module to an infinite-dimensional irreducible E ₇-module is obtained. Our general framework also gives a direct polynomial extension from irreducible E ₆-modules to irreducible E ₇-modules, which can be used to derive Gel'fand–Zetlin bases for E ₇ from those for E ₆ that can be obtained from those for D ₅ according to our earlier work.     

Superdecomposable E(R)-Algebras

ABSTRACT

An E-ring is a ring that is naturally isomorphic to the endomorphism ring of its additive group. E-rings with various properties have been constructed in the literature; we now consider superdecomposable E-rings. More generally, we construct superdecomposable algebras A over integral domains R, which are at the same time E(R)-algebra in the sense that the ring of R-endomorphisms of the underlying R-module structure is canonically isomorphic to A. We also establish the existence of arbitrarily large superdecomposable modules over such algebras.     

Zagier's conjecture on L(E,2)

In this paper we introduce an elliptic analog of the Bloch-Suslin complex and prove that it (essentially) computes the weight two parts of the groups K ₂(E) and K ₁(E) for an elliptic curve E over an arbitrary field k. Combining this with the results of Bloch and Beilinson we proved Zagier's conjecture on L(E,2) for modular elliptic curves over ℚ. Oblatum 3-VI-1996 & 16-V-1997     

W(R)-splines

In [3] Golomb describes, for 1 < p < ∞, the H^r,p(R)-extremal extension F_* of a function ƒ:E → R (i.e., the H^r,p-spline with knots in E) and studies the cone H_*E^r,p of all such splines. We study the problem of determining when F_* is in W^r,p ≡ H^r,p ∩ L^p. If F_* ε W^r,p, then F_* is called a W^r,p-spline, and we denote by W_*E^r,p the cone of all such splines. If E is quasiuniform, then F_* ε W^r,p if and only if {ƒ(t_i)}_tiεE ε l^p. The cone W_*E^r,p with E quasiuniform is shown to be homeomorphic to l^p. Similarly, H_*E^r,p is homeomorphic to h^r,p. Approximation properties of the W^r,p-splines are studied and error bounds in terms of the mesh size ¦ E ¦ are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of ¦ E ¦ are derived.     

Thompson’s conjecture for Lie type groups E 7(q)

In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type E ₇(q). More precisely, we show that every finite group G with the properties Z(G) = 1 and cs(G) = cs(E ₇(q)) is necessarily isomorphic to E ₇(q), where cs(G) and Z(G) are the set of lengths of conjugacy classes of G and the center of G respectively.     

The kernel of the Rost invariant, Serre''s Conjecture II and the Hasse principle for quasi-split groups 3,6 D 4 , E 6 , E 7

 We prove that for a simple simply connected quasi-split group of type ^3,6 D ₄ ,E ₆ ,E ₇ defined over a perfect field F of characteristic ≠=2,3 the Rost invariant has trivial kernel. In certain cases we give a formula for the Rost invariant. It follows immediately from the result above that if cd F≤2 (resp. vcd F≤2) then Serre's Conjecture II (resp. the Hasse principle) holds for such a group. For a (C ₂ )-field, in particular ℂ(x,y), we prove the stronger result that Serre's Conjecture II holds for all (not necessary quasi-split) exceptional groups of type ^3,6 D ₄ ,E ₆ ,E ₇ . Received: 27 March 2002 / Published online: 28 March 2003 The author gratefully acknowledge the support of TMR ERB FMRX CT-97-0107 and Forschungsinstitut für Mathematik, ETH in Zürich     

Index Reduction Formulas for Twisted Flag Varieties, II

This is a continuation of the determination begun in K-Theory 10 (1996), 517–596, of explicit index reduction formulas for function fields of twisted flag varieties of adjoint semisimple algebraic groups. We give index reduction formulas for the varieties associated to the classical simple groups of outer type A _n-1 and D _n, and the exceptional simple groups of type E ₆ and E ₇. We also give formulas for the varieties associated to transfers and direct products of algebraic groups. This allows one to compute recursively the index reduction formulas for the twisted flag varieties of any semi-simple algebraic group.     

A characterization of some geometries of Lie type

For geometries associated with permutation representations of the groups of Lie type E ₆, E ₇, E ₈ on certain maximal parabolic subgroups (e.g. the stabilizers of root subgroups), axiom systems are given that characterize them in terms of points and lines.     

Minimal permutation representations of finite simple exceptional groups of typesE 6,E 7, andE 8

A minimal permutation representation of a group is its faithful permutation representation of least degree. We will find degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, for groups of types E₆, E₇, and E₈. This brings to a close the study of minimal permutation representations of finite simple Chevalley groups. Supported by RFFR grant No. 93-01-01501, through the program “Universities of Russia,” and by grant No. RPC300 of ISF and the Government of Russia. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 518–530, September–October, 1997.     

On Besselian Schauder bases inl p (E)

By complex interpolation and tensor products, Schauder bases are constructed of the Banach sequence spacesl _p (E). In a general result, we study the Besselian property of the basis, and ifE is assumed to be theL _p (Lebesgue) andS _p (v. Neumann-Schatten) space, we obtain inequalities for the coefficient functionals associated to the basis which generalise other results given by Hausdorff—Young and Gohberg—Marcus. Finally, we construct non-Besselian and conditional bases ofl _p (E).     

A remark on the unconditional structure ofL (E,F)

We present a sequence of symmetric Banach spacesE _n ,n εN, withd (E _n , ),n ∈N, unbounded and ubc(L(E*_n,E _n )),n ∈N, uniformly bounded. During the preparation of this paper the author was supported by the Deutsche Forschungsgemeinschaft.     

On the Overgroups of EO(2l,R)

Let R be a commutative ring with 1, let 2 R ^*, and let l 3. We describe the subgroups of the general linear group GL(n,R) that contain the split elementary orthogonal group EO(2l,R). For every intermediate subgroup H, there exists a unique maximal ideal A R such that E(2l,R,A) H and, moreover, H normalizes EO(2l,R)E(2l,R,A). In the case where R = K is a field, similar results were obtained earlier by Dye, King, Li Shangzhi, and Bashkirov. Bibliography: 31 titles.     

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for ² F ₄(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ² F ₄(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.     

Derived Equivalence Classification of the Cluster-Tilted Algebras of Dynkin Type E

We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E ₆, E ₇ and E ₈ which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as supplementary material to this paper. From a structural point of view the remarkable outcome of our classification is that two cluster-tilted algebras of Dynkin type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers which in turn happens if and only if the two algebras are connected by a sequence of “good” mutations. This is reminiscent of the derived equivalence classification of cluster-tilted algebras of Dynkin type A, but quite different from the situation in Dynkin type D where a far-reaching classification has been obtained using similar methods as in the present paper but some very subtle questions are still open.     

Elliptic curves and their torsion subgroups over number fields of type (2, 2, ..., 2)

Suppose thatE: y ² =x(x + M) (x + N) is an elliptic curve, whereM N are rational numbers (#0, ±1), and are relatively prime. LetK be a number field of type (2,...,2) with degree 2′. For arbitrary n, the structure of the torsion subgroup E(K) _tors of theK-rational points (Mordell group) ofE is completely determined here. Explicitly given are the classification, criteria and parameterization, as well as the groups E(K) _tors themselves. The order of E( K)_tors is also proved to be a power of 2 for anyn. Besides, for any elliptic curveE over any number field F, it is shown that E( L)_tors = E( F) _tors holds for almost all extensionsL/F of degree p(a prime number). These results have remarkably developed the recent results by Kwon about torsion subgroups over quadratic fields.     

Experimental evidence for the occurrence of E 8 in nature and the radii of the Gosset circles

A recent experimental discovery involving the spin structure of electrons in a cold one-dimensional magnet points to a validation of a (1989) Zamolodchikov model involving the exceptional Lie group E ₈. The model predicts 8 particles and predicts the ratio of their masses. The conjectures have now been validated experimentally, at least for the first five masses. The Zamolodchikov model was extended in 1990 to a Kateev–Zamolodchikov model involving E ₆ and E ₇ as well. In a seemingly unrelated matter, the vertices of the 8-dimensional Gosset polytope identifies with the 240 roots of E ₈. Under the famous two-dimensional (Peter McMullen) projection of the polytope, the images of the vertices are arranged in eight concentric circles, hereafter referred to as the Gosset circles. The McMullen projection generalizes to any complex simple Lie algebra (in particular not restricted to A-D-E types) whose rank is greater than 1. The Gosset circles generalize as well, using orbits of the Coxeter element on roots. Applying results in Kostant (Am J Math 81:973–1032, 1959), I found some time ago a very easily defined operator A on a Cartan subalgebra, the ratio of whose eigenvalues is exactly the ratio of squares of the radii r _i of the generalized Gosset circles. The two matters considered above relate to one another in that the ratio of the masses in the E ₆, E ₇, E ₈ Kateev–Zamolodchikov models are exactly equal to the ratios of the radii of the corresponding generalized Gosset circles.     

The existence of J 3 and its embeddings in E 6

We determine the embeddings of the third sporadic group J ₃ of Janko in simple Chevalley groups of type E ₆ over finite and algebraically closed fields. As a corollary we obtain a short elegant existence proof of J ₃. This is of interest as J ₃ is one of the few sporadic groups not contained in the Monster, so its existence cannot be verified within that group. Previous existence proofs were highly computational; cf. [4] and [6].To Jacques Tits on his sixtieth birthdayPartially supported by NSF DMS-8721480 and NSA MDA90-88-H-2032.     

Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique ≤2; Galois cohomology of quasi-split groups over fields of cohomological dimension ≤ 2

Let k be a perfect field with cohomological dimension 2. Serre's conjecture II claims that the Galois cohomology set H ¹(k,G) is trivial for any simply connected semi-simple algebraic G/k and this conjecture is known for groups of type ¹ A _n after Merkurjev–Suslin and for classical groups and groups of type F ₄ and G ₂ after Bayer–Parimala. For any maximal torus T of G/k, we study the map H ¹(k, T) H ¹(k, G) using an induction process on the type of the groups, and it yields conjecture II for all quasi-split simply connected absolutely almost k-simple groups with type distinct from E ₈. We also have partial results for E ₈ and for some twisted forms of simply connected quasi-split groups. In particular, this method gives a new proof of Hasse principle for quasi-split groups over number fields including the E ₈-case, which is based on the Galois cohomology of maximal tori of such groups.