a general theory of canonical induction formulae abstract so far there exist three independent constructions of two different canonical versions of brauer s induction theorem for complex characters due to v snaith p symonds and the author canonical in this context means functorial with respect to restrictions along group homomorphisms in this article we axiomatize the situation in which the above canonical induction formulae are constructed mackey functors and related structures arise in this way naturally as a convenient language this approach allows us to construct canonical induction formulae for arbitrary mackey functors in particular we obtain canonical induction formulae for the brauer character ring the group of projective characters the ring of trivial source modules and the ring of linear source modules in most cases it is not difficult to construct such formulae over the rational numbers a much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers we give a sufficient condition in the general framework of mackey functors for a canonical induction formula to be integral as an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings as for example the span of linear characters in the case of the canonical brauer induction formula this will be used in a subsequent article in the case of adams operations and chern classes

geometric properties of the sections of solutions to the monge ampere equation in this paper we establish several geometric properties of the cross sections of generalized solutions 4 to the monge ampere equation det d2 when the measure t satisfies a doubling property a main result is a characterization of the doubling measures t in terms of a geometric property of the cross sections of 4 this is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations 0 introduction let ri r be a convex function a supporting hyperplane to q at the point xo q xo is an affine function x q xo p x xo such that q x x for all x e r given t 0 a section of 0 at height t is the convex set so xo p t x e r x p ri defined by vq xo p q x xo p x xo vx e rn if e c rn then v e uxee vq x by a classical theorem of aleksandrov the class of sets e such that v e is lebesgue measurable is a o algebra that contains the borel sets and one can define the monge ampere measure associated with q as the borel measure u given by pt v e see ch y the purpose in this paper to analyze in detail several important geometric properties of the sections of the convex function q when its associated monge ampere measure u satisfies a doubling condition the interest in these properties comes from the study of the solutions of the monge ampere equation and its linearizations both elliptic and parabolic and from real harmonic analysis see cl c2 c g1 c g2 and h some of these properties have been used in those references to prove a lemma of besicovitch s type and a calderon zygmund decomposition ir terms of sections which allows us to establish estimates of the solutions to the linearized monge ampere equation also some of these properties imply that r received by the editors june 9 1997 1991 mathematics subject classification primary 35j60 35d10 secondary 26b25

crossed burnside rings and bouc s construction of green functors abstract we use a formula for primitive idempotents of the crossed burnside ring given by f oda and t yoshida and a result from the theory of green functors obtained by s bouc to prove an induction theorem for the drinfel d double for a finite group over the complex field we obtain artin s induction theorem for group algebras as a corollary of the theorem

brauer relations for finite groups in the ring of semisimplified modular representations abstract let g be a finite group and p be a prime we study the kernel of the map between the burnside ring of g and the grothendieck ring of f p g modules taking a g set to its associated permutation module we are able for all finite groups to classify the primitive quotient of the kernel that is for each g the kernel modulo elements coming from the kernel for proper subquotients of g we are able to identify exactly which groups have non trivial primitive quotient and we give generators for the primitive quotient in the soluble case

some remarks on g functors and the brauer morphism keywords finite group g functor restriction maps transfer maps mackey s formula frobenius reciprocity brauer morphism defect base nilpotent ideal burnside ring idempotent formula induction formula symmetric g functor character ring green ring reference ctg article 1988 003 record created on 2008 12 16 modified on 2017 05 12

properties of the solutions of the linearized monge ampere equation let r n r be a function strictly convex and smooth and det d 2 is the monge ampere generated by given x r n and t 0 let s x t y r n y x x y x t the purpose of this paper is to study the properties of the solutions of the linearized monge ampere equation given by a ij x d ij u 0 where the coefficients a ij x are the cofactors of the matrix d 2 x it is assumed that satisfies a doubling condition on the sets s x t and a uniform continuity condition at every scale with respect to lebesgue measure we establish that the distribution functions of nonnegative solutions u at altitude t decay like a negative power of t and prove an invariant harnack s inequality on the sections s x t all the estimates are independent of the regularity of and depend only on the constants in the hypotheses made on the measure

on certain degenerate and singular elliptic pdes ii divergence form operators harnack inequalities and applications abstract by means of the monge ampere real analysis and pde techniques associated to certain convex functions an approach towards harnack inequalities is developed that simultaneously extends the one for uniformly elliptic operators from the de giorgi nash moser theory and the one for the linearized monge ampere operator from the caffarelli gutierrez theory applications include regularity properties for solutions to divergence form elliptic equations with power like singularities and c 2 estimates for solutions to the monge ampere equation

elementary construction of the quiver of the mackey algebra for groups with cyclic normal p sylow subgroup through recent investigations of j thrvenaz and p j webb 4 8 the concept of mackey functors being originally introduced by j a green 3 and studied by a w m dress t yoshida h sasaki and others has regained considerable interest an important reason for this is i new interpretation of a conjecture of aiperin in terms of mackey functors in 8 using results of 7 given a commutative ring 4 by its definition it is immediate that the category of mackey functors over 4 for a finite group g is an abelian category even that it is the category of 4 representations of a quiver with relations 8 namely a mackey functor for g associates to each subgroup of g a module and moreover certain linear maps between these spaces satisfying certain relations involving g thus a mackey functor for g can be interpreted as a module over a path algebra of a quiver with relations the so called mackey algebra 6 9 the main difference to the usual representation theory of quivers and thus the main problem of interpreting mackey functors as such representations lies in the fact that the maps between these spaces do not lie in the radical of this path algebra and this path algebra is not basic in general so it is a natural problem to compute the quiver in the usual sense for the category of mackey functors for a group g in particular in the case where 4 is a field of characteristic p dividing the order of g the question of when there are up to isomorphism only finitely many indecomposable mackey functors for g led to recent investigations of thrvenaz and webb solving this problem in general and it also led to the investigations we present in this paper which presents in a special situation namely if g has a normal cyclic p sylow subgroup an explicit computation of the quiver of the now so called mackcy algebra by definition the mackey algebra is the basic algebra whose category of finitely generated

properties of the solutions to the monge ampere equation abstract we consider solutions to the equation det d 2 when has a doubling property we prove new geometric characterizations for this doubling property by means of the so called engulfing property and deduce the quantitative behaviour of also a constructive approach to the celebrated c1 estimates proved by l caffarelli is presented settling one of the open questions posed by villani amer math soc 58 2003

maximum likelihood array processing in spatially correlated noise fields using parameterized signals this paper deals with the problem of estimating signal parameters using an array of sensors this problem is of interest in a variety of applications such as radar and sonar source localization a vast number of estimation techniques have been proposed in the literature during the past two decades most of these can deliver consistent estimates only if the covariance matrix of the background noise is known in many applications the aforementioned assumption is unrealistic recently a number of contributions have addressed the problem of signal parameter estimation in unknown noise environments based on various assumptions on the noise herein a different approach is taken we assume instead that the signals are partially known the received signals are modeled as linear combinations of certain known basis functions the exact maximum likelihood ml estimator for the problem at hand is derived as well as computationally more attractive approximation the cramer rao lower bound crb on the estimation error variance is also derived and found to coincide with the crb assuming an arbitrary deterministic model and known noise covariance

rigidity and exotic models for v1 local g equivariant stable homotopy we prove that the v 1 local g equivariant stable homotopy category for g a finite group has a unique g equivariant model at p 2 this means that at the prime 2 the homotopy theory of g spectra up to fixed point equivalences on k theory is uniquely determined by its triangulated homotopy category and basic mackey structure the result combines the rigidity result for k local spectra of the second author with the equivariant rigidity result for g spectra of the first author further when the prime p is at least 5 and does not divide the order of g we provide an algebraic exotic model as well as a g equivariant exotic model for the v 1 local g equivariant stable homotopy category showing that for primes p ge 5 equivariant rigidity fails in general

on geometric characterizations for monge ampere doubling measures abstract in this article we prove a theorem on the size of the image of sections of a convex function under its normal mapping when the sections satisfy a geometric property we apply this result to get new geometric characterizations for monge ampere doubling measures

induction formulae for mackey functors with applications to representations of the twisted quantum double of a finite group abstract in the theory of canonical induction formulae for mackey functors boltje 4 demonstrated that the plus constructions together with the mark morphism are useful for the study of canonical versions of induction theorems analogous to those in representation theory of finite groups in this paper we present a short exact sequence for the plus constructions derived from cauchy frobenius lemma and apply it to the proof of boltje s integrality result for canonical induction formulae the methods appearing in boltje s theory combined with the dress construction for mackey functors are applicable to induction theorems on representations of the twisted quantum double of a finite group as a sequel to such a research we describe canonical versions of two induction theorems whose origins are artin s induction theorem and brauer s induction theorem on c characters of a finite group

crossed burnside rings ii the dress construction of a green functor abstract we present some results about the dress construction of green functor associated to a g monoid in particular we see that the crossed burnside rings the representation rings of quantum double and hochschild cohomology rings are constructed by the dress construction of some green functors

the dade group of mackey functors for p groups abstract we introduce endo permutation mackey functors and the mackey dade group of a p group and study their basic properties we exhibit relations between the mackey dade group of a finite p group p and the dade groups of the weyl groups of all subgroups of p as a result we show that the mackey dade group tensored with q over z is the kernel of the linearization map for mackey functors introduced by the author

duality in g algebras keywords finite group complete discrete valuation ring symmetric g algebra symmetric bilinear form projective ideal symmetric algebra rg lattice unimodular symmetric g algebra almost projective ideal almost split sequences block defect groups virtually irreducible lattices splitting trace lattices reference ctg article 1988 005doi 10 1007 bf01161746 record created on 2008 12 16 modified on 2017 05 12

on the axiomatic approach to harnackʼs inequality in doubling quasi metric spaces abstract we introduce a novel approach towards harnack s inequality in the context of spaces of homogeneous type this approach based on the so called critical density property and doubling properties for weights avoids the explicit use of covering lemmas and bmo

a note on green functors with inflation abstract this note is motivated by the problem to understand given a commutative ring f which g sets x y give rise to isomorphic f g representations f x f y a typical step in such investigations is an argument that uses induction theorems to give very general sufficient conditions for all such relations to come from proper subquotients of g in the present paper we axiomatise the situation and prove such a result in the generality of mackey functors and green functors with inflation our result includes as special cases a result of deligne on monomial relations a result of the first author and tim dokchitser on brauer relations in characteristic 0 and a new result on brauer relations in characteristic p 0 we will need the new result in a forthcoming paper on brauer relations in positive characteristic

harnackʼs inequality for solutions to the linearized monge ampere operator with lower order terms abstract under minimal geometric assumptions we prove a harnack inequality for non negative solutions to the non homogeneous linearized monge ampere equation with bounded lower order terms compatible with the maximum principle

finite groups acting linearly hochschild cohomology and the cup product abstract when a finite group acts linearly on a complex vector space the natural semi direct product of the group and the polynomial ring over the space forms a skew group algebra this algebra plays the role of the coordinate ring of the resulting orbifold and serves as a substitute for the ring of invariant polynomials from the viewpoint of geometry and physics its hochschild cohomology predicts various hecke algebras and deformations of the orbifold in this article we investigate the ring structure of the hochschild cohomology of the skew group algebra we show that the cup product coincides with a natural smash product transferring the cohomology of a group action into a group action on cohomology we express the algebraic structure of hochschild cohomology in terms of a partial order on the group modulo the kernel of the action this partial order arises after assigning to each group element the codimension of its fixed point space we describe the algebraic structure for coxeter groups where this partial order is given by the reflection length function a similar combinatorial description holds for an infinite family of complex reflection groups

mackey functors induction from restriction functors and coinduction from transfer functors abstract boltje s plus constructions extend two well known constructions on mackey functors the fixed point functor and the fixed quotient functor in this paper we show that the plus constructions are induction and coinduction functors of general module theory as an application we construct simple mackey functors from simple restriction functors and simple transfer functors we also give new proofs for the classification theorem for simple mackey functors and semisimplicity theorem of mackey functors

socles and radicals of mackey functors we study the socle and the radical of a mackey functor m for a finite group g over a field k usually of characteristic p 0 for a subgroup h of g we construct bijections between some classes of the simple subfunctors of m and some classes of the simple kn g h submodules of m h we relate the multiplicity of a simple mackey functor sh vg in the socle of m to the multiplicity of v in the socle of a certain kn g h submodule of m h we also obtain similar results for the maximal subfunctors of m we then apply these general results to some special mackey functors for g including the functors obtained by inducing or restricting a simple mackey functor mackey functors for a p group the fixed point functor and the burnside functor bkg for instance we find the first four top factors of the radical series of bkg for a p group g and assuming further that g is an abelian p group we find the radical series of bkg

the representation ring of the quantum double of a finite group abstract we present some results about the representation ring of the quantum double of a finite group over fields of arbitrary characteristic we give a direct sum decomposition of this representation ring into ideals involving green rings of subgroups given characters of such green rings we construct characters of the representation ring of the quantum double and show that all characters arise in this fashion we prove that the characters thereby obtained from brauer characters separate modules of the quantum double up to composition factors

quillen stratification for hochschild cohomology of blocks we decompose the maximal ideal spectrum of the hochschild cohomology ring of a block of a finite group into a disjoint union of subvarieties corresponding to elementary abelian p subgroups of a defect group these subvarieties are described in terms of group cohomological varieties and the alperin broue correspondence on blocks our description leads in particular to a homeomorphism between the hochschild variety of the principal block and the group cohomological variety the proofs require a result of stephen f siegel given in the appendix which states that nilpotency in hochschild cohomology is detected on elementary abelian p subgroups

categorifying induction formulae via divergent series abstract we show how to get explicit induction formulae for finite group representations and more generally for rational green functors by summing a divergent series over dwyer s subgroup and centralizer decomposition spaces this results in formulae with rational coefficients the former space yields a well known induction formula the latter yields a new one as essentially immediate corollaries of the existing literature we get similar formulae in group cohomology and stable splittings of classifying spaces

g algebras jacobson radical and almost split sequences keywords finite group indecomposable kg module sas sequence almost split exact sequences restriction relative jacobson radicals local criterion multiplicity module vertex source indecomposable projective module twisted group algebra induction reference ctg article 1988 006doi 10 1007 bf01393690 record created on 2008 12 16 modified on 2017 05 12

on certain degenerate and singular elliptic pdes i nondivergence form operators with unbounded drifts and applications to subelliptic equations abstract we prove a harnack inequality for nonnegative strong solutions to degenerate and singular elliptic pdes modeled after certain convex functions and in the presence of unbounded drifts our main theorem extends the harnack inequality for the linearized monge ampere equation due to caffarelli and gutierrez and it is related although under different hypotheses to a recent work by n q le since our results are shown to apply to the convex functions x p with p 2 and their tensor sums the degenerate elliptic operators that we can consider include subelliptic grushin and grushin like operators as well as a recent example by a montanari of a nondivergence form subelliptic operator arising from the geometric theory of several complex variables in the light of these applications it follows that the monge ampere quasi metric structure can be regarded as an alternative to the usual carnot caratheodory metric in the study of certain subelliptic pdes

mackey functors with chain conditions abstract letabe a green functor for a finite groupgand letmbe a lefta module mis callednoetherian artinian if the lattice of all submodules ofmsatisfies acc respectively dcc mis calledtotally noetherian totally artinian ifmxis noetherian respectively artinian for every finiteg setx in this paper it is shown that the totally noetherian totally artinian lefta modules behave in the same way as the classical noetherian artinian modules the left noetherian left artinian green functors are less well behaved but some properties from classical algebra still hold it is shown that cohen s theorem holds in this setting our main result shows that if a commutative green functorais artinian then the ringsa h jac a h are semisimple artinian for all subgroupshofg

defect theory for maximal ideals and simple functors keywords green functor source green correspondence maximal ideal finite group fixed point algebra defect theory for g algebras defect group restriction transfer conjugation defect pair functorial ideal simple g functors reference ctg article 1991 002doi 10 1016 0021 8693 91 90168 8 record created on 2008 12 16 modified on 2017 05 12