Symmetric Presentations for the Fischer Groups II: The Sporadic Groups

We give symmetric presentations for groups closely related to the three sporadic Fischer groups Fi₂₂, Fi₂₃ and Fi₂₄.Mathematics Subject Classiffications (2000). 20D08, 20D06, 20F05This research was supported by an EPSRC grant ref. GR/N27491/01     

Fischer subspaces of Hermitian manifolds

The finite groups generated by 3-transpositions, studied by B.Fischer [5]have a geometrical interpretation given by F.Buekenhout [1] under the name of Fischer spaces.This geometrical concept allows us to study the Fischer subgroups of primitive groups classified by Fischer, and in particular those of unitary groups PSU(n,4), symplectic groups PSp(n,2), orthogonal groups PO(n,2) and symmetric groups Sym(n).This problem is linked to the determination of groups of projectivities generated by clations in characteristic 2 studied by Wagner [9] and McLaughlin[8], and is related to Kantor's work [7] on classical groups generated by long root elements, and to Enright's recent note on Fi₂₂ and Fi₂₃ [5].     

The 2-modular characters of the Fischer group

We determine the 2-modular character table of the Fischer group Fi₂₃.     

More on Geometries of the Fischer Group Fi 22

We give a new, purely combinatorial characterization of geometries with diagram identifying each under some natural conditions—but not assuming any group action a priori—with one of the two geometries and related to the Fischer 3-transposition group Fi ₂₂ and its non-split central extension 3 · Fi ₂₂, respectively. As a by-product we improve the known characterization of the c-extended dual polar spaces for Fi ₂₂ and 3 · Fi ₂₂ and of the truncation of the c-extended 6-dimensional unitary polar space.     

A design and a geometry for the group Fi 22

The Fischer group Fi ₂₂ acts as a rank 3 group of automorphisms of a symmetric 2-(14080,1444,148) design. This design does not have a doubly transitive automorphism group, since there is a partial linear space with lines of size 4 defined combinatorially from the design and preserved by its automorphism group. We investigate this geometry and determine the structure of various subspaces of it.      

Présentations des groupes de Fischer (II)

In this paper we give Coxeter presentation (X, ) for the three Fischer groupsG=Fi₂₂, Fi₂₃, Fi₂₄; we apply methods exposed in the first part. Each of these groups is generated by a class of 3-transpositions (named here a Fischer class) in which elements ofX are chosen. A subset of is the set of all the relations (xy) ^m(x,y)=1, wherex andy are inX and wherem(x,y) means the order ofxy inG. We obtainG as a specified quotient of the Coxeter group (X, ) with the appropriate diagram .     

Irreducible characters of the group S n that are semiproportional on A n

Previously, we dubbed the conjecture that the alternating group A_n has no semiproportional irreducible characters for any natural n [1]. This conjecture was then shown to be equivalent to the following [3]. Let α and β be partitions of a number n such that their corresponding characters χ^α and χ^β in the group S_n are semiproportional on A_n. Then one of the partitions α or β is self-associated. Here, we describe all pairs (α, β) of partitions satisfying the hypothesis and the conclusion of the latter conjecture. Supported by RFBR (grant No. 07-01-00148) and by RFBR-NSFC (grant No. 05-01-39000). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 135–156, March–April, 2008.     

nX-complementary generations of the Fischer group Fi 23

Let G be a finite group andnX a conjugacy class of elements of order n inG. G is callednX—complementary generated if, for everyx ∈ G?{1}, there is ay ∈nX such thatG = 〈hx,y〉. In [27] the question of finding all positive integersn such that a given non-abelian finite simple groupG isnX—-complementary generated was posed. In this paper we answer this question for the sporadic groupFi ₂₃. In fact, we prove that the Fischer groupFi ₂₃ isnX-complementary generated if and only ifn < 12 orn ∈ {7, 8, 10, 11} ornX ∈ {6N, 6O, 9D, 9E, 12C, 12D, …,12O}.     

Exceptional Extended Dual Polar Spaces

We address the classification problem of flag-transitive geometries with diagrams of the form where the leftmost edge symbolizes the geometry of vertices and edges of a complete graph ons + 2 vertices and the residue of an element of the leftmost type is a finite thick classical dual polar space. These geometries are known asextended dual polar spaces. An extended dual polar space is called affine if it possesses a flag-transitive automorphism group which contains a normal subgroup acting regularly on the set of elements of the leftmost type. For a dual polar space D with three points per line there exists a unique 2-simply connected affine extension A(D) of D. We show that a flag-transitive extended dual polar space is either a quotient of A(D) for some D or isomorphic to one of 19 exceptional geometries whose full automorphism groups are isomorphic respectively to Sym₈,U₄(2).2,Sp₆(2) × 2,Sp₆(2), 3 · U4(3).2²,U₄(3).2²,U₅(2).2,McL.2,HS.2,Suz.2,Sp₈(2), 3 · Fi₂₂.2,Fi₂₂.2,Co₂ × 2,Co₂,Fi₂₄(s = 4,t = 2),Fi₂₄(s = t = 3),F₁andFi₂₃.     

Cap representations of G2 and the spin L-function of PGSp6

We give a construction of a family of CAP representations of the exceptional group G ₂, whose existence is predicted by Arthur’s conjecture. These are constructed by lifting certain cuspidal representations of PGS _p6. To show that the lifting is non-zero, we establish a Rankin-Selberg integral for the degree 8 Spin L-function of these cuspidal representations of PGS _p6.     

K2 and K3 of the circle

Let k be $\text{Z}\text{[}\text{1}\text{2}\text{]}$, $\text{Q}$ or $\text{R}$, and set $\text{A =}\text{k[x,y]}\text{(x}{}^2\text{+ y}{}^2\text{? 1)}$. We compute K₂(A) and K₃(A). Our method is to construct a map $\text{? : K}{}_1\text{(k[i])→K}{}_{\mathrm{1\; +\; 1}}\text{(A)}$ and compare this to a localization sequence.We give three applications. We show that ? accounts for the primitive elements in K₂(A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec(A). Third, we can iterate the construction to compute the K-theory of the torus ring A ?_kA.     

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     

Some group actions on K(x 1 , x 2 , x 3 )

LetK be any field which may not be algebraically closed,K(x ₁ ,x ₂ ,x ₃ ) be the rational function field of three variables overK, and σ:K(x ₁ ,x ₂ ,x ₃ ) → K(x ₁ ,x ₂ ,x ₃ ) be aK-automorphism defined by wherea _i ,b _i ,c _i ,d _i ∈K anda _i d _i −b _i c _i ≠0. Let ,f _i (T)=T ² −(a _i +d _i )T+(a _i d _i −b _i c _i )∈K[T] be the “characteristic polynomial” of σ _i . Theorem:Assume that charK≠2.Then the fixed field K(x ₁ ,x ₂ ,x ₃ ) ^<σ> is not rational (=purely transcendental) over K if and only if (i) for each 1≤i≤3, f _i (T) is irreducible; (ii) the Galois group of f ₁ (T)f ₂ (T)f ₃ (T) over K is of order 8; and (iii) for each 1≤i≤3,ord (σ _[itn] )is an even integer.     

Units in totally complex S3 fields

Let B be a totally complex number field, Galois over the rational field Q, with Galois group S₃, the symmetric group on three elements. The group of units of B has torsion free rank 2. In this paper, we determine the various inequivalent representations that occur of S₃ acting on the group of units and determine arithmetic criteria for deciding which representation occurs for a particular field. As a result, we can give a relatively simple computational procedure for determining a pair of fundamental units of B given a fundamental unit in a cubic subfield.     

Isotonic Approximation in L1

Let (Ω, ,P) be a measurable space and a sub-σ-lattice of the σ-algebra . For XL₁(Ω, ,P) we denote by P X the set of conditional 1-mean (or best approximants) of X given L₁( ) (the set of all -measurable and integrable functions). In this paper, we obtain characterizations of the elements in P X, similar to those obtained by Landers and Rogge for conditional s-means with 1<s<∞. Moreover, using these characterizations we can extend the operator P to a bigger space L₀(Ω, ,P). When, in certain sense, _n goes to _∞, we will be able to prove theorems about convergence and we will obtain bounds for the maximal function sup_nP _nX. A sharper characterization of conditional 1-means for certain particular σ-lattice was proved in previous papers. In the last section of this paper we generalize those results to all totally ordered σ-lattices.     

On the sum ∑k=13 | 2πk − αk |, αk algebraic numbers

If α₁, α₂, α₃ are algebraic numbers satisfying (i) the height of α₁, α₂, α₃ do not exceed H (ii) the degree of the field generated by α₁, α₂, α₃ over the field of rational numbers do not exceed D, then a positive lower bound for

$\text{∑}\text{k=1}\text{3}\text{|2π}{}^{\mathrm{k}}\text{?αk|}$

is determined explicitly (except for an absolute constant) in terms of D and H.     

When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

We discuss subsetsS of ℝⁿ such that every real valued functionf onS is of the formf(x₁, x₂, ..., x_n) =u ₁(x₁) +u ₂(x₂) +...+u _n(x_n), and the related concepts and situations in analysis.     

Quasirecognition by the set of element orders of the groups E 6( q ) and 2 E 6( q )

We prove that if L is one of the simple groups E ₆(q) and ² E ₆(q) and G is some finite group with the same spectrum as L, then the commutant of G/F(G) is isomorphic to L and the quotient G/G′ is a cyclic {2,3}-group. Original Russian Text Copyright ? 2007 Kondrat’ev A. S. The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00463) and the RFBR-NSFC (Grant 05-01-39000). __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 6, pp. 1250–1271, November–December, 2007.     

Differential-geometric Newton method for the best rank-(R1, R2, R3) approximation of tensors

An increasing number of applications are based on the manipulation of higher-order tensors. In this paper, we derive a differential-geometric Newton method for computing the best rank-(R ₁, R ₂, R ₃) approximation of a third-order tensor. The generalization to tensors of order higher than three is straightforward. We illustrate the fast quadratic convergence of the algorithm in a neighborhood of the solution and compare it with the known higher-order orthogonal iteration (De Lathauwer et al., SIAM J Matrix Anal Appl 21(4):1324–1342, 2000). This kind of algorithms are useful for many problems. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. Research supported by: (1) Research Council K.U.Leuven: GOA-Ambiorics, CoE EF/05/006 Optimization in Engineering (OPTEC), (2) F.W.O.: (a) project G.0321.06, (b) Research Communities ICCoS, ANMMM and MLDM, (3) the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, “Dynamical systems, control and optimization”, 2007–2011), (4) EU: ERNSI. M. Ishteva is supported by a K.U.Leuven doctoral scholarship (OE/06/25, OE/07/17, OE/08/007), L. De Lathauwer is supported by “Impulsfinanciering Campus Kortrijk (2007–2012)(CIF1)” and STRT1/08/023.