On the number of conjugacy class sizes and character degrees in finite -groups

In this note we prove that for any two integers 1$"> there exist finite -groups of class such that and .

     

Low-dimensional unitary representations of

We characterize all simple unitarizable representations of the braid group on complex vector spaces of dimension . In particular, we prove that if and denote the two generating twists of , then a simple representation (for ) is unitarizable if and only if the eigenvalues of are distinct, satisfy and 0$"> for , where the are functions of the eigenvalues, explicitly described in this paper.

     

Extending Baire Property by countably many sets

We prove that if ZFC is consistent so is ZFC + ``for any sequence of subsets of a Polish space there exists a separable metrizable topology on with , and Borel in for all .' This is a category analogue of a theorem of Carlson on the possibility of extending Lebesgue measure to any countable collection of sets. A uniform argument is presented, which gives a new proof of the latter as well.

Some consequences of these extension properties are also studied.

     

A remark on a theorem of J. Tits

Let be a rank two Chevalley group and be the corresponding Moufang polygon. J. Tits proved that is the universal completion of the amalgam formed by three subgroups of : the stabilizer of a point of , the stabilizer of a line incident with , and the stabilizer of an apartment passing through and . We prove a slightly stronger result, in which the exact structure of is not required. Our result can be used in conjunction with the ``weak -pair" theorem of Delgado and Stellmacher in order to identify subgroups of finite groups generated by minimal parabolics.

     

Stable rank of corner rings

B. Blackadar recently proved that any full corner in a unital C*-algebra has K-theoretic stable rank greater than or equal to the stable rank of . (Here is a projection in , and fullness means that .) This result is extended to arbitrary (unital) rings in the present paper: If is a full idempotent in , then . The proofs rely partly on algebraic analogs of Blackadar's methods and partly on a new technique for reducing problems of higher stable rank to a concept of stable rank one for skew (rectangular) corners . The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if where is a finitely generated projective generator, and can be generated by elements, then .

     

Mod Galois representations of solvable image

It is proved that, for a number field and a prime number , there exist only finitely many isomorphism classes of continuous semisimple Galois representations of into of fixed dimension and bounded Artin conductor outside which have solvable images. Some auxiliary results are also proved.

     

Characterization of completions of reduced local rings

We find necessary and sufficient conditions for a complete local ring to be the completion of a reduced local ring. Explicitly, these conditions on a complete local ring with maximal ideal are (i) or , and (ii) for all , if is an integer of , then .

     

The groups of quasiconformal homeomorphisms on Riemann surfaces

Suppose is a connected Riemann surface. Let denote the homeomorphism group of with the compact-open topology, and denote the subgroup of quasiconformal mappings of onto itself, and let and denote the identity components of and respectively. In this paper we show that the pair is an -manifold, and determine their topological types.

     

On the interpolation constant for Orlicz spaces

In this paper we deal with the interpolation from Lebesgue spaces and , into an Orlicz space , where and for some concave function , with special attention to the interpolation constant . For a bounded linear operator in and , we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,

where the interpolation constant depends only on and . We give estimates for , which imply . Moreover, if either or , then . If , then , and, in particular, for the case this gives the classical Orlicz interpolation theorem with the constant .

     

Character degree graphs that are complete graphs

Let be a finite group, and write for the set of degrees of irreducible characters of . We define to be the graph whose vertex set is , and there is an edge between and if . We prove that if is a complete graph, then is a solvable group.

     

Fuchs' problem 34 for mixed Abelian groups

This paper investigates the extent to which an Abelian group is determined by the homomorphism groups . A class of Abelian groups is a Fuchs 34 class if and in are isomorphic if and only if for all . Two -groups and satisfy for all groups if and only if they have the same -Ulm-Kaplansky-invariants and the same final rank. The mixed groups considered in this context are the adjusted cotorsion groups and the class introduced by Glaz and Wickless. While is a Fuchs 34 class, the class of (adjusted) cotorsion groups is not.     

On an asymptotic behavior of elements of order in irreducible representations of the classical algebraic groups with large enough highest weights

The behavior of the images of a fixed element of order in irreducible representations of a classical algebraic group in characteristic with highest weights large enough with respect to and this element is investigated. More precisely, let be a classical algebraic group of rank over an algebraically closed field of characteristic 2$">. Assume that an element of order is conjugate to that of an algebraic group of the same type and rank naturally embedded into . Next, an integer function on the set of dominant weights of and a constant that depend only upon , and a polynomial of degree one are defined. It is proved that the image of in the irreducible representation of with highest weight contains more than Jordan blocks of size if and are not too small and .

     

Invariant projections and convolution operators

We prove the existence of invariant projections from the Banach space of -pseudomeasures onto with for closed neutral subgroup of a locally compact group . As a main application we obtain that every closed neutral subgroup is a set of -synthesis in and in fact locally -Ditkin in . We also obtain an extension theorem concerning the Fourier algebra.

     

On minimal lengths of expressions of Coxeter group elements as products of reflections

It is shown that the absolute length of a Coxeter group element (i.e. the minimal length of an expression of as a product of reflections) is equal to the minimal number of simple reflections that must be deleted from a fixed reduced expression of so that the resulting product is equal to , the identity element. Also, is the minimal length of a path in the (directed) Bruhat graph from the identity element to , and is determined by the polynomial of Kazhdan and Lusztig.

     

Serial subalgebras of finitary Lie algebras

A Lie subalgebra of is said to be finitary if it consists of elements of finite rank. We show that, if acts irreducibly on , and if is infinite-dimensional, then every non-trivial ascendant Lie subalgebra of acts irreducibly on too. When , it follows that the locally solvable radical of such is trivial. In general, locally solvable finitary Lie algebras over fields of characteristic are hyperabelian.

     

Combinatorics of rank jumps in simplicial hypergeometric systems

Let be an integer matrix, and assume that the convex hull of its columns is a simplex of dimension  not containing the origin. It is known that the semigroup ring is Cohen-Macaulay if and only if the rank of the GKZ hypergeometric system equals the normalized volume of for all complex parameters (Saito, 2002). Our refinement here shows that has rank strictly larger than the volume of if and only if lies in the Zariski closure (in  ) of all -graded degrees where the local cohomology is nonzero. We conjecture that the same statement holds even when is not a simplex.

     

Blocks with -power character degrees

Let be a -block of a finite group . If is a -power for all , then is nilpotent.

     

Semilinear transformations

In previous papers, nice trinomial equations were given for unramified coverings of the once punctured affine line in nonzero characteristic with the projective general group and the general linear group as Galois groups where is any integer and is any power of . These Galois groups were calculated over an algebraically closed ground field. Here we show that, when calculated over the prime field, as Galois groups we get the projective general semilinear group and the general semilinear group . We also obtain the semilinear versions of the local coverings considered in previous papers.

     

A sheaf of Hochschild complexes on quasi-compact opens

For a scheme , we construct a sheaf of complexes on such that for every quasi-compact open , is quasi-isomorphic to the Hochschild complex of (Lowen and Van den Bergh, 2005). Since is moreover acyclic for taking sections on quasi-compact opens, we obtain a local to global spectral sequence for Hochschild cohomology if is quasi-compact.

     

Asymptotic cones and Assouad-Nagata dimension

We prove that the dimension of any asymptotic cone over a metric space does not exceed the asymptotic Assouad-Nagata dimension of . This improves a result of Dranishnikov and Smith (2007), who showed for all separable subsets of special asymptotic cones , where is an exponential ultrafilter on natural numbers.

We also show that the Assouad-Nagata dimension of the discrete Heisenberg group equals its asymptotic dimension.