Fixed point spaces, primitive character degrees and conjugacy class sizes

Let be a finite group that acts on a nonzero finite dimensional vector space over an arbitrary field. Assume that is completely reducible as a -module, and that fixes no nonzero vector of . We show that some element has a small fixed-point space in . Specifically, we prove that we can choose so that , where is the smallest prime divisor of .

     

Characters of -degree with cyclotomic field of values

If is a prime number and is a finite group, we show that has an irreducible complex character of degree not divisible by with values in the cyclotomic field .

     

Minimal index of a -crossed product by a finite group

Let be the -crossed product of a simple unital -algebra by a finite group . In this paper we show that the canonical conditional expectation from to has the minimal index if is simple. It is also proved that if is an outer action, then the canonical one is the unique conditional expectation of index-finite type from to , while there are infinitely many conditional expectations when a nontrivial subgroup of acts innerly on .

     

Poincaré duality algebras and rings of coinvariants

Let be a faithful representation of a finite group over the field . Via the group acts on and hence on the algebra of homogenous polynomial functions on the vector space . R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field has characteristic 0, then is a Poincaré duality algebra if and only if is a pseudoreflection group. The purpose of this note is to extend this result to the case (i.e. the order of is relatively prime to the characteristic of ).

     

Periodic groups whose simple modules have finite central endomorphism dimension

Theorem. If is an uncountable field and is a periodic group with no elements of order the characteristic of and if all simple modules have finite central endomorphism dimension, then has an abelian subgroup of finite index.

     

Two characterizations of pure injective modules

Let be a commutative ring with identity and an -module. It is shown that if is pure injective, then is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if is Noetherian, then is pure injective if and only if is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that is pure injective if and only if there is a family of -algebras which are finitely presented as -modules, such that is isomorphic to a direct summand of a module of the form , where for each , is an injective -module.

     

Universal localization of triangular matrix rings

If is a triangular matrix ring, the columns and are f.g. projective -modules. We describe the universal localization of which makes invertible an -module morphism , generalizing a theorem of A. Schofield. We also describe the universal localization of -modules.

     

Open subgroups and the centre problem for the Fourier algebra

Let be the Fourier algebra of a locally compact group and the -algebra of uniformly continuous linear functionals on . We study how the centre problem for the algebra (resp. ) is related to the centre problem for the algebras (resp. ) of -compact open subgroups of . We extend some results of Lau-Losert on the centres of and .

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

On stable equivalences induced by exact functors

Let and be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence between and such that is induced by exact functors. We present a nice correspondence between indecomposable modules over and . As a consequence, we have the following: (1) If is a self-injective algebra, then so is ; (2) If and are finite dimensional algebras over an algebraically closed field , and if is of finite representation type such that the Auslander-Reiten quiver of has no oriented cycles, then and are Morita equivalent.

     

Hodge structures on posets

Let be a poset with unique minimal and maximal elements and . For each , let be the vector space spanned by -chains from to in . We define the notion of a Hodge structure on which consists of a local action of on , for each , such that the boundary map intertwines the actions of and according to a certain condition.

We show that if has a Hodge structure, then the families of Eulerian idempotents intertwine the boundary map, and so we get a splitting of into Hodge pieces.

We consider the case where is , the poset of subsets of with cardinality divisible by is fixed, and is a multiple of . We prove a remarkable formula which relates the characters of acting on the Hodge pieces of the homologies of the to the characters of acting on the homologies of the posets of partitions with every block size divisible by .

     

A strong hot spot theorem

A real number is said to be -normal if every -long string of digits appears in the base- expansion of with limiting frequency . We prove that is -normal if and only if it possesses no base- ``hot spot'. In other words, is -normal if and only if there is no real number such that smaller and smaller neighborhoods of are visited by the successive shifts of the base- expansion of with larger and larger frequencies, relative to the lengths of these neighborhoods.

     

A short proof of a conjecture on the connectivity of graph coloring complexes

The -complexes were introduced by Lovász to study topological obstructions to graph colorings. It was conjectured by Babson and Kozlov, and proved by Cukic and Kozlov, that is -connected, where is the maximal degree of a vertex of , and the number of colors. We give a short proof of the conjecture.

     

The rank of elliptic curves with rational 2-torsion points over large fields

Let be a number field, an algebraic closure of , the absolute Galois group , the maximal abelian extension of and an elliptic curve defined over . In this paper, we prove that if all 2-torsion points of are -rational, then for each , has infinite rank, and hence has infinite rank.

     

Wreath products and Kaluzhnin-Krasner embedding for Lie algebras

The wreath product of groups is one of basic constructions in group theory. We construct its analogue, a wreath product of Lie algebras.

Consider Lie algebras and over a field . Let be the universal enveloping algebra. Then has the natural structure of a Lie algebra, where the multiplication is defined via the comultiplication in . Also, acts by derivations on via the (left) coregular action. The semidirect sum we call the wreath product and denote by . As a main result, we prove that an arbitrary extension of Lie algebras can be embedded into the wreath product .

     

The weak Dirichlet problem for Baire functions

Let be a simplex and a compact subset of the set of all extreme points of . We show that any bounded function of Baire class on can be extended to a function of affine class on . Moreover, can be chosen in such a way that .

     

The spherical Paley-Wiener theorem on the complex Grassmann manifolds SU

We prove the Paley-Wiener theorem for the spherical transform on the complex Grassmann manifolds SUSU   U. This theorem characterizes the -biinvariant smooth functions on the group that are supported in the -invariant ball of radius , with less than the injectivity radius of , in terms of holomorphic extendability, exponential growth, and Weyl invariance properties of the spherical Fourier transforms , originally defined on the discrete set of highest restricted spherical weights.

     

Irreducible characters which are zero on only one conjugacy class

Suppose that is a finite solvable group which has an irreducible character which vanishes on exactly one conjugacy class. Then we show that has a homomorphic image which is a nontrivial -transitive permutation group. The latter groups have been classified by Huppert. We can also say more about the structure of depending on whether is primitive or not.

     

Isomorphic -subspaces in Orlicz-Lorentz sequence spaces

Given a decreasing weight and an Orlicz function satisfying the -condition at zero, we show that the Orlicz-Lorentz sequence space contains an -isomorphic copy of , if and only if the Orlicz sequence space does, that is, if , where and are the Matuszewska-Orlicz lower and upper indices of , respectively. If does not satisfy the -condition, then a similar result holds true for order continuous subspaces and of and , respectively.

     

Minimizing functions for an uncertainty principle on locally compact groups of bounded representation dimension

Let be a locally compact group of bounded representation dimension . Then, for any integrable function on , the product of the measures of the support of and the support of its operator-valued Fourier transform on the dual space of is bounded below by . We classify all functions for which equality holds and prove criteria for when such functions exist.