Constructive recognition of

Existing black box and other algorithms for explicitly recognising groups of Lie type over have asymptotic running times which are polynomial in , whereas the input size involves only . This has represented a serious obstruction to the efficient recognition of such groups. Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises explicitly.

The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising in its natural representation in polynomial time, given a discrete logarithm oracle for . The algorithm presented here takes as input a generating set for a subgroup of that is isomorphic modulo scalars to , where is a finite field of the same characteristic as ; it returns the natural representation of modulo scalars. Since a faithful projective representation of in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in rather than in , elementary algorithms will recognise explicitly in polynomial time in these cases. Given a discrete logarithm oracle for , our algorithm thus provides the required polynomial time oracle for recognising explicitly in the remaining case, namely for representations in the natural characteristic.

This leads to a partial solution of a question posed by Babai and Shalev: if is a matrix group in characteristic , determine in polynomial time whether or not is trivial.

     

Associahedra, cellular -construction and products of -algebras

The aim of this paper is to construct a functorial tensor product of -algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These constructions in fact already appeared (Saneblidze and Umble, 2000 and 2002); we will try to give a more conceptual presentation. We also prove that there does not exist a coassociative diagonal.

     

On the -compact groups corresponding to the -adic reflection groups

There exists an infinite family of -compact groups whose Weyl groups correspond to the finite -adic pseudoreflection groups of family 2a in the Clark-Ewing list. In this paper we study these -compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical -homomorphism. Finally, we also describe a faithful complexification homomorphism from these -compact groups to the -completion of unitary compact Lie groups.

     

Invariance in and

We define , a substructure of (the lattice of classes), and show that a quotient structure of , , is isomorphic to . The result builds on the isomorphism machinery, and allows us to transfer invariant classes from to , though not, in general, orbits. Further properties of and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.

     

The fourth power moment of automorphic -functions for over a short interval

In this paper we will prove bounds for the fourth power moment in the aspect over a short interval of automorphic -functions for on the central critical line Re. Here is a fixed holomorphic or Maass Hecke eigenform for the modular group , or in certain cases, for the Hecke congruence subgroup with . The short interval is from a large to . The proof is based on an estimate in the proof of subconvexity bounds for Rankin-Selberg -function for Maass forms by Jianya Liu and Yangbo Ye (2002) and Yuk-Kam Lau, Jianya Liu, and Yangbo Ye (2004), which in turn relies on the Kuznetsov formula (1981) and bounds for shifted convolution sums of Fourier coefficients of a cusp form proved by Sarnak (2001) and by Lau, Liu, and Ye (2004).

     

Crystals of type and Young walls

We give a new realization of arbitrary level perfect crystals and arbitrary level irreducible highest weight crystals of type , in the language of Young walls. We refine the notions of splitting of blocks and slices that have appeared in our previous works, and these play crucial roles in the construction of crystals. The perfect crystals are realized as the set of equivalence classes of slices, and the irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls which, in turn, are concatenations of slices.

     

Non-Moishezon twistor spaces of with non-trivial automorphism group

We show that a twistor space of a self-dual metric on with -isometry is not Moishezon iff there is a -orbit biholomorphic to a smooth elliptic curve, where the -action is the complexification of the -action on the twistor space. It follows that the -isometry has a two-sphere whose isotropy group is . We also prove the existence of such twistor spaces in a strong form to show that a problem of Campana and Kreußler is affirmative even though a twistor space is required to have a non-trivial automorphism group.

     

Constant mean curvature surfaces in

The subject of this paper is properly embedded surfaces in Riemannian three manifolds of the form , where is a complete Riemannian surface. When , we are in the classical domain of surfaces in . In general, we will make some assumptions about in order to prove stronger results, or to show the effects of curvature bounds in on the behavior of surfaces in .

     

A version of Bourgain's theorem

Let satisfy We construct an orthonormal basis for such that and are both uniformly bounded in . Here . This generalizes a theorem of Bourgain and is closely related to recent results on the Balian-Low theorem.

     

Exponents for -stable ideals

Let be a simple algebraic group over the complex numbers containing a Borel subgroup . Given a -stable ideal in the nilradical of the Lie algebra of , we define natural numbers which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types and some other types.

When , we recover the usual exponents of by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.

     

The invariant factors of the incidence matrices of points and subspaces in and

We determine the Smith normal forms of the incidence matrices of points and projective -dimensional subspaces of and of the incidence matrices of points and -dimensional affine subspaces of for all , , and arbitrary prime power .

     

The -theory of is undecidable

The three quantifier theory of , the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of that lies between the two and three quantifier theories with but includes function symbols.

Theorem. The two quantifier theory of , the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on ) is undecidable.

The same result holds for various lattices of ideals of which are natural extensions of preserving join and infimum when it exits.

     

Quivers with relations arising from clusters case)

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let be a cluster algebra of type . We associate to each cluster of an abelian category such that the indecomposable objects of are in natural correspondence with the cluster variables of which are not in . We give an algebraic realization and a geometric realization of . Then, we generalize the ``denominator theorem' of Fomin and Zelevinsky to any cluster.

     

On the mod cohomology of

We study the mod  cohomology of the classifying space of the projective unitary group . We first prove that conjectures due to J.F. Adams and Kono and Yagita (1993) about the structure of the mod  cohomology of the classifying space of connected compact Lie groups hold in the case of . Finally, we prove that the classifying space of the projective unitary group is determined by its mod  cohomology as an unstable algebra over the Steenrod algebra for 3$">, completing previous work by Dwyer, Miller and Wilkerson (1992) and Broto and Viruel (1998) for the cases .

     

On and Gevrey regularity of sublaplacians

In this paper we consider zero order perturbations of a class of sublaplacians on the two-dimensional torus and give sufficient conditions for global regularity to persist. In the case of analytic coefficients, we prove Gevrey regularity for a general class of sublaplacians when the finite type condition holds.

     

Homomorphisms between Weyl modules for

We classify all homomorphisms between Weyl modules for when is an algebraically closed field of characteristic at least three, and show that the -spaces are all at most one dimensional. As a corollary we obtain all homomorphisms between Specht modules for the symmetric group when the labelling partitions have at most three parts and the prime is at least three. We conclude by showing how a result of Fayers and Lyle on Hom-spaces for Specht modules is related to earlier work of Donkin for algebraic groups.

     

Square -lattice paths and

The combinatorial -Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The -Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the 'th -Catalan number is the Hilbert series for the module of diagonal harmonic alternants in variables; it is also the coefficient of in the Schur expansion of . Using -analogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of and the Hilbert series of the diagonal harmonics modules.

This article extends the combinatorial constructions of Haglund et al. to the case of lattice paths contained in squares. We define and study several -analogues of these lattice paths, proving combinatorial facts that closely parallel corresponding results for the -Catalan polynomials. We also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator. In particular, we conjecture combinatorial formulas for the monomial expansion of , the ``Hilbert series' , and the sign character .