Semisimple infinitesimal q-Schur algebras

In this paper, we shall classify the semisimple infinitesimal q-Schur algebras. Received: 2 May 2007, Revised: 27 September 2007     

Universal characters from the MacDonald identities

The main result of [Kostant, Invent. Math. 158 (1) (2004) 181-226, arXiv:math.GR/0309232] shows that a sequence of complexes associated with the MacDonald identities gives a sequence of universal characters which extends and generalises some representations of the exceptional series given in [Deligne, C. R. Acad. Sci. Paris Sér. Math. 322(4) (1996) 321-326]. This proves the conjectures in [Macfarlane, Pfeiffer, J. Phys. A 36 (2003) 230 5-2317. arxiv.math-ph/0208014].In this paper, we use a twisted version of this definition to define a sequence of complexes for the first row of the Freudenthal magic square.We also calculate the dimensions of these characters for the special linear groups and for the symplectic groups. This gives two new explicit expressions for the D’Arcais polynomials.     

Variations on seven points: An introduction to the scope and methods of coding theory and finite geometries

We use a simple example (the projective plane on seven points) to give an introductory survey on the problems and methods in finite geometries — an area of mathematics related to geometry, combinatorial theory, algebra, group theory and number theory as well as to applied mathematics (e.g., coding theory, information theory, statistical design of experiments, tomography, cryptography, etc.). As this list already indicates, finite geometries is — both from the point of view of pure mathematics and from that of applications related to computer science and communication engineering — one of the most interesting and active fields of mathematics. It is the aim of this paper to introduce the nonspecialist to some of these aspects.To Professor Günter Pickert on the occasion of his 65th birthday     

A generalization of group difference sets and the matrix equationA m =dI+λJ

LetG be a finite group. In this paper, we will study the-group matrices forG which satisfy the matrixA ^m =dI+J and we will show that the existence of such a solution is equivalent to the existence of a combinatorial structure inG which is a generalization of group difference sets.     

Second cohomology groups for Frobenius kernels and related structures

Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U) with coefficients in k.     

Roots,iterations and logarithms of formal automorphisms

In this paper it is proved that having a logarithm is equivalent to having roots of arbitrary order in the group of automorphisms of a formal power series ring, and in algebraic subgroups too     

Roots,iterations and logarithms of formal automorphisms

In this paper it is proved that having a logarithm is equivalent to having roots of arbitrary order in the group of automorphisms of a formal power series ring, and in algebraic subgroups too     

Graph factorization,general triple systems,and cyclic triple systems

In this self-contained exposition, results are developed concerning one-factorizations of complete graphs, and incidence matrices are used to turn these factorization results into embedding theorems on Steiner triple systems. The result is a constructive graphical proof that a Steiner triple system exists for any order congruent to 1 or 3 modulo 6. A pairing construction is then introduced to show that one can also obtain triple systems which are cyclically generated.     

On the Radon-Nikodym property for operator valued measures

A theorem about the Radon-Nikodym property and the convergence of bounded martingales is proved for a bilinear integral in locally convex spaces.     

All triples for orthogonal designs of order 40

The use of amicable sets of eight circulant matrices and four negacyclic matrices to generate orthogonal designs is relatively new. We find all 1841 possible orthogonal designs of order 40 in three variables, using only these new techniques.     

Representations and cohomology for Frobenius-Lusztig kernels

Let U_ζ be the quantum group (Lusztig form) associated to the simple Lie algebra g, with parameter ζ specialized to an ?-th root of unity in a field of characteristic p>0. In this paper we study certain finite-dimensional normal Hopf subalgebras U_ζ(G_r) of U_ζ, called Frobenius-Lusztig kernels, which generalize the Frobenius kernels G_r of an algebraic group G. When r=0, the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible U_ζ(G_r)-modules and discuss their characters. We then study the cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of G. We prove that the cohomology ring for the first Frobenius-Lusztig kernel is finitely-generated when g has type A or D, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.     

A survey of pseudo (v,k, λ)-designs

This work is an attempt to give a complete survey of all known results about pseudo (v, k, )-designs. In doing this, the author hopes to bring more attention to his conjecture given in Section 6; an affirmative answer to this conjecture would settle completely the existence and construction problem for a pseudo (v, k, )-design in terms of the existence of an appropriate (v, k, )-design.     

Factoring operators over Hilbert-Schmidt maps and vector measures

We study the structure of Banach spaces X determined by the coincidence of nuclear maps on X with certain operator ideals involving absolutely summing maps and their relatives. With the emphasis mainly on Hilbert-space valued mappings, it is shown that the class of Hilbert—Schmidt spaces arises as a ‘solution set’ of the equation involving nuclear maps and the ideal of operators factoring through Hilbert—Schmidt maps. Among other results of this type, it is also shown that Hilbert spaces can be characterised by the equality of this latter ideal with the ideal of 2-summing maps. We shall also make use of this occasion to give an alternative proof of a famous theorem of Grothendieck using some well-known results from vector measure theory.     

Further examples of maximal curves

It is shown that the curve over F_q²ⁿ with n≥3 odd, that generalizes Serre’s curve y⁴+y=x³ over F₆₄, is also maximal. We also investigate a family of maximal curves over F_q²ⁿ and provide isomorphisms between these curves.