generalized blocks for symmetric groups we study via character theoretic methods an l analogue of the modular representation theory of the symmetric group for an arbitrary integer l 2 we find that many of the invariants of the usual block theory ie when l is prime generalize in a natural fashion to this new context

classifying spaces of kac moody groups we study the structure of classifying spaces of kac moody groups from a homotopy theoretic point of view they behave in many respects as in the compact lie group case the mod p cohomology algebra is noetherian and lannes t functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p subgroups also spaces of maps from classifying spaces of finite p groups to classifying spaces of kac moody groups are described in terms of classifying spaces of centralizers while the classifying space of a kac moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p subgroups up to p completion we show that these properties are common to a larger class of groups also including parabolic subgroups of kac moody groups and centralizers of finite p subgroups

a note on noncausality in this note the relationship between alternative concepts of noncausality is analyzed using the tool of conditional independence among a fields for the reader who is unfamiliar with this technique the appendix sketches the proofs and the basic technical apparatus along with some basic motivations furthermore the relationship between the concepts of noncausality and transitivity is made explicit in order to facilitate in econometric modelling the use of results already obtained in sequential analysis

binary models for marginal independence summary log linear models are a classical tool for the analysis of contingency tables in particular the subclass of graphical log linear models provides a general framework for modelling conditional independences however with the exception of special structures marginal independence hypotheses cannot be accommodated by these traditional models focusing on binary variables we present a model class that provides a framework for modelling marginal independences in contingency tables the approach that is taken is graphical and draws on analogies with multivariate gaussian models for marginal independence for the graphical model representation we use bidirected graphs which are in the tradition of path diagrams we show how the models can be parameterized in a simple fashion and how maximum likelihood estimation can be performed by using a version of the iterated conditional fitting algorithm finally we consider combining these models with symmetry restrictions

reduced fusion systems over 2 groups of small order we prove when s is a 2 group of order at most 29 that each reduced fusion system over s is the fusion system of a finite simple group and is tame it then follows that each saturated fusion system over a 2 group of order at most 29 is realizable what is most interesting about this result is the method of proof we show that among 2 groups with order in this range the ones which can be sylow 2 subgroups of finite simple groups are almost completely determined by criteria based on bender s classification of groups with strongly 2 embedded subgroups

granger causality and path diagrams for multivariate time series in this paper we introduce path diagrams for multivariate time series which visualize the dynamic relationships among the variables in these path diagrams the vertices represent the components of the time series and are connected by arrows or lines according to the nonvanishing parameters in the autoregressive representation of the time series we show that these path diagrams provide a framework for the analysis of the dependence structure of the time series in particular we give sufficient graphical conditions for granger noncausality and granger noncausality up to a certain horizon

subcentric linking systems for part of this research the author was supported by the danish national research foundation through the centre for symmetry and deformation dnrf92

transfer and characteristic idempotents for saturated fusion systems abstract we construct a well behaved transfer map from the p local burnside ring of the underlying p group s to the p local burnside ring of a saturated fusion system f using this transfer map we give new results on the characteristic idempotent of f the unique idempotent in the p local double burnside ring of s satisfying properties of linckelmann and webb we describe this idempotent explicitly both in terms of fixed points and as a linear combination of transitive bisets additionally using fixed points we determine the map for burnside rings given by multiplication with the characteristic idempotent and show that this is the transfer map previously constructed applying these results we show that for every saturated fusion system the ring generated by all not necessarily idempotent characteristic elements in the p local double burnside ring is isomorphic as rings to the p local single burnside ring of the fusion system and we disprove a conjecture by park ragnarsson stancu on the composition product of fusion systems

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

stratification and duality for homotopical groups abstract we generalize quillen s f isomorphism theorem quillen s stratification theorem the stable transfer and the finite generation of cohomology rings from finite groups to homotopical groups as a consequence we show that the category of module spectra over c b g f p is stratified and costratified for a large class of p local compact groups g including compact lie groups connected p compact groups and p local finite groups thereby giving a support theoretic classification of all localizing and colocalizing subcategories of this category moreover we prove that p compact groups admit a homotopical form of gorenstein duality

dwork cohomology de rham cohomology and hypergeometric functions in the 1960s dwork developed a p adic cohomology theory of de rham type for varieties over finite fields based on a trace formula for the action of a frobenius operator on certain spaces of p analytic functions one can consider a purely algebraic analogue of dwork s theory for varieties over a field of characteristic zero and ask what is the connection between this theory and ordinary de rham cohomology katz showed that dwork cohomology coincides with the primitive part of de rham cohomology for smooth projective hypersurfaces but the exact relationship for varieties of higher codimension has been an open question in this article we settle the case of smooth affine complete intersections

broue s conjecture holds for principal 3 blocks with elementary abelian defect group of order 9 abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a finite group g has an abelian sylow p subgroup p then the derived categories of the principal p blocks of g and of the normalizer ng p of p in g are equivalent we prove in this paper that broue inage kus conjecture holds for the principal 3 block of an arbitrary finite group g with an elementary abelian sylow 3 subgroup p of order 9 by using initiated works for the case where g is simple which are due to puig okuyama waki miyachi and the authors the result depends on the classification of finite simple groups

a test for second order stationarity of a multivariate time series it is well known that the discrete fourier transforms dft of a second order stationary time series between two distinct fourier frequencies are asymptotically uncorrelated in contrast for a large class of second order nonstationary time series including locally stationary time series this property does not hold in this paper these starkly differing properties are used to define a global test for stationarity based on the dft of a vector time series it is shown that the test statistic under the null of stationarity asymptotically has a chi squared distribution whereas under the alternative of local stationarity asymptotically it has a noncentral chi squared distribution further if the time series is gaussian and stationary the test statistic is pivotal however in many econometric applications the assumption of gaussianity can be too strong but under weaker conditions the test statistic involves an unknown variance that is extremely difficult to directly estimate from the data to overcome this issue a scheme to estimate the unknown variance based on the stationary bootstrap is proposed the properties of the stationary bootstrap under both stationarity and nonstationarity are derived these results are used to show consistency of the bootstrap estimator under stationarity and to derive the power of the test under nonstationarity the method is illustrated with some simulations the test is also used to test for stationarity of ftse 100 and dax 30 stock indexes from january 2011 december 2012

qualitative threshold arch models abstract in this paper we consider a class of dynamic models in which both the conditional mean and the conditional variance are endogenous stepwise functions we first consider the probabilistic properties of these models stationarity conditions leptokurtic effect linear representation optimal prediction in this first part most results are based on markov chains theory then we derive statistical properties of this class of models pseudo maximum likelihood estimators conditional homoscedasticity tests tests of weak or strong white noise capm test factors determination arch m effects we also discuss the introduction of exogenous variables and the case of multiple lags finally an application to the paris stock index is proposed

saturated fusion systems over 2 groups we develop methods for listing for a given 2 group s all nonconstrained centerfree saturated fusion systems over s these are the saturated fusion systems which could potentially include minimal examples of exotic fusion systems fusion systems not arising from any finite group to test our methods we carry out this program over four concrete examples two of order 2 7 and two of order 2 10 our long term goal is to make a wider more systematic search for exotic fusion systems over 2 groups of small order

a characteristic subgroup for fusion systems abstract as a counterpart for the prime 2 to glauberman s zj theorem stellmacher proves that any nontrivial 2 group s has a nontrivial characteristic subgroup w s with the following property for any finite 4 free group g with s a sylow 2 subgroup of g and with o 2 g self centralizing the subgroup w s is normal in g we generalize stellmacher s result to fusion systems a similar construction of w s can be done for odd primes and gives rise to a glauberman functor

trivial source bimodule rings for blocks and p permutation equivalences we associate with any p block of a finite group a grothendieck ring of certain p permutation bimodules we extend the notion of p permutation equivalences introduced by boltje and xu 4 to source algebras of p blocks of finite groups we show that a p permutation equivalence between two source algebras a b of blocks with a common defect group and same local structure induces an isotypy

directed information graphs for the granger causality of multivariate time series abstract in this paper we investigate the links between strong granger causality and directed information theory for multivariate time series based on the decomposition of conditional directed information we propose a definition of granger causality including instantaneous variables in the conditional set which can avoid the spurious causality the directed information graphs are presented to describe the granger causality and instantaneous coupling the structure learning of the graph models is based on the leonenko s k nn estimator of the statistics and a permutation test of the significant finally we demonstrate the numerical implementation of these techniques on linear and nonlinear time series

minimal fusion systems with a unique maximal parabolic abstract we define minimal fusion systems in a way that every non solvable fusion system has a section which is minimal minimal fusion systems can also be seen as analogs of thompson s n groups in this paper we consider a minimal fusion system f on a finite p group s that has a unique maximal p local subsystem containing n f s for an arbitrary prime p we determine the structure of a certain explicitly described p local subsystem of f if p 2 this leads to a complete classification of the fusion system f

saturated fusion systems with parabolic families abstract let g be a group a finite p subgroup s of g is a sylow p subgroup if every finite p subgroup of g is conjugate to a subgroup of s in this paper we examine the relations between the fusion system over s which is given by conjugation in g and a certain chamber system c on which g acts chamber transitively with chamber stabilizer n g s next we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system we determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system we give an application to fusion systems with parabolic families of classical type

graphical modelling of multivariate time series we introduce graphical time series models for the analysis of dynamic relationships among variables in multivariate time series the modelling approach is based on the notion of strong granger causality and can be applied to time series with non linear dependences the models are derived from ordinary time series models by imposing constraints that are encoded by mixed graphs in these graphs each component series is represented by a single vertex and directed edges indicate possible granger causal relationships between variables while undirected edges are used to map the contemporaneous dependence structure we introduce various notions of granger causal markov properties and discuss the relationships among them and to other markov properties that can be applied in this context examples for graphical time series models include nonlinear autoregressive models and multivariate arch models

conditional independence graph for nonlinear time series and its application to international financial markets abstract conditional independence graphs are proposed for describing the dependence structure of multivariate nonlinear time series which extend the graphical modeling approach based on partial correlation the vertexes represent the components of a multivariate time series and edges denote direct dependence between corresponding series the conditional independence relations between component series are tested efficiently and consistently using conditional mutual information statistics and a bootstrap procedure furthermore a method combining information theory with surrogate data is applied to test the linearity of the conditional dependence the efficiency of the methods is approved through simulation time series with different linear and nonlinear dependence relations finally we show how the method can be applied to international financial markets to investigate the nonlinear independence structure

a note on olssonʼs conjecture abstract for a p block b of a finite group g with defect group d olsson conjectured that k 0 b d d where k 0 b is the number of characters in b of height 0 and d denotes the commutator subgroup of d brauer deduced olsson s conjecture in the case where d is a dihedral 2 group using the fact that certain algebraically conjugate subsections are also conjugate in g we generalize brauer s argument for arbitrary primes p and arbitrary defect groups this extends two results by robinson for p 3 we show that olsson s conjecture is satisfied for defect groups of p rank 2 and for minimal non abelian defect groups

splendid morita equivalences for principal 2 blocks with dihedral defect groups given a dihedral 2 group p of order at least 8 we classify the splendid morita equivalence classes of principal 2 blocks with defect groups isomorphic to p to this end we construct explicit stable equivalences of morita type induced by specific scott modules using brauer indecomposability and gluing methods we then determine when these stable equivalences are actually morita equivalences and hence automatically splendid morita equivalences finally we compute the generalised decomposition numbers in each case

graphical gaussian modelling of multivariate time series with latent variable in time series analysis inference about causeeffect relationships among multiple times series is commonly based on the concept of granger causality which exploits temporal structure to achieve causal ordering of dependent variables one major problem in the application of granger causality for the identification of causal relationships is the possible presence of latent variables that affect the measured components and thus lead to so called spurious causalities in this paper we describe a new graphical approach for modelling the dependence structure of multivariate stationary time series that are affected by latent variables to this end we introduce dynamic maximal ancestral graphs dmags in which each time series is represented by a single vertex for gaussian processes this approach leads to vector autoregressive models with errors that are not independent but correlated according to the dashed edges in the graph we discuss identifiability of the parameters and show that these models can be viewed as graphical arma models that satisfy the granger causality restrictions encoded by the associated dynamic maximal ancestral graph

fusion systems over non resistant a2 groups abstract in this paper we study the saturated fusion systems f over potential non resistant a 2 groups of order greater than p 4 for each case in the classification f is constrained and realized by a finite group

generalized perfect isometries in some groups of lie rank one abstract in a recent paper b kulshammer j b olsson and g r robinson introduced notions of generalized blocks and generalized perfect isometries and studied them in the case of the symmetric group s n it is the purpose of this paper to investigate some families of groups of lie rank one exhibiting generalized perfect isometries where there are none in the sense given by m broue and thus proving a weaker analogue of one of broue s conjectures in these cases

generation of fusion systems of characteristic 2 type we prove that if is a saturated fusion system on a finite 2 group s then either is known or is generated by the normalizers of two canonically defined characteristic subgroups of s there are various corollaries for finite groups of characteristic 2 type

causal conditioning and instantaneous coupling in causality graphs this study aims at providing the definitive links between massey and kramer s directed information theory and granger causality graphs recently formalized by eichler this naturally leads to consider two concepts of causality that can occur in physical systems dynamical causality and instantaneous coupling although it is well accepted that the former is assessed by conditional transfer entropy the latter is often overlooked even if it was clearly introduced and understood in seminal studies in the bivariate case we show that directed information is decomposed into the sum of transfer entropy and instantaneous information exchange in the multivariate case encountered for conditional graph modeling such a decomposition does not hold anymore we provide a new insight into the problem of instantaneous coupling and show how it affects the estimated structure of a graphical model that should provide a sparse representation of a complex system this result is discussed and analyzed for the inference of causality graphs two synthetic examples are developed to illustrate the theoretical concepts practical issues are also briefly discussed on these examples and an extension of leonenko s k nearest neighbors based entropy estimators is used to derive a nonparametric estimator of directed information

separation and completeness properties for amp chain graph markov models pearl s well known d separation criterion for an acyclic directed graph adg is a pathwise separation criterion that can be used to efficiently identify all valid conditional independence relations in the markov model determined by the graph this paper introduces p separation a pathwise separation criterion that efficiently identifies all valid conditional independences under the andersson madigan perlman amp alternative markov property for chain graphs adicyclic graphs which include both adgs and undirected graphs as special cases the equivalence of p separation to the augmentation criterion occurring in the amp global markov property is established and p separation is applied to prove completeness of the global markov property for amp chain graph models strong completeness of the amp markov property is established that is the existence of markov perfect distributions that satisfy those and only those conditional independences implied by the amp property equivalently by p separation a linear time algorithm for determining p separation is presented

control of transfer and weak closure in fusion systems we show that k1 and k 1 control transfer in every fusion system on a finite p group when p 5 and that they control weak closure of elements in every fusion system on a finite p group when p 3 this generalizes results of g glauberman concerning finite groups

edge exclusion tests for graphical model selection complex gaussian vectors and time series we consider the problem of inferring the conditional independence graph cig of a stationary multivariate gaussian time series a p variate gaussian time series graphical model associated with an undirected graph with p vertices is defined as the family of time series that obey the conditional independence restrictions implied by the edge set of the graph in some existing methods partial coherence has been used as a test statistic for graphical model selection to test inclusion exclusion of a given edge in the graph the test is applied at distinct frequencies requiring multiple tests and leading to a loss in test power the nonparametric method of matsuda uses the kullback leibler divergence measure to define a test statistic which does not need multiple testing to test between two competing models in this paper we propose a generalized likelihood ratio test glrt based edge exclusion test statistic that also does not need multiple testing it is computationally significantly faster than the method of matsuda and simulations show that we achieve comparable power levels our proposed approach is based on a novel formulation of a glrt based edge exclusion test for p variate complex gaussian graphical models cggms this result is also of independent interest the computational complexity of the proposed statistic is mathcal o p 3 compared to mathcal o p 5 for the existing result we also apply our time series graphical model selection method to a foreign exchange data set consisting of monthly trends of foreign exchange rates of 10 countries

splendid equivalences derived categories and permutation modules we introduce the notion of a splendid equivalence between blocks of finite groups which is a strengthening of the notion of an equivalence of derived categories such an equivalence is always induced by a two sided tilting complex whose terms are summands of permutation modules a feature that is compatible with the belief that these complexes should be related to the adic cohomology of deligne lusztig varities in the case of reductive groups over finite fields we show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence in particular it gives for principal blocks a structural explanation of brou s concept of an isotypy a compatible family of perfect isometries we give several examples including examples of morita equivalences that are splendid in an unexpectedly non trivial way

representations of the double burnside algebra and cohomology of the extraspecial p group ii abstract let e be the extraspecial p group of order p 3 and exponent p where p is an odd prime we determine the mod p cohomology h x f p of a summand x in the stable splitting of p completed classifying space be in the previous paper hida and yagita 2014 7 we determined this cohomology modulo nilpotence in this paper we consider the whole of the cohomology moreover we consider the stable splittings of bg for some finite groups with sylow p subgroup e related to the three dimensional linear group l 3 p

products in fusion systems abstract we revisit the notion of a product of a normal subsystem with a p subgroup as defined by aschbacher 2011 asc11 chapter 8 in particular we give a previously unknown more transparent construction

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

wreath products and representations of p local finite groups abstract given two finite p local finite groups and a fusion preserving morphism between their sylow subgroups we study the question of extending it to a continuous map between their classifying spaces the results depend on the construction of the wreath product of p local finite groups which is also used to study p local permutation representations

new refinements of the mckay conjecture for arbitrary finite groups let g be an arbitrary finite group and fix a prime number p the mckay conjecture asserts that g and the normalizer in g of a sylow p subgroup have equal numbers of irreducible characters with degrees not divisible by p the alperin mckay conjecture is version of this as applied to individual brauer p blocks of g we offer evidence that perhaps much stronger forms of both of these conjectures are true

on saturated fusion systems and brauer indecomposability of scott modules let p be a prime number g a finite group p a p subgroup of g and k an algebraically closed field of characteristic p we study the relationship between the category ff p g and the behavior of p permutation kg modules with vertex p under the brauer construction we give a sufficient condition for ff p g to be a saturated fusion system we prove that for scott modules with abelian vertex our condition is also necessary in order to obtain our results we prove a criterion for the categories arising from the data of b g brauer pairs in the sense of alperin brou e and brou e puig to be saturated fusion systems on the underlying p group

saturated fusion systems over mathcal a _2 groups in this paper we study the saturated fusion systems mathcal f over mathcal a 2 groups s of order greater than p 4 this study requires the analysis of the possible saturated fusion systems in terms of the mathcal f automorphisms of the possible mathcal f centric and mathcal f radical subgroups also for each case in the classification either s is resistant or mathcal f is constrained the classification identifies a large family of resistant groups complementing the results on resistant groups found by various authors such as green minh stancu and diaz ruiz viruel moreover the classification provides some interesting examples of saturated fusion systems which have an mathcal f essential abelian subgroup of index p

amalgams blocks weights fusion systems and finite simple groups abstract we associate an iterated amalgam of finite groups to a certain class of fusion systems on finite p groups p a prime in such a way that the p group of the fusion system is a maximal finite p subgroup of the resulting group unique up to conjugacy and furthermore the conjugation action of the resulting usually infinite group on p subgroups induces the original fusion system on the p group 1 in view of earlier work of puig and of broto castellana grodal levi and oliver c broto n castellana j grodal r levi r oliver subgroup families controlling p local finite groups proc london math soc 3 91 2005 325 354 the fusion systems we deal with include all saturated fusion systems the resulting amalgam has free normal subgroups of finite index and we examine the images of the group by its maximal free normal subgroups of finite index these images all contain isomorphic copies of the original p group and are generated by the images of the finite groups used in the amalgamation if there is no non trivial normal p subgroup of the fusion system equivalently if the iterated amalgam constructed has no non trivial normal p subgroup then the generalised fitting subgroup of each of these homomorphic images is a direct product of non abelian simple groups each of order divisible by p and if there is no proper non trivial strongly closed p subgroup in the fusion system then the generalised fitting subgroup of any of these finite groups is characteristically simple we note that the finite dimensional representation theory of this amalgam is almost by construction p locally determined in the case of the fusion system associated to a p block of a finite group we suggest strong links between block theoretic invariants of the above finite epimorphic images of the associated amalgam and of the original block we believe that the results of this paper offer the prospect of at least the beginnings of a structural explanation for some of the current numerical conjectures in block theory e g j l alperin weights for finite groups in proc sympos pure math vol 47 amer math soc providence 1987 pp 369 379 e c dade counting characters in blocks i invent math 109 1 1992 187 210

structural learning with time varying components tracking the cross section of financial time series summary when modelling multivariate financial data the problem of structural learning is compounded by the fact that the covariance structure changes with time previous work has focused on modelling those changes by using multivariate stochastic volatility models we present an alternative to these models that focuses instead on the latent graphical structure that is related to the precision matrix we develop a graphical model for sequences of gaussian random vectors when changes in the underlying graph occur at random times and a new block of data is created with the addition or deletion of an edge we show how a bayesian hierarchical model incorporates both the uncertainty about that graph and the time variation thereof

representations of categories and their applications we define for each small category c a category algebra rc over a base ring r and study its representations when c is an ei category we develop a theory of vertices and sources for rc mod which parameterizes the indecomposable rc modules as a main application we use our theory to find formulas for computing higher inverse limits over c

extra special defect groups of order p3 and exponent p2 abstract in this paper we give results on p blocks with the defect groups isomorphic to an extra special group of order p3 we are particularly interested in the number of irreducible ordinary characters and the number of irreducible brauer characters in the block the situation splits naturally into two cases according to the exponent of the extra special group in this paper we concentrate on the exponent p2 case we prove that olsson s conjecture theorem 4 6 and brauer s k b conjecture proposition 5 13 hold for the exponent p2 case we are also able to calculate two important block invariants k b l b and l b m g b 1 1 in this case

isometries and extra special sylow groups of order p3 abstract a new type of isometry between the sets of irreducible characters of blocks is introduced and a conjecture is stated using the classifications of finite simple groups and saturated fusion systems over an extra special group p of order p 3 and exponent p we check that the conjecture is affirmatively answered for the principal blocks of finite groups with sylow p subgroup p

a reduction theorem for fusion systems of blocks let k be an algebraically closed field of characteristic p and g a finite group an interesting question for fusion systems is whether they can be obtained from the local structure of a block of the group algebra kg in this paper we develop some methods to reduce this question to the case when g is a central p extension of a simple group as an application of our result we obtain that the exotic examples of fusion systems discovered by ruiz and viruel a ruiz a viruel the classification of p local finite groups over the extra special group of order p3 and exponent p math z 248 2004 45 65 do not occur as fusion systems of p blocks of finite groups

testing causality between two vectors in multivariate autoregressive moving average models abstract in the analysis of economic time series a question often raised is whether a vector of variables causes another one in the sense of granger most of the literature on this topic is concerned with bivariate relationships or uses finite order autoregressive specifications the purpose of this article is to develop a causality analysis in the sense of granger for general vector autoregressive moving average arma models we give a definition of granger noncausality between vectors which is a natural and simple extension of the notion of granger noncausality between two variables in our context this definition is shown to be equivalent to a more complex definition proposed by tjostheim for the class of linear invertible processes we derive a necessary and sufficient condition for noncausality between two vectors of variables when the latter do not necessarily include all the variables considered in the analysis this result is then specialized to the class of stationary invertible arma process

homotopy equivalences of p completed classifying spaces of finite groups we study homotopy equivalences of p completions of classifying spaces of finite groups to each finite group g and each prime p we associate a finite category p c g with the following properties two p completed classifying spaces bg p and bg p have the same homotopy type if and only if the associated categories p c g and p c g are equivalent furthermore the topological monoid aut bg p of self equivalences is determined by the self equivalences of the associated category p c g

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated

the classification of p local finite groups over the extraspecial group of order p 3 and exponent p in this paper we classify the p local finite groups over p1 2 the extraspecial group of order p3 and exponent p for odd p this study reduces to the classification of the saturated fusion systems over p1 2 which will be characterized by the outer automorphism group the number of open image in new window radical subgroups and the automorphism group of each nontrivial open image in new window radical subgroup as part of this classification we obtain three new exotic 7 local finite groups