glauberman correspondence of p blocks of finite groups abstract let g and a be finite groups with coprime orders suppose that a is solvable and that it acts on g by automorphisms let c cg a by irr g and irra g we denote the set of all irreducible and all a invariant irreducible characters of g respectively let d c be a fixed p subgroup of g for a prime p using the glauberman correspondence g a irr a g irr c a watanabe j algebra216 1999 548 565 recently established a glauberman correspondence between a invariant p blocks b of g with defect group d and p blocks b1 of c with defect group d let br b and br b1 be the brauer correspondents of b and b1 in ng d and nc d respectively the main result of this article asserts that the block algebras br b and br b1 are morita equivalent furthermore if g is p solvable and d is abelian then the block algebras b and b1 are morita equivalent

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

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

a geometric classification of lagrangian functions and the reduction of evolution space a geometric classification of degenerate time dependent lagrangian systems is given and the reduction of evolution space is analysed general properties of semiregular lagrangians type ii are discussed and particular attention is paid to the reduction of completely degenerate lagrangians type iii which are considered in detail

multivector field formulation of hamiltonian field theories equations and symmetries we state the intrinsic form of the hamiltonian equations of first order classical field theories in three equivalent geometrical ways using multivector fields jet fields and connections thus these equations are given in a form similar to that in which the hamiltonian equations of mechanics are usually given then using multivector fields we study several aspects of these equations such as the existence and non uniqueness of solutions and the integrability problem in particular these problems are analysed for the case of hamiltonian systems defined in a submanifold of the multimomentum bundle furthermore the existence of first integrals of these hamiltonian equations is considered and the relation between cartan noether symmetries and general symmetries of the system is discussed noether s theorem is also stated in this context both the classical version and its generalization to include higher order cartan noether symmetries finally the equivalence between the lagrangian and hamiltonian formalisms is also discussed

invariant blocks under coprime action let a and g be finite groups with a g 1 assume that a acts on g via automorphisms let b be an a invariant block of g if each irreducible character in b is a invariant and b has cyclic defect groups then a centralizes a defect group of b

poly poisson sigma models and their relational poly symplectic groupoids the main idea of this paper is to describe the integration procedure for poly poisson structures that is to find a poly symplectic groupoid integrating a poly poisson structure in terms of topological field theories namely via the path space construction this will be given in terms of the poly poisson sigma model and we prove that every poly poisson structure has a natural integration via relational poly symplectic groupoids extending known results for poisson structures we provide familiar examples trivial linear constant and symplectic within this formulation and give some applications of this construction regarding the classification of poly symplectic integrations as well as morita equivalence of poly poisson manifolds the main idea of this paper is to describe the integration procedure for poly poisson structures that is to find a poly symplectic groupoid integrating a poly poisson structure in terms of topological field theories namely via the path space construction this will be given in terms of the poly poisson sigma model and we prove that every poly poisson structure has a natural integration via relational poly symplectic groupoids extending known results for poisson structures we provide familiar examples trivial linear constant and symplectic within this formulation and give some applications of this construction regarding the classification of poly symplectic integrations as well as morita equivalence of poly poisson manifolds

multiplicative forms at the infinitesimal level we describe arbitrary multiplicative differential forms on lie groupoids infinitesimally i e in terms of lie algebroid data this description is based on the study of linear differential forms on lie algebroids and encompasses many known integration results related to poisson geometry we also revisit multiplicative multivector fields and their infinitesimal counterparts drawing a parallel between the two theories

perfect generalized characters inducing the alperin mckay conjecture it is well known that the perfect isometries predicted in broue s conjecture do not always exist when the defect groups are non abelian even when the blocks have equivalent brauer categories we consider perfect generalized characters which induce bijections between the sets of irreducible characters of height zero of a block and of its brauer correspondent in the normalizer of a defect group in this way the perfect isometries predicted in broue s conjecture for blocks with abelian defect groups are generalized whilst such generalized characters do not exist in general we show that they do exist when the defect groups are non abelian trivial intersection subgroups of order p 3 as well as for 2b 2 q for q a power of two and psu 3 q for all q further we show that these blocks satisfy a generalized version of an isotypy

n symplectic algebra of observables in covariant lagrangian field theory n symplectic geometry on the adapted frame bundle l e e of an n m k dimensional fiber bundle e m is used to set up an algebra of observables for covariant lagrangian field theories using the principle bundle l e j1 we lift a lagrangian l j1 r to a lagrangian l l l e r and then use l to define a modified n symplectic potential l on l e the cartan hamilton poincare chp rn valued 1 form if the lifted lagrangian is nonzero then l e d l is an n symplectic manifold to characterize the observables we define a lifted legendre transformation l from l e into le the image ql l l e is a submanifold of le and ql d ql is shown to be an n symplectic manifold we prove the theorem that l l ql and pull back the reduced canonical n symplectic geometry on ql to l e to define the algebras of observables on the n symplectic manifold l e d l to find the reduced n symplectic algebra on ql we set up the equations of n symplectic reduction and apply the general theory to the mo

automizers as extended reflection groups abstract let g be a finite group having an abelian sylow p subgroup p broue malle and michel have shown that if g is a simple chevalley group then the automizer of p is an irreducible complex reflection group for p not too small and different from the defining characteristic the aim in this note is to show that a suitable version of this property holds for general finite groups

on equivalences between blocks of group algebras reduction to the simple components abstract a conjecture of michel broue states that if d is an abelian sylow p subgroup of a finite group g and h n g d then the principal blocks of g and h are rickard equivalent the structure of groups with abelian sylow p subgroups as determined by p fong and m e harris raises the following question assuming that broue s conjecture holds for the simple components of g under what conditions does it hold for g itself due to the structure of g this problem requires mainly the lifting of rickard complexes to p extensions of the simple components and the construction of complexes over wreath products we give here these reduction steps which may be regarded as a clifford theory of tilting complexes

geometry of multisymplectic hamiltonian first order field theories in the jet bundle description of field theories multisymplectic models in particular there are several choices for the multimomentum bundle where the covariant hamiltonian formalism takes place as a consequence several proposals for this formalism can be stated and on each one of them the differentiable structures needed for setting the formalism are obtained in different ways in this work we make an accurate study of some of these hamiltonian formalisms showing their equivalence in particular the geometrical structures canonical or not needed for the hamiltonian formalism are introduced and compared and the derivation of hamiltonian field equations from the corresponding variational principle is shown in detail furthermore the hamiltonian formalism of systems described by lagrangians is performed both for the hyper regular and almost regular cases finally the role of connections in the construction of hamiltonian field theories is clarified

euler poincare reduction for discrete field theories in this note we develop a theory of euler poincare reduction for discrete lagrangian field theories we introduce the concept of euler poincare equations for discrete field theories as well as a natural extension of the moser veselov scheme and show that both are equivalent the resulting discrete field equations are interpreted in terms of discrete differential geometry an application to the theory of discrete harmonic mappings is also briefly discussed

k cosymplectic manifolds and lagrangian field theories a geometrical description of classical field theories of first order is given the underlying k cosymplectic structure permits to derive the corresponding field equations

on donovan s conjecture abstract this article contains reduction theorems for some weaker variants of donovan s conjecture which supposes that for every finite group d of prime order p there are only finitely many morita equivalence classes of p blocks of finite groups having defect groups isomorphic to d i when restricting the conjecture to the case of abelian defect groups it can be reduced to only considering p blocks of a class of groups that are close to being direct products of simple groups ii when restricting the conjecture to the case of principal blocks of finite groups having an abelian sylow p subgroup it can be reduced to only considering p blocks of simple groups a weaker version of donovan s conjecture would suppose that there are only finitely many cartan matrices of p blocks of finite groups with defect group isomorphic to a given one iii this conjecture can be reduced to only considering p blocks of quasisimple groups with centers having orders prime to p

reduction of polysymplectic manifolds the aim of this paper is to generalize the classical marsden weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which inherit the polysymplectic structure this generalization allows us to reduce polysymplectic hamiltonian systems with symmetries such as those appearing in certain kinds of classical field theories as an application of this technique an analogue to the kirillov kostant souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed our procedure corrects some mistakes and inaccuracies in previous papers gunther 1987 j differ geom 25 23 53 munteanu et al 2004 j math phys 45 1730 51 on this subject

glauberman watanabe corresponding p blocks of finite groups with normal defect groups are morita equivalent let g be a finite group and let a be a solvable finite group that acts on g such that the orders of g and a are relatively prime let b be a p block of g with normal defect group d such that a stabilizes b and d c g a then there is a morita equivalence between the block b and its watanabe correspondent block w b of c g a given by a bimodule m with vertex d and trivial source that on the character level induces the glauberman correspondence and which is an isotypy by a theorem of watanabe

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

symmetries in optimal control it is argued that the existence of symmetries may simplify as in classical mechanics the solution of optimal control problems a procedure for obtaining symmetries for the optimal hamiltonian resulting from the maximum principle is given this avoids the actual calculation of the optimal hamiltonian this procedure is based upon the notion of symmetry for the hamiltonian system with inputs and outputs associated with an optimal control problem

cosymplectic reduction for singular momentum maps in this paper we prove that the cosymplectic reduction of cosymplectic manifolds with symmetry due to c albert 1989 may be obtained from the marsden weinstein reduction theory we also study the reduction of cosymplectic manifolds with singular momentum map by using the results of sjamaar and lerman 1991 for the symplectic case

a note on perfect isometries between finite general linear and unitary groups at unitary primes abstract let q be a power of a prime l a prime not dividing q d a positive integer coprime to both l and the multiplicative order of q mod l and n a positive integer a watanabe proved that there is a perfect isometry between the principal l blocks of gl n q and gl n q d where the correspondence of characters is given by shintani descent in the same paper watanabe also proved that if l and q are odd and l does not divide gl n q 2 u n q then there is a perfect isometry between the principal l blocks of u n q and gl n q 2 with the correspondence of characters also given by shintani descent r kessar extended this first result to all unipotent blocks of gl n q and gl n q d in this paper we extend this second result to all unipotent blocks of u n q and gl n q 2 in particular this proves that any two unipotent blocks of u n q at unitary primes for possibly different n with the same weight are perfectly isometric we also prove that this perfect isometry commutes with deligne lusztig induction at the level of characters

gunther s formalism k symplectic formalism in classical field theory skinner rusk approach and the evolution operator the first aim of this paper is to extend the skinner rusk formalism on classical mechanics for first order field theories the second is to generalize the definition and properties of the evolution k operator on classical mechanics for first order field theories using in both cases gunther s formalism k symplectic formalism

morita equivalences related to the glauberman correspondence abstract starting with p interior algebras where p is a finite p group we prove two theorems establishing certain group graded morita equivalences these apply in particular to the case of blocks with normal defect groups and defect zero blocks of normal subgroups respectively

constraint field systems in multimomentum canonical variables to describe constraint field models we apply the multimomentum hamiltonian formalism where momenta correspond to derivatives of fields with respect to all world coordinates not only time if a lagrangian density is degenerate the euler lagrange equations are underdetermined and need additional gauge type conditions which remain elusive in general one gets these conditions automatically as a part of the hamilton equations but must consider a family of hamiltonian forms associated with the same lagrangian density in order to exhaust solutions of the euler lagrange equations the case of degenerate quadratic and affine lagrangian densities is elaborated as a result we get the universal procedure of describing constraint field systems

reduction in principal bundles covariant lagrange poincare equations a general reduction theory for field theoretical lagrangians on principal fiber bundles is presented the reduced variational problem as well as the associated equations are formulated the link between the solutions of the original problem and that of the reduced problem is discussed and an obstruction to the reconstruction of the solutions is isolated the important case of semidirect products is discussed in detail and several concrete examples are presented

hamiltonian systems on k cosymplectic manifolds the hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories to do this the notion of k cosymplectic manifold is introduced and a suitable hamiltonian formalism is developed so that the field equations for scalar and vector hamiltonian functions are derived

multi moment maps we introduce a notion of moment map adapted to actions of lie groups that preserve a closed three form we show existence of our multi moment maps in many circumstances including mild topological assumptions on the underlying manifold such maps are also shown to exist for all groups whose second and third lie algebra betti numbers are zero we show that these form a special class of solvable lie groups and provide a structural characterisation we provide many examples of multi moment maps for different geometries and use them to describe manifolds with holonomy contained in g2 preserved by a two torus symmetry in terms of tri symplectic geometry of four manifolds

on a kind of noether symmetries and conservation laws in k cosymplectic field theory this paper is devoted to studying symmetries of certain kinds of k cosymplectic hamiltonian systems in first order classical field theories thus we introduce a particular class of symmetries and study the problem of associating conservation laws to them by means of a suitable generalization of noether s theorem

perfect isometries for principal blocks with abelian defect groups and elementary abelian 2 inertial quotients abstract letbbe the principalp block of a finite groupgwith an abelian defect grouppandea root ofbincg p if the inertial quotiente ng p e p cg p is an elementary abelian 2 group respectively a dihedral group of order 8 andp 3 thenband its brauer correspondent considered as blocks ofgandng p are isotypic and in particular there is a perfect isometry between them

hyperkahler and quaternionic kahler geometry a quaternion hermitian manifold of dimension at least 12 with closed fundamental 4 form is shown to be quaternionic kahler a similar result is proved for 8 manifolds hyperkahler metrics are constructed on the fundamental quaternionic line bundle with the zero section removed of a quaternionic kahler manifold indefinite if the scalar curvature is negative this construction is compatible with the quaternionic kahler and hyperkahier quotient constructions and allows quaternionic kahler geometry to be subsumed into the theory of hyperkahler manifolds it is shown that the hyperkahler metrics that arise admit a certain type of su 2 action possess functions which are kahler potentials for each of the complex structures simultaneously and determine quaternionic kahler structures via a variant of the moment map construction quaternionic kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperkahler metrics with circle actions on complex line bundles over kahler einstein complex contact manifolds nilpotent orbits in a complex semi simple lie algebra with the hyperkahler metrics defined by kronheimer are shown to give rise to quaternionic kahler metrics and various examples of these metrics are identified it is shown that any quaternionic kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds the twistor space structure of the projectivised nilpotent orbits is studied

the polysymplectic hamiltonian formalism in field theory and calculus of variations i the local case on presente un formalisme hamiltonien geometrique invariant pour des problemes variationnels a integrale multiple et des theories de champ le formalisme est base sur la notion de forme polysymplectique qui est une generalisation a valeur vectorielle des formes symplectiques on etudie des equations hamiltoniennes des transformations canoniques des systemes de lagrange des symetries des applications en pulsion de theorie du champ et une classification des systemes de theorie du champ g homogenes sur une generalisation des orbites coadjointes

a rational function approximation of the singular eigenfunction of the monoenergetic neutron transport equation demonstrates how a proper rational fraction approximation to the singular eigenfunction of the neutron transport theory can be constructed based on the properties of generalised functions and singular integral equations the parameters of the approximant are determined by a proper use of the orthogonality integrals satisfied by the case eigenfunctions this ensures the convergence of the approximant to its exact singular distributional form use of lebesgue integrable spaces made in the analysis leads to a new possibility of approximating functions in lp 1 p infinity and also of finding approximate solutions of singular integral equations

on perfect isometries and isotypies in alternating groups perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the brauer correspondents of these blocks these perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended broue conjecture holds for the principal block of an almost simple group with an abelian sylow psubgroup and a generalized fitting subgroup isomorphic to an alternating group let g be a finite group and let 0 be a complete discrete valuation ring with field of quotients k of characteristic 0 and residue class field k of characteristic p 0 we suppose that k contains a primitive igi th root of unity in 4 6 1 michel broue posed the following isotypy conjecture let e be a block of og with abelian defect group d and let f be the brauer correspondent of e in 9ng d then e and f are isotypic blocks if g has an abelian sylow p subgroup then the conjecture can be posed for the principal block of 0g in this case the authors have shown that the conjecture holds provided an extended conjecture holds for the principal block of almost simple groups with an abelian sylow p subgroup see 10 5e in this paper the isotypy conjecture for an arbitrary block with abelian defect group is proved for alternating groups in addition the extended conjecture is proved for the principal p block of almost simple groups with abelian sylow p subgroups and generalized fitting subgroup isomorphic to an alternating group we recall the basic definitions let 0 k be the canonical quotient mapping and let og kg be the induced 0 algebra homomorphism of the group algebras in particular e e induces a bijection between central idempotents of og and kg if e is a block idempotent of og let kgemod be the category of left kge modules of finite type and let zk g e be the grothendieck group of kgemod let gv be the set of irreducible characters of g over k we identify r k g e with the free abelian group on g e v x e gv x ge x g for all g e g let cf g k be the k space of k valued class functions on g and let cf g e k be the k subspace of class functions ae in cf g k such that ao ge a g the received by the editors march 26 1996 1991 mathematics subject classification primary 20c15 20c20 secondary 20c30 the first author was supported in part by nsf grant dms 9100310 the second author was supported in part by nsa grant mda 904 92 h 3027 1997 american mathematical society

non standard connections in classical mechanics in the jet bundle description of first order classical field theories there are some elements such as the lagrangian energy and the construction of the hamiltonian formalism which require the prior choice of a connection bearing these facts in mind we analyse the situation in the jet bundle description of time dependent classical mechanics we prove that this connection dependence also occurs in this case although it is usually hidden by the use of the natural connection given by the trivial bundle structure of the phase spaces under consideration however we also prove that this dependence is dynamically irrelevant except where the dynamical variation of the energy is concerned in addition the relationship between first integrals and connections is shown for a sufficiently large class of lagrangians

generalised cartan invariants of symmetric groups kulshammer olsson and robinson developed an analogue of modular representation theory of symmetric groups where is not necessarily a prime they gave a conjectural combinatorial description for invariant factors of the cartan matrix in this context we confirm their conjecture by proving a more precise blockwise version of the conjecture due to bessenrodt and hill

the gunther s formalism in classical field theory momentum map and reduction the polysymplectic formalism in local field theory developed by gunther j diff geom 25 23 1987 is revised a new approach and new results on momentum maps and reduction are given

some geometric structures associated with a k symplectic manifold a canonical connection is attached to any k symplectic manifold we study the properties of this connection and its geometric applications to k symplectic manifolds in particular we prove that under some natural assumptions any k symplectic manifold admits an ehresmann connection discuss some corollaries of this result and find vanishing theorems for characteristic classes on a k symplectic manifold

k symplectic structures the basic properties of k symplectic structures are introduced and developed in analogy with the well known symplectic differential geometry examples of such structures related to k symplectic lie algebras are given

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

orthogonal polynomials and differential equations in neutron transport and radiative transfer theories abstract there is a set of orthogonal polynomials gn x which plays a relevant role in the treatment of the case of anisotropic scattering in neutron transport and radiative transfer theories they appear also in the spherical harmonics treatment of the isotropic scattering these polynomials are orthogonal with respect to a weight function which is continuous in the interval 1 1 and has a finite number of symmetric dirac masses although some other structural properties of these polynomials e g the three term recurrence relation as well as some properties of their zeros have been published much more need to be known in particular neither the second order differential equation nor the density of zeros i e the number of zeros per unit of interval of the polynomial gn x have been found here we obtain the second order differential equation in the case that these polynomials are hypergeometric so leaving open the general case furthermore the exact expressions of the moments around the origin of the density of zeros of gn x are given in the general case the asymptotic density of zeros is also pointed out finally these polynomials are shown to belong to the nevai s class

on the glauberman and watanabe correspondences for blocks of finite p solvable groups if g is a finite p solvable group for some prime p a a solvable subgroup of the automorphism group of g of order prime to g such that a stabilises a p block b of g and acts trivially on a defect group p of b then there is a morita equivalence between the block b and its watanabe correspondent w b of c g a given by a bimodule m with vertex p and an endo permutation module as source which on the character level induces the glauberman correspondence and which is an isotypy by watanabe s results

perfect isometries for blocks with abelian defect groups and klein four inertial quotients abstract let b be a p block of a finite group g with an abelian defect group p and e a root of b in cg p if the inertial quotient e ng p e cg p is a klein four group there is a so called perfect isometry from the group of generalized characters of a suitable twisted group algebra of the semidirect product of e and p onto the group of generalized characters of g in b

a new set of orthogonal polynomials for use in neutron transport theory the proper set of orthogonal polynomials to use in approximation work depends on the weight function of the problem the author introduces a new set of polynomials orthogonal with respect to a simple closed form approximate representation of the weight function of case s half range neutron transport theory 1967 the accuracy of this approximate weight function is shown to be very good and the construction procedure of the new set of orthogonal polynomials ci mu c or mu or 1 is demonstrated

perfect isometries between blocks of complex reflection groups acknowledgements the authors are grateful to m broue for asking the question this article settles at the end of a talk given by the first author at the beijing center for mathematical research during the third international symposium on groups algebras and related topics celebrating the 50th anniversary of the journal of algebra part of this work was done at the cirm in luminy during a research in pairs stay the authors wish to thank the cirm gratefully for their financial and logistical support the first author is supported by agence nationale de la recherche projet gerepmod anr 16 ce40 00010 01 the second author also acknowledges financial support from the engineering and physical sciences research council grant combinatorial representation theory ep m019292 1

on the brauer glauberman correspondence abstract the purpose of the present paper is two fold on the one hand to show the existence of a correspondence unifying brauer s and glauberman s ones see theorem 4 6 and on the other hand to give an alternative proof of watanabe s equivalence in atumi watanabe the glauberman character correspondence and perfect isometries for blocks of finite groups j algebra 216 1999 548 565 by the way we give a short proof of the coincidence of the clifford extensions associated with a pair of glauberman correspondent irreducible representations see corollary 4 16 a question that surprisingly enough has only been partially solved recently see morton harris markus linckelmann on the glauberman and watanabe correspondences for blocks of finite p solvable groups trans amer math soc 354 2002 3435 3453 and shigeo koshitani gerhard michler glauberman correspondence of p blocks of finite groups j algebra 243 2001 504 517

on the isotypy induced by the glauberman dade correspondence between blocks of finite groups abstract we show that under some conditions the character correspondence introduced by dade which partly generalizes the glauberman correspondence induces isotypy between blocks of finite groups

reduction of constrained mechanical systems and stability of relative equilibria a mechanical system with perfect constraints can be described under some mild assumptions as a constrained hamiltonian system m h d w m thephase space is a symplectic manifold h thehamiltonian a smooth function onm d theconstraint submanifold a submanifold ofm andw theprojection bundle a vector sub bundle oft d m the reduced tangent bundle alongd we prove that when these data satisfy some suitable conditions the time evolution of the system is governed by a well defined differential equation ond we define constrained hamiltonian systems with symmetry and prove a reduction theorem application of that theorem is illustrated on the example of a convex heavy body rolling without slipping on a horizontal plane two other simple examples show that constrained mechanical systems with symmetry may have an attractive or repulsive set of relative equilibria

k symplectic lie systems theory and applications abstract a lie system is a system of first order ordinary differential equations describing the integral curves of a t dependent vector field taking values in a finite dimensional real lie algebra of vector fields a so called vessiot guldberg lie algebra we suggest the definition of a particular class of lie systems the k symplectic lie systems admitting a vessiot guldberg lie algebra of hamiltonian vector fields with respect to the presymplectic forms of a k symplectic structure we devise new k symplectic geometric methods to study their superposition rules t independent constants of motion and general properties our results are illustrated through examples of physical and mathematical interest as a byproduct we find a new interesting setting of application of the k symplectic geometry systems of first order ordinary differential equations

on a discretised spectral approximation in neutron transport theory an approximation method significantly different from existing ones is developed for the one speed neutron transport equation the theory is based on the singular eigenfunction method and is founded on the new concepts of i a rational function approximation of the case singular eigenfunction and ii discretisation of the continuous spectrum 0 1 of the half range transport problem using the roots of a set of polynomials orthogonal in 0 1 with respect to a simple closed form accurate approximation of the half range weight function w mu a sample numerical example is given to illustrate the application of the theory

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

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

the isaacs navarro conjecture for the alternating groups abstract a recent refinement of the mckay conjecture is verified for the case of the alternating groups the argument builds upon the verification of the conjecture for the symmetric groups p fong the isaacs navarro conjecture for symmetric groups j algebra 250 1 2003 154 161

perverse equivalences and broueʼs conjecture abstract we give a new approach to the construction of derived equivalences between blocks of finite groups based on perverse equivalences in the setting of broue s abelian defect group conjecture we provide in particular local and global perversity data describing the principal blocks and the derived equivalences for a number of finite simple groups with sylow subgroups elementary abelian of order 9 we also examine extensions to automorphism groups in a general setting

extended hamiltonian systems in multisymplectic field theories we consider hamiltonian systems in first order multisymplectic field theories we review the properties of hamiltonian systems in the so called restricted multimomentum bundle including the variational principle which leads to the hamiltonian field equations in an analogous way to how these systems are defined in the so called extended symplectic formulation of nonautonomous mechanics we introduce hamiltonian systems in the extended multimomentum bundle the geometric properties of these systems are studied the hamiltonian equations are analyzed using integrable multivector fields the corresponding variational principle is also stated and the relation between the extended and the restricted hamiltonian systems is established all these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost regular field theories

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

cartan invariants and central ideals of group algebras abstract in this paper we investigate general properties of cartan invariants of a finite group g in characteristic 2 one of our results shows that the cartan matrix of g in characteristic 2 contains an odd diagonal entry if and only if g contains a real element of 2 defect zero we also apply these results to 2 blocks of symmetric groups and to blocks with normal or abelian defect groups the second part of the paper deals with annihilators of certain ideals in centers of group algebras and blocks

perfect isometries and murnaghan nakayama rules acknowledgement the authors wish to thank the referee for a very careful and precise reading of several earlier versions of this manuscript they are grateful for the number quality and helpfulness of comments and suggestions received

a transport theoretic approximation method for the monoenergetic neutron transport equation introduces the basic features of a proposed transport theoretic approximation of the one speed neutron transport equation the method utilises the exact results of case et al 1967 and retains the use of the eigenfunctions phi 0 or mu together with a properly constructed rational approximation phi nu mu of the singular eigenfunction phi nu mu as the complete set of basis functions a major contribution of this work is the adaption of case s orthogonality relations to this basis phi 0 or mu phi nu mu for obtaining the desired solution