Proceedings of the London Mathematical Society - Current Issue

Volume 93 Issue 03

Wed, 01 Nov 2006 00:00:00 GMT

Proceedings of the London Mathematical Society, Volume 93 Issue 03

An eminent international mathematics journal, the Proceedings of the London Mathematical Society has been published since 1865. In 2005, the London Mathematical Society is pleased to announce, the introduction of an open access policy, in which the most recent two issues will be freely available online as part of a one year experiment. Papers from the Proceedings, cover a wide range of mathematical topics that include real and complex analysis, differential equations and related areas, topology, geometry, logic, probability, statistics, algebra, number theory and combinatorial theory. ***Latest three issues now available FREE ONLINE!***

SPARSE PARTITION REGULARITY

Research Articles
IMRE LEADER, PAUL A. RUSSELL,
Proceedings of the London Mathematical Society, Volume 93 Issue 03 , pp 545-569

Abstract
Our aim in this paper is to prove Deuber s conjecture on sparse partition regularity, that for every $m$, $p$ and $c$ there exists a subset of the natural numbers whose $(m,p,c)$-sets have high girth and chromatic number. More precisely, we show that for any $m$, $p$, $c$, $k$ and $g$ there is a subset $S$ of the natural numbers that is sufficiently rich in $(m,p,c)$-sets that whenever $S$ is $k$-coloured there is a monochromatic $(m,p,c)$-set, yet is so sparse that its $(m,p,c)$-sets do not form any cycles of length less than $g$.Our main tools are some extensions of Ne et il R dl amalgamation and a Ramsey theorem of Bergelson, Hindman and Leader. As a sideline, we obtain a Ramsey theorem for products of trees that may be of independent interest.

FROBENIUS SPLITTING OF EQUIVARIANT CLOSURES OF REGULAR CONJUGACY CLASSES

Research Articles
JESPER FUNCH THOMSEN,
Proceedings of the London Mathematical Society, Volume 93 Issue 03 , pp 570-592

Abstract
Let $G$ denote a connected semisimple and simply connected algebraic group over an algebraically closed field $k$ of positive characteristic and let $g$ denote a regular element of $G$. Let $X$ denote any equivariant embedding of $G$. We prove that the closure of the conjugacy class of $g$ within $X$ is normal and Cohen Macaulay. Moreover, when $X$ is smooth we prove that this closure is a local complete intersection. As a consequence, the closure of the unipotent variety within $X$ shares the same geometric properties.

DETECTING $K$-THEORY BY CYCLIC HOMOLOGY

Research Articles
WOLFGANG LÜCK, HOLGER REICH,
Proceedings of the London Mathematical Society, Volume 93 Issue 03 , pp 593-634

Abstract
We discuss which part of the rationalized algebraic $K$-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology.

KAZHDAN–LUSZTIG CELLS AND THE MURPHY BASIS

Research Articles
MEINOLF GECK,
Proceedings of the London Mathematical Society, Volume 93 Issue 03 , pp 635-665

Abstract
Let $H$ be the Iwahori Hecke algebra associated with $S_n$, the symmetric group on $n$ symbols. This algebra has two important bases: the Kazhdan Lusztig basis and the Murphy basis. We establish a precise connection between the two bases, allowing us to give, for the first time, purely algebraic proofs for a number of fundamental properties of the Kazhdan Lusztig basis and Lusztig s results on the $a$-function.

INTRANSITIVE GEOMETRIES

Research Articles
RALF GRAMLICH, HENDRIK VAN MALDEGHEM,
Proceedings of the London Mathematical Society, Volume 93 Issue 03 , pp 666-692

Abstract
A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion of an amalgam induced by a sufficiently transitive group of automorphisms of that geometry. In the present paper, we generalize this lemma to intransitive geometries, thus opening the door for numerous applications. We treat ourselves some amalgams related to intransitive actions of finite orthogonal groups, as a first class of examples.

AMENABILITY, FREE SUBGROUPS, AND HAAR NULL SETS IN NON-LOCALLY COMPACT GROUPS

Research Articles
SLAWOMIR SOLECKI,
Proceedings of the London Mathematical Society, Volume 93 Issue 03 , pp 693-722

Abstract
The paper has two objectives. On the one hand, we study left Haar null sets, a measure-theoretic notion of smallness on Polish, not necessarily locally compact, groups. On the other hand, we introduce and investigate two classes of Polish groups which are closely related to this notion and to amenability. We show that left Haar null sets form a $\sigma$-ideal and have the Steinhaus property on Polish groups which are amenable at the identity , and that they lose these two properties in the presence of appropriately embedded free subgroups. As an application we prove an automatic continuity result for universally measurable homomorphisms from inverse limits of sequences of amenable, locally compact, second countable groups to second countable groups.

ON A GENERALIZATION OF SZEMERÉDI'S THEOREM

Research Articles
I. D. SHKREDOV,
Proceedings of the London Mathematical Society, Volume 93 Issue 03 , pp 723-760

Abstract
Let $N$ be a natural number and $A \subseteq [1, \dots, N]^2$ be a set of cardinality at least $N^2 / (\log \log N)^c$, where $c > 0$ is an absolute constant. We prove that $A$ contains a triple $\{(k, m), (k+d, m), (k, m+d) \}$, where $d > 0$. This theorem is a two-dimensional generalization of Szemer di s theorem on arithmetic progressions.

CYCLES AND 1-UNCONDITIONAL MATRICES

Research Articles
STEFAN NEUWIRTH,
Proceedings of the London Mathematical Society, Volume 93 Issue 03 , pp 761-790

Abstract
We characterise the 1-unconditional subsets $(\mathrm{e}_{rc})_{(r,c) \in I}$ of the set of elementary matrices in the Schatten von-Neumann class $\mathrm{S}^p$. The set of couples $I$ must be the set of edges of a bipartite graph without cycles of even length $4 \lel \le p$ if $p$ is an even integer, and without cycles at all if $p$ is a positive real number that is not an even integer. In the latter case, $I$ is even a Varopoulos set of V-interpolation of constant 1. We also study the metric unconditional approximation property for the space $\mathrm{S}^p_I$ spanned by $(\mathrm{e}_{rc})_{(r,c) \in I}$ in $\mathrm{S}^p$.

$G$-STRUCTURES ON SPHERES

Research Articles
MARTIN CADEK, MICHAEL CRABB,
Proceedings of the London Mathematical Society, Volume 93 Issue 03 , pp 791-816

Abstract
A generalization of classical theorems on the existence of sections of real, complex and quaternionic Stiefel manifolds over spheres is proved. We obtain a complete list of Lie group homomorphisms $\rho : G \to G_n$, where $G_n$ is one of the groups $SO(n)$, $SU(n)$ or $Sp(n)$ and $G$ is one of the groups $SO(k)$, $SU(k)$ or $Sp(k)$, which reduce the structure group $G_n$ in the fibre bundle $G_n \to G_{n + 1} \to G_{n + 1} / G_n$.