1. Hermitian presentations of Chevalley groups II.
    A Chevalley group is called Hermitian if its root system is 3-graded. In this case, the roots of degree 0 are called compact and the remaining ones (those of degree 1 or -1) are called noncompact. Here, we modify the classical Steinberg presentation of a Chevalley group in a way that its generators become the symbols indexed by noncompact roots (the noncompact symbols); the new presentation is referred to as Hermitian. Either a compact symbol (that is, a symbol indexed by a compact root) or the commutator of a compact symbol and a noncompact one can be expressed as a product of noncompact symbols by Chevalley's commutator formula. Combining these expressions, one obtains a formula that displays a pair of concatenated commutators and involves noncompact symbols only. We get a presentation of the same Chevalley gr oup when we replace Chevalley's commutator formula by this new double commut ator formula and restrict the other relations to noncompact symbols. The si mply-laced case has already been treated in our paper Hermitian presentations of Chevalley groups I , J. Algebra 276 (2004) 371--382. Here we proceed with an intrinsic investigatio n of the general case. In the process, we give a detailed analysis of the struct ure constants as well as higher order constants of the Chevalley algebra associa ted with our Chevalley group. In particular, we see that there are fewer choices for the signs of the coefficients appearing in the double commutator formula th an there are in Chevalley's commutator formula; actually, we show that the forme r are in one-to-one correspondence with the choice of signs produced by the nonc ompact vectors in a Chevalley basis of the above Chevalley algebra when seen as a basis for the Lie triple system they span. In the end we give examples of Herm itian presentations for the types B_n and C_n and a review of basic properties of 3-graded root systems.
  2. The Coxeter Legacy---Reflections and Projections. (Editor, together with Chan dler Davis). American Mathematical Society and Fields Institute for Research in Mathematical Sciences; Providence, RI; Toronto, Ontario; March 2006
    Donald Coxeter infused enthusiasm, even passion, for mathematics in people of an y age, any background, any profession, any walk of life. Enchanted by Euclidean geometry, he was interested in the beauty, the description, and the exploration of the world around us. His involvement in art and with artists earned him admir ation and friends in the intellectual community all over the globe. Coxeter's de votion to polytopes and his interest in the theory of configurations live on in his students and followers. Coxeter groups arise in various subjects in applied mathematics, and they have a permanent place in some of the most demanding and f ascinating branches of abstract mathematics, such as Lie algebras, algebraic gro ups, Chevalley groups, and Kac-Moody groups. This collection of articles by outs tanding researchers and expositors is intended to capture the essence of the Cox eter Legacy. It is a mixture of surveys, up-to-date information, history, story telling, and personal memories; and it includes a rich variety of beautiful ill ustrations.
  3. Siegel transformations for even characteristic. (with O. Villa) Linear Al gebra Appl. 395 (2005) 163--174
    Let V be a vector space over a field K of even characteristic containing more th an 2 elements. Suppose K is perfect and pi is an element in the special o rthogonal group SO(V) with path dimension B(pi) equal to 2d. Then pi is a product of d-1 Siegel transformations and one transformation kappa in SO(V) with path dimension B(kappa) equal to 2. The length of pi with respect to the Siegel transformations is d if pi is unipotent or i f the dimension of the quotient group B(pi) over its radical is greater o r equal to 4; in all other cases it is d+1.
  4. Hermitian presentations of Chevalley groups I.
    We give a presentation for a Chevalley group arising from a Hermitian Lie algebra whose roots have all the same length. This is a variant of Steinberg's presentation of a general Chevalley group, using only noncompact roots.
  5. Conjugacy classes of involutions in the Lorentz group Omega(V) and in SO(V).
    The Lorentz group Omega(V) is bireflectional and all involutions in Omega(V) are conjugate. More generally, we give conditions for two involutions to be conjugate in SO(V), provided that V is a vector space over a finite field or over an ordered field.
  6. The special orthogonal group is trireflectional.
    Let K be a field of even characteristic, V a finite-dimensional vector space over K, and SO(V) the special orthogonal group. Then SO(V) is trireflectional, provided dim(V)>2 and SO(V) is distinct from O+(4,2).
  7. Intersection of conjugacy classes with Bruhat cells in Chevalley groups.
    Let G be a simple and simply-connected algebraic group that is defined and quasi-split over a field K. We investigate properties of intersections of Bruhat cells of G with conjugacy classes C of G, in particular, we consider the question, when is such an intersection not empty.
  8. Coxetergruppen - ein Beispiel.
  9. Products of involutions in the finite Chevalley groups of type F4(K).
    Let K be a finite field of odd characteristic and let G be a Chevalley group of type F4(K). We find sufficient conditions for an element in G to be a product of two or three involutions.
  10. Products of transvections in one conjugacy class of the symplectic group over the p-adic numbers.
    Every element in the symplectic group over the field of p-adic numbers (p>3) is a product of transvections in a single conjugacy class. We determine the minimal number of factors needed in any such product for transformations with path dimensions 1, 2, and 3. For indecomposable symplectic transformations with path dimensions 4, 5, and 6 we find upper bounds for the minimal number of factors. Results of Knüppel can now be applied to obtain similar upper bounds for transformations with higher path dimensions.
  11. Products of involutions in simple Chevalley groups
    Let G be a Chevalley group defined over a field K. If K contains enough elements, then every element in G is a product of five or fewer involutions. The subgroup N of G is generated by involutions provided G is not of type Cr or B2.
  12. Gauss decomposition with prescribed semisimple part: short proof
    We give a uniform short proof of the fact that the intersection of every noncentral conjugacy class in a Chevalley group and a big Gauss cell is nonempty and that this intersection contains elements with any prescribed semisimple part. The interest in this property stems at least in part from its relation to Ore's and Thompson's conjectures in the theory of finite groups.
  13. Bireflectionality of orthogonal and symplectic groups of characteristic 2
    Let V be a finite dimensional vector space over a field K of characteristic 2. Let O(V) be the orthogonal group defined by a nondegenerate quadratic form. Then every element in O(V) is a product of two elements of order 2, unless all nonsingular subspaces of V are at most 2-dimensional. If V is a nonsingular symplectic space, then every element in the symplectic group Sp(V) is a product of two elements of order 2, except if dim V=2 and |K|=2.
  14. Intersection of conjugacy classes of Chevalley groups with Gauss cells
    Let G be a proper Chevalley group or a finite twisted Chevalley group. We give some description of the intersections of noncentral conjugacy classes of G with certain Gauss cells, which we call Coxeter cells. This generalizes a previous result of the authors. It is also the basis of yet another generalization of the same result involving weight functions on conjugacy classes.
  15. Covering numbers for Chevalley groups
    Let G be a quasisimple Chevalley group. We give an upper bound for the covering number cn(G) which is linear in the rank of G, i. e., we give a constant d such that for every noncentral conjugacy class C of G we have Crd=G, where r = rankG.
  16. A generalization of Sourour's theorem
    Let X be an invertible n by n matrix, n > 1, with entries in some field K. Assume X is not equal to diag (a,...,a) for any a in K. Then for every sequence (a1,..., an-1), where ai in K, there is a matrix Y with det Y=1 such that the n-1 principal minors of YXY-1 have the values a1,...,an-1 respectively.
  17. On the conjectures of J. Thompson and O. Ore
    If G is a finite simple group of Lie type over a field containing more than 8 elements (for twisted groups lXn(ql) we require q > 8, except for 2B2(q2), 2G2(q2), and 2F4(q2), where we assume q2 > 8), then G is the square of some conjugacy class and consequently every element in G is a commutator.
  18. Groups satisfying Scherk's length theorem
    We consider subgroups G of the general linear group GL(n,K), where char K is distinct from 2. If G is generated by the set S of its simple involutions, if -1V is an element in G, and if Scherk's length theorem holds for G, then G is a subgroup of an orthogonal group.

Updated Jan 2006

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