In this paper we conduct a systematic computerized search for groups generated by small, but highly symmetric, sets of elements of order 3. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. Firstly, we introduce monomial modular representations as these will prove useful later in the paper. Then the techniques of symmetric generation developed elsewhere are described afresh. The results we obtain are presented in a convenient tabular form, together with relevant character tables.

Let k be an algebraically closed field of arbitrary characteristic. Lines in ℙ3 are parametrized by the Grassmannian G(2, 4), which is isomorphic to a smooth quadric in ℙ5. We can consider the configuration space Xn = G(2, 4)n / PGL4(k) parametrizing ordered n-tuples of lines in ℙ3 up to projective equivalence. dim PGL4(k) = 15 and for n [ges ] 5, the stabilizer of a general n-tuple of lines is trivial, so for n [ges ] 5, Xn has the expected dimension 4n − 15.

The question of rationality of Xn was posed by Dolgachev. The space Xn is clearly unirational, since there is a dominant rational map to it from the rational variety G(2, 4)n. The following results are known in characteristic 0: it is a special case of a theorem by Dolgachev and Boden [1] for configuration spaces in greater generality that if Xn is rational for some n [ges ] 5 then so is XN for any N [ges ] n. They also proved that the configuration space of lines in ℙm is rational if m is odd and recently Zaitsev [2] proved this for all m.

Our proof uses different methods and it also has the advantage that it is valid in any characteristic. The main result is the following:

Soient k un corps de nombres et f: X → P1k un k-morphisme surjectif de k-variétés projectives lisses, à fibre générique géométriquement intègre. Supposons que les fibres de f au-dessus d'un sous-ensemble Hilbertien de P1(k) satisfont le principe de Hasse (resp. le principe de Hasse et l'approximation faible). Supposons que toutes les fibres géométriques de f ont au moins une composante de multiplicité un. Le rang de f est la somme des degrés des points fermés P dont la fibre XP = f−1(P) ne possède pas de composante de multiplicité un géométriquement intègre. Supposons le rang au plus égal à 2. Alors l'obstruction de Brauer–Manin au principe de Hasse (resp. à l'approximation faible) sur X est la seule. Dans des articles antérieurs nous n'avions obtenu ce résultat que sous des hypothèses plus fortes sur la nature des fibres. Le présent énoncé, obtenu grâce à une descente sur des variétés ouvertes, permet d'étudier des variétés données par des équations affines simples, sans calcul explicite d'un modè le projectif et lisse.

We classify finite Moufang loops which are centrally nilpotent of class 2 in terms of certain cubic forms, concentrating on small Frattini Moufang loops, or SFMLs, which are Moufang loops L with a central subgroup Z of order p such that L/Z is an elementary abelian p-group. (For example, SFM 2-loops are precisely the class of code loops, in the sense of Griess.)

More specifically, we first show that the nuclearly-derived subloop (normal associator subloop) of a Moufang loop of class 2 has exponent dividing 6. It follows that the subloop of elements of p-power order is associative for p > 3. Next, we show that if L is an SFML, then L/Z has the structure of a vector space with a symplectic cubic form. We then show that every symplectic cubic form is realized by some SFML and that two SFMLs are isomorphic in a manner preserving the central subgroup Z if and only if their symplectic cubic spaces are isomorphic up to scalar multiple. Consequently, we also obtain an explicit characterization of isotopy in SFM 3-loops. Finally, we extend many of our results to all finite Moufang loops of class 2.

The McLaughlin sporadic simple group McL is the flag-transitive automorphism group of a Petersen-type geometry [Gscr ] = [Gscr ](McL) with the diagram

diagram here

where the edge in the middle indicates the geometry of vertices and edges of the Petersen graph. The elements corresponding to the nodes from the left to the right on the diagram P33 are called points, lines, triangles and planes, respectively. The residue in [Gscr ] of a point is the P3-geometry [Gscr ](Mat22) of the Mathieu group of degree 22 and the residue of a plane is the P3-geometry [Gscr ](Alt7) of the alternating group of degree 7. The geometries [Gscr ](Mat22) and [Gscr ](Alt7) possess 3-fold covers [Gscr ](3Mat22) and [Gscr ](3Alt7) which are known to be universal. In this paper we show that [Gscr ] is simply connected and construct a geometry [Gscr ]˜ which possesses a 2-covering onto [Gscr ]. The automorphism group of [Gscr ]˜ is of the form 323McL; the residues of a point and a plane are isomorphic to [Gscr ](3Mat22) and [Gscr ](3Alt7), respectively. Moreover, we reduce the classification problem of all flag-transitive Pmn-geometries with n, m [ges ] 3 to the calculation of the universal cover of [Gscr ]˜.

In this paper, we construct the Ext-quivers of the principal blocks of F[Sfr ]11 and F[Sfr ]12, where F is an algebraically closed field of characteristic 3. We also obtain the Loewy structures of three principal indecomposable modules of the principal block of F[Sfr ]11.

Code loops are certain Moufang loop extensions of doubly even binary codes which are useful in finite group theory (e.g. Conway's construction of the Monster). We give several methods for explicitly constructing code loops as centrally twisted products. More specifically, after establishing some preliminary examples, we show how to use decompositions of codes to build code loops out of more familiar pieces, such as abelian groups, extraspecial groups, or the octonion loop. In particular, we use Turyn's construction of the Golay code to give a simple explicit construction of the Parker loop, one which may have applications to the study of the Monster.

Let G be a connected real Lie group and let [gfr ] be its Lie algebra. We shall denote by [qfr ] ⊂ [gfr ] the radical of [gfr ]. Let [gfr ] = [qfr ] [ltimes ] [sfr ] (where [sfr ] is semisimple or 0) be a Levi decomposition of [gfr ] (cf. [11]). When [sfr ] ≠ 0 we can apply the Iwasawa decomposition on [sfr ] (cf. [8]) [sfr ] = [nfr ] [oplus ] [afr ] [oplus ] [kfr ], where [nfr ] is nilpotent and [afr ] is abelian and normalizes [nfr ] so that [nfr ] [oplus ] [afr ] is a soluble algebra. Since [nfr ] [oplus ] [afr ] normalizes [qfr ] it is clear that [rfr ] = [qfr ] [oplus ] [nfr ] [oplus ] [afr ] is a soluble Lie algebra of [gfr ]. By Lie's theorem (cf. [11]) we can find a basis on [rfr ]c = [rfr ] [otimes ] C for which the adjoint action of [rfr ] on [rfr ]c takes a triangular form. Let us denote by λ1(x); λ2(x), …, λn(x), x ∈ [rfr ] the corresponding eigenvalues. The λj's can be identified with elements of Homℝ([rfr ], C) and are called the roots of the adjoint action of [rfr ]. Let us denote by [Lscr ] = {L1, …, Lk} the set of the non zero real parts of the λj's. We shall say that the group G is a B-group if [Lscr ] ≠ &0slash; and if there exist α1, …, αk [ges ] 0, [sum ]kj=1 αj = 1, such that [sum ]kj=1 αjLj = 0. Otherwise we say that G is an NB-group. It can be shown that the above definition is independent of the particular choice of the Levi and Iwasawa decompositions that are used (cf. [13]).

We shall denote by dlg = dg (resp. drg) the left (resp. right) Haar measure on G and by m(g) = drg/dlg the modular function.

Let [Xscr ] = {X1, X2, …, Xn} be left invariant fields on G that verify the Hörmander condition (cf. [15]) and let Δ = −[sum ]X2j be the corresponding sub-Laplacian. Δ is formally self adjoint on the Hilbert space L2(G, drg) and the spectral gap of Δ is defined by

formula here

In analogy with the setting of homogeneous, locally finite trees we introduce the Radon transforms on affine, locally finite buildings with diagram of type Ãn. We study the problem of the inversion of these transforms and in some cases we show how inversion algorithms and formulas are available.

In the study of highest weight categories, the class of Weyl modules Δ(λ) and their duals ∇(λ) are of central interest; this is for example motivated by the problem of finding the characters of the simple modules. Weyl modules form the building blocks for the category [Fscr ](Δ), whose objects have a filtration 0 = M0 [les ] M1 [les ] … [les ] Mi−1 [les ] Mi = M with quotients isomorphic to Δ(λ) for various λ. Knowing Extr(Δ(λ), Δ(μ)) is essential for the understanding of this category.

In [3, 13], we determined Ext1 for Weyl modules of SL(2, k) and q-GL(2, k) over an infinite field k of characteristic p > 0. Here we are able to extend these results to determine Ext2 for Weyl modules in both these cases (see Theorem 4 · 6). Moreover, this also gives Ext2 between any pair of Weyl modules Δ(λ), Δ(μ) for q-GL(n, k) (where n [ges ] 2) such that both λ and μ have at most two rows or two columns, or where they differ by some multiple of a simple root (see Section 7).

Consider (for simplicity) polynomial representations of degree d for GL(n, k). A partition of d which has at most two rows is uniquely determined by the difference in the row lengths, which we use as a label for the partition. In this case our main result is

We show that for an arbitrary locally compact group G and for E in a certain class of closed subsets of the primitive ideal space of L1(G), the kernel k(E) has a bounded approximate unit. This generalizes some well-known previous results.

In this paper we extend the concept of the group of covering automorphisms associated to a universal covering space ϕ: U → X (where X is a connected topological manifold), to the case of left (or right) minimal approximations. In the case of torsion-free coverings of abelian groups we exhibit a topology on these groups which makes them into topological groups and we give necessary and sufficient conditions for these groups to be compact. Finally we prove that when these groups are compact they are pronilpotent (Theorem 5·3). We also characterize when these groups are torsion-free (Proposition 5·4).

A two-generator group 〈f, g〉 of Möbius maps is determined, up to conjugacy, by the numbers β = tr2(f) − 4, β′ = tr2(g) − 4 and γ = tr (fgf−1g−1) − 2, provided that γ ≠ 0. We study the subset D of C3 of those (β, β′, γ) which arise from discrete groups. In particular, we identify precisely D[setmn ]D.

Recently Calabi–Yau threefolds have been studied intensively by physicists and mathematicians. They are used as physical models of superstring theory [Y] and they are one of the building blocks in the classification of complex threefolds [KMM]. These are three dimensional analogues of K3 surfaces. However, there is a fundamental difference as is to be expected. For K3 surfaces, the moduli space N of K3 surfaces is irreducible of dimension 20, inside which a countable number of families Ng with g [ges ] 2 of algebraic K3 surfaces of dimension 19 lie as a dense subset. More explicitly, an element in Ng is (S, H), where S is a K3 surface and H is a primitive ample divisor on S with H2 = 2g − 2. For a generic (S, H), Pic (S) is generated by H, so that the rank of the Picard group of S is 1. A generic surface S in N is not algebraic and it has Pic (S) = 0, but dim N = h1(S, TS) = 20 [BPV]. It is quite an interesting problem whether or not the moduli space M of all Calabi–Yau threefolds is irreducible in some sense [R]. A Calabi–Yau threefold is algebraic if and only if it is Kaehler, while every non-algebraic K3 surface is still Kaehler. Inspired by the K3 case, we define Mh,d to be {(X, H)[mid ]H3 = h, c2(X) · H = d}, where H is a primitive ample divisor on a smooth Calabi–Yau threefold X. There are two parameters h, d for algebraic Calabi–Yau threefolds, while there is only one parameter g for algebraic K3 surfaces. (Note that c2(S) = 24 for every K3 surface.) We know that Ng is of dimension 19 for every g and is irreducible but we do not know the dimension of Mh,d and whether or not Mh,d is irreducible. In fact, the dimension of Mh,d = h1(X, TX), where (X, H) ∈ Mh,d. Furthermore, it is well known that χ(X) = 2 (rank of Pic (X) − h1(X, TX)), where χ(X) is the topological Euler characteristic of X. Calabi–Yau threefolds with Picard rank one are primitive [G] and play an important role in the moduli spaces of all Calabi–Yau threefolds. In this paper we give a bound on c3 of Calabi–Yau threefolds with Picard rank 1.

Modify the usual percolation process on the infinite binary tree by forbidding infinite clusters to grow further. The ultimate configuration will consist of both infinite and finite clusters. We give a rigorous construction of a version of this process and show that one can do explicit calculations of various quantities, for instance the law of the time (if any) that the cluster containing a fixed edge becomes infinite. Surprisingly, the distribution of the shape of a cluster which becomes infinite at time t > ½ does not depend on t; it is always distributed as the incipient infinite percolation cluster on the tree. Similarly, a typical finite cluster at each time t > ½ has the distribution of a critical percolation cluster. This elaborates an observation of Stockmayer [12].

In this paper we investigate the relationship between genus and localization of finitely generated (torsion-free abelian)-by-finite groups and genus and localization in integral representation theory. In particular when such a group has finite commutator subgroup we can determine the genus exactly.

The Kähler case of Riemannian homogeneous structures [3, 15, 18] has been studied in [1, 2, 6, 7, 13, 16], among other papers. Abbena and Garbiero [1] gave a classification of Kähler homogeneous structures, which has four primitive classes [Kscr ]1, …, [Kscr ]4 (see [6, theorem 5·1] for another proof and Section 2 below for the result). The purpose of the present paper is to prove the following result:

THEOREM 1·1. A simply connected irreducible homogeneous Kähler manifold admits a nonvanishing Kähler homogeneous structure in Abbena–Garbiero's class [Kscr ]2 [oplus ] [Kscr ]4if and only if it is the complex hyperbolic space equipped with the Bergman metric of negative constant holomorphic sectional curvature.

We use the theory of the integral closure of an ideal to study the equivalence of map-germs under C0 coordinate changes in the target. We also derive a formula for the number of double points of a map germ from Cn → C2n in terms of the Segre number of dimension 0 of an ideal associated to the double point locus of f, and the number of Whitney umbrellas of the composition of f with a generic projection to C2n−1.

Throughout this paper, let M be a finitely generated module over a Noether local ring (A, [mfr ]) with dim M = d. Let x = (x1, …, xd) be a system of parameters of M and n = (n1, …, nd) ∈ ℕd a d-tuple of positive integers. This paper is concerned with the following two points of view.

First, it is well-known that, the difference between lengths and multiplicities

formula here

considered as a function in n, gives a lot of information on the structure of M. This function in general is not a polynomial in n for all n1, …, nd large enough (n [Gt ] 0 for short). But, it was shown in [C3] that the least degree of all polynomials in n bounding above IM (n, x) is independent of the choice of x. This numerical invariant is denoted by p(M) and called the polynomial type of the module M. By [C2] and [C3] this polynomial type does not change by the [mfr ]-adic completion Mˆ of M and p(M) is just equal to the dimension of the non-Cohen–Macaulay locus of Mˆ (see [C2, C3, C4, CM] for more details). Therefore a module M is Cohen–Macaulay or generalized Cohen–Macaulay if and only if p(M) = − ∞ or p(M) [les ] 0, respectively, where we set by − ∞ the degree of the zero-polynomial. However, one knows little about the structure of M when p(M) > 0.

Second, following Sharp and Hamieh ([SH]), we consider the difference

formula here

as a function in n, where

formula here

is the cyclic submodule of the module of generalized fractions U(M)−d−1d+1M defined in [SZ1].

In order to model biological reactions, distances between knots/links based on tangle replacement are defined. Given a tangle R, an R-move is defined as the replacement of the zero tangle in a link L with the tangle R. The R-distance between the link L and the link M is defined to be the minimum number of R-moves required to change L into M where the minimum is taken over all diagrams of the link L. A formula is given to determine when one 4-plat knot/link can be obtained from another 4-plat via one R-move when R is a rational tangle and not equal to the 1/n tangle. The formula can also be used to find all rational tangles R for which the R-distance between two given 4-plats is 1 except in the case R = 1/n tangle. These results are also generalized to the case when any rational tangle P is replaced with any rational tangle R, P not necessarily the zero tangle.