high accuracy cubic spline finite difference approximation for the solution of one space dimensional non linear wave equations abstract in this paper we propose a new three level implicit nine point compact cubic spline finite difference formulation of order two in time and four in space directions based on cubic spline approximation in x direction and finite difference approximation in t direction for the numerical solution of one space dimensional second order non linear hyperbolic partial differential equations we describe the mathematical formulation procedure in details and also discuss how our formulation is able to handle wave equation in polar coordinates the proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable numerical results are provided to justify the usefulness of the proposed method

numerical study of fisher s reaction diffusion equation by the sinc collocation method fisher s equation which describes a balance between linear diffusion and nonlinear reaction or multiplication is studied numerically by the sinc collocation method the derivatives and integrals are replaced by the necessary matrices and a system of algebraic equations is obtained to approximate solution of the problem the error in the approximation of the solution is shown to converge at an exponential rate numerical examples are given to illustrate the accuracy and the implementation of the method the results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large both the limiting wave fronts and the limiting speed are independent of the initial values

classifying blocks with abelian defect groups of rank 3 for the prime 2 abstract in this paper we classify all blocks with defect group c 2 n c 2 c 2 up to morita equivalence together with a recent paper of wu zhang and zhou this completes the classification of morita equivalence classes of 2 blocks with abelian defect groups of rank at most 3 the classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field the case considered in this paper is significant because it involves comparison of morita equivalence classes between a group and a normal subgroup of index 2 so requires novel reduction techniques which we hope will be of wider interest we note that this also completes the classification of blocks with abelian defect groups of order dividing 16 up to morita equivalence a consequence is that broue s abelian defect group conjecture holds for all blocks mentioned above

algebras of quotients of graded lie algebras abstract in this paper we explore graded algebras of quotients of lie algebras with special emphasis on the 3 graded case and answer some natural questions concerning its relation to maximal jordan systems of quotients

strong primeness of hermitian jordan systems this paper establishes the strong primeness of all jordan systemsjof hermitian type trapped between ample hermitian elements of a prime associative systemrand its martindale system of symmetric quotientsq r h0 r j h q r this completes the converse of zelmanov s classification of strongly prime jordan systems providing if as well as only if classifications of strongly prime and primitive jordan systems

minimal ideals of jordan systems we show that minimal ideals of jordan systems algebras triple systems and pairs are either simple or trivial so answering the question posed by nam and mccrimmon in 1983

abelian sylow subgroups in a finite group ii abstract let p 3 5 be a prime we prove that sylow p subgroups of a finite group g are abelian if and only if the class sizes of the p elements of g are all coprime to p this gives a solution to a problem posed by r brauer in 1956 for p 3 5

trivial minimal ideals of jordan systems abstract we show that in jordan systems algebras triple systems and pairs monomials containing two elements of a trivial minimal ideal vanish so improving the answer given by anquela and cortes in j a anquela t cortes minimal ideals of jordan systems invent math 168 2007 83 90 to the problem of nam and mccrimmon in n s nam k mccrimmon minimal ideals in quadratic jordan algebras proc amer math soc 88 4 1983 579 583 inspired by the problem posed in 1971 by zhevlakov in the dniester notebook unsolved problems in the theory of rings and modules second ed 1976 see filippov et al 1993 5

high accuracy non polynomial spline in compression method for one space dimensional quasi linear hyperbolic equations with significant first order space derivative term abstract in this paper we propose a new three level implicit nine point compact non polynomial spline in compression finite difference method of order two in time and four in space directions based on non polynomial spline approximation in x direction and central difference approximation in t direction for the numerical solution of one space dimensional second order quasi linear hyperbolic partial differential equations with first order space derivative term we describe the mathematical details of the method and also discuss how our method is able to handle wave equation in polar coordinates the proposed method when applied to a linear hyperbolic equation is shown to be unconditionally stable numerical results are provided to justify the usefulness of the proposed method

the development version of the chevie package of gap3 abstract i describe the current state of the development version of the chevie package which allows to transform the theories of coxeter groups reductive algebraic groups complex reflection groups hecke algebras braid monoids etc into actual computations examples are given showing the code to check some results of lusztig

vers une decomposition de jordan des blocs des groupes reductifs finis abstract let g be a reductive algebraic group over an algebraic closure of a prime field f p defined over f q with frobenius endomorphism f let g f be the subgroup of rational points let l be a prime number assume that l is different from p if g f is in duality with g f then by a theorem of m broue and j michel m broue j michel blocs et series de lusztig dans un groupe reductif fini j reine angew math 395 1989 56 67 for any l bloc b of g f there exists a unique g f conjugacy class s of l semi simple elements such that some irreducible representation of b is in the rational lusztig s series associated to s if s 1 b is said to be unipotent if g is not connected with identity component g define the unipotent l blocs of g f as the l blocks that cover some unipotent l bloc of g f assuming l is good for g we construct from g f and s some reductive algebraic group g s f defined over f q and a one to one map from the set of l blocks of g f with associated class s onto the set of unipotent l blocks of g s f such that if b corresponds to b then there is a significant height preserving one to one map from the set of irreducible representations irr b onto the set irr b the respective defect groups of b and b are isomorphic the associated brauer categories are equivalent one do not assume that the center of g is connected the class of s may be isolated in g

local and subquotient inheritance of simplicity in jordan systems abstract in this paper we prove that the local algebras of a simple jordan pair are simple jordan pairs all of which local algebras are simple are also studied showing that they have a nonzero simple heart which is described in terms of powers of the original pair similar results are given for jordan triple systems and algebras finally we characterize the inner ideals of a simple pair which determine simple subquotients answering the question posed by o loos and e neher 1994 j algebra166 255 295

brauer s height conjecture for p solvable groups we complete the proof of the height conjecture for p solvable groups using the classification of finite simple groups introduction the height conjecture is the statement that a p block of a finite group has an abelian defect group if and only if all ordinary irreducible characters in the block have height zero while a proof of this conjecture for general finite groups seems remote considerable progress has been made toward proving it for p solvable groups fong 5 proved that all characters in a block with abelian defect group have height zero in a p solvable group and he proved the converse direction for the principal block 5 and for solvable groups in the case that p is the largest prime divisor of the group order 6 recently 24 8 the converse direction has been established for all solvable groups in this paper we prove the converse direction for all p solvable groups assuming the classification of finite simple groups in its general outline this paper resembles 8 where we proved the height conjecture for solvable groups the reader is assumed to have some familiarity with 8 now we state our main results the analogs of the main results of 8 theorem a suppose that n g that g n is p solvable that t e irr n and that p 1 x i w l for all x e irr g i p then the p sylow subgroups of g n are abelian theorem b let b be a p block of a p solvable group with defect group d if every ordinary irreducible character in b has height zero then d is abelian theorem c suppose that n g that g n is p solvable and that w e irr n suppose that e is an integer such that pe l does not divide x i w l for all x e irr g ip then the derived length of a p sylow subgroup of g n is at most 2e 1 theorem d let b and d be as in theorem b if every ordinary irreducible character in b has height at most e then the derived length of d is at most 2e 1 received by the editors january 10 1983 1980 mathematics subject classification primary 20c20 20c 15 1984 american mathematical society 0002 9947 84 1 00 25 per page

character tables and defect groups abstract let b be a block of a finite group g with defect group d we prove that the exponent of the center of d is determined by the character table of g in particular we show that d is cyclic if and only if b contains a large family of irreducible p conjugate characters more generally for abelian d we obtain an explicit formula for the exponent of d in terms of character values in small cases even the isomorphism type of d is determined in this situation moreover it can read off from the character table whether d d 4 where d denotes the commutator subgroup of d we also propose a new characterization of nilpotent blocks in terms of the character table

tension spline method for solution of non linear fisher equation tension spline method is proposed for the non linear fisher equation with initial boundary values the three time level implicit method based on the non polynomial cubic tension spline is developed for the solution of the non linear reaction diffusion equation the method involves the parameters and its order can be increased from o k 2 k 2 h 2 h 2 to o k 2 k 2 h 2 h 4 by an appropriate choice of the parameters the stability of proposed method is analyzed finally numerical results are presented to demonstrate the accuracy and efficiency of this method

on heights of characters of finite groups abstract in this paper we first state a conjecture on the lower bound of the maximal height of characters in a p block of a finite group then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group for all blocks of a quasi simple group g with g z g isomorphic to a 6 a 7 or a simple group of lie type with an exceptional covering group for all blocks of a symmetric group as well as for all blocks of finite general linear or unitary groups for the proof of the case with symmetric groups it relates to an open question of olsson on the existence of t core partitions where t 3 is an integer as a byproduct our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the isaacs moreto navarro tiep conjecture claiming that the number of distinct irreducible character degrees of a sylow p subgroup p of an arbitrary finite group g is at most one more than the number of irreducible character degrees of g that are multiples of p

conjectures of alperin and broue for 2 blocks with elementary abelian defect groups of order 8 using the classification of finite simple groups we prove alperin s weight conjecture and the character theoretic version of broue s abelian defect group conjecture for 2 blocks of finite groups with an elementary abelian defect group of order 8

blocks with frobenius inertial quotients abstract in the paper we prove that a perfect isometry between a block of a finite group and its brauer correspondent can be extended to a perfect isometry between a block of a larger finite group and its brauer correspondent with suitable hypotheses we also obtain a similar result for derived equivalences

brauer s height zero conjecture for quasi simple groups abstract we show that brauer s height zero conjecture holds for blocks of finite quasi simple groups this result is used in navarro spath s reduction of this conjecture for general groups to the inductive alperin mckay condition for simple groups

characters of relative p degree over normal subgroups let z be a normal subgroup of a finite group g let irr z be an irreducible complex character of z and let p be a prime number if p does not divide the integers 1 1 for all irr g lying over then we prove that the sylow p subgroups of g z are abelian this theorem which generalizes the gluck wolf theorem to arbitrary finite groups is one of the principal obstacles to proving the celebrated brauer height zero conjecture

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

the decomposition of lusztig induction in classical groups abstract we give a short combinatorial proof of asai s decomposition formula for lusztig induction of unipotent characters in groups of classical type relying solely on the mackey formula

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

simple endotrivial modules for quasi simple groups we investigate simple endotrivial modules of finite quasi simple groups and classify them in several important cases this is motivated by a recent result of robinson 41 showing that simple endotrivial modules of most groups come from quasi simple groups

brauer s height zero conjecture for two primes let p and q be two primes we propose that brauer s height zero conjecture for the principal p blocks of finite groups can naturally be extended from the perspective of q we prove one direction of this new conjecture and show the reverse direction assuming that the inductive alperin mckay condition holds for the finite simple groups

quadratic jordan systems of hermitian type abstract in this paper we extend zel manov s classification of linear jordan systems triple systems and pairs of hermitian type to quadratic jordan systems over an arbitrary ring of scalars using associative triple systems of the first kind for what we believe to be a more natural tool to describe special jordan triples we show that an i special prime quadratic jordan triple system the ideals of which remain semiprime with a nonzero hermitian part lies between an ample subspace of hermitian elements in a prime associative triple system and those in its martindale system of symmetric quotients modulo some extra definitions the structure of strongly prime jordan pairs of hermitian type follows almost immediately from the jordan triple result

on symmetric quotients of symmetric algebras we investigate symmetric quotient algebras of symmetric algebras with an emphasis on finite group algebras over a complete discrete valuation ring o using elementary methods we show that if an ordinary irreducible character of a finite group g gives rise to a symmetric quotient over o which is not a matrix algebra then the decomposition numbers of the row labelled by are all divisible by the characteristic p of the residue field of o

equivariant splendid rickard equivalences abstract broue s conjecture says that if a finite group h has an abelian sylow p subgroup p then the principal blocks of h and n h p are splendidly rickard equivalent in this paper broue s conjecture is reduced to simple groups with a stability condition a structure theorem on finite groups with abelian sylow p subgroups due to kimmerle and sandling is used

nilpotent and abelian hall subgroups in finite groups the research of the first second fourth fifth and sixth authors was supported by the prometeo generalitat valenciana proyectos mtm2010 15296 mtm2010 19938 c03 02 fundacio bancaixa p11b2010 47 and fondos feder the third author gratefully acknowledges financial support by erc advanced grant 291512 the seventh author was supported by the nsa grant h98230 13 1 0229 the eighth author gratefully acknowledges the support of the nsf grants dms 0901241 and dms 1201374

trivial intersection of blocks and nilpotent subgroups abstract let p q be different primes and suppose that the principal p and the principal q block of a finite group have only one irreducible complex character in common namely the trivial one we conjecture that this condition implies the existence of a nilpotent hall p q subgroup and prove that a minimal counter example must be an almost simple group where pq divides the order of its simple nonabelian normal subgroup as an immediate consequence we obtain that the conjecture holds true for p solvable or q solvable groups furthermore we prove the conjecture in case 2 p q using the classification theorem of finite simple groups finally we consider the situation that the intersection of an arbitrary p block with an arbitrary q block contains only one irreducible character

quasi isolated blocks and brauer s height zero conjecture this paper has two main results firstly we complete the parametrisation of all p blocks of nite quasi simple groups by nding the so called quasi isolated blocks of exceptional groups of lie type for bad primes this relies on the explicit decomposition of lusztig induction from suitable levi subgroups our second major result is the proof of one direction of brauer s long standing height zero conjecture on blocks of nite groups using the

grassmann speciality of jordan supersystems abstract the grassmann envelope s s 0 0 s 1 1 is a useful tool for deciding varietal questions about supersystems s s 0 s 1 it is not useful in deciding simplicity since the envelope is always fraught with trivial ideals coming from trivial superscalars at first glance it would not seem useful in deciding superspeciality either but we will show that it is a more sensitive tool than it seems we say a jordan supersystem js algebra triple or pair is grassmann special if its grassmann envelope is special as an ordinary jordan system over 0 certainly superspeciality js as over implies grassmann speciality js as over 0 but it is not obvious how grassmann speciality influences superspeciality of js nevertheless we show how to transform obstacles to speciality of js over anti superspecial elements into obstacles to speciality of js over 0 anti special elements using grassmann boosters so that non superspeciality js over implies non speciality of js over 0 thus grassmann speciality is the same as superspeciality a supersystem j is a superspecial jordan supersystem over iff js is a special jordan system over 0 it is not clear this is the same as speciality of js over since in general speciality of quadratic jordan systems depends on the scalars we give an example of a jordan algebra which is not special but is special over a scalar subring as a corollary any jordan superalgebra with zero linear extreme radical which is viably evenly 4 interconnected is superspecial thus exceptional superalgebras must be built over even parts of degree 3

derived categories and deligne lusztig varieties ii this paper is a continuation and a completion of boro1 we extend the jordan decomposition of blocks we show that blocks of finite groups of lie type in non describing characteristic are morita equivalent to blocks of subgroups associated to isolated elements of the dual group the key new result is the invariance of the part of the cohomology in a given modular series of deligne lusztig varieties associated to a given levi subgroup under certain variations of parabolic subgroups we also show that the equivalence arises from a splendid rickard equivalence even in the setting of boro1 the finer homotopy equivalence was unknown as a consequence the equivalence preserves defect groups and categories of subpairs we finally determine when deligne lusztig induced representations of tori generate the derived category of representations

tits kantor koecher algebras of strongly prime hermitian jordan pairs abstract in this paper we describe the tits kantor koecher algebras of strongly prime hermitian jordan pairs in terms of prime associative pairs and their martindale pairs of symmetric quotients

local pi theory of jordan systems ii abstract in this paper we study jordan systems having nonzero local algebras that satisfy a polynomial identity we prove that in nondegenerate jordan systems the set of elements at which the local algebra is pi is an ideal and that if it is nonzero it coincides with the socle when the system is primitive

sylow numbers and nilpotent hall subgroups let be a set of primes and g a finite group we characterize the existence of a nilpotent hall subgroup of g in terms of the number of sylow subgroups for the primes in

2 blocks with abelian defect groups we give a classification up to morita equivalence of 2 blocks of quasi simple groups with abelian defect groups as a consequence we show that donovan s conjecture holds for elementary abelian 2 groups and that the entries of the cartan matrices are bounded in terms of the defect for arbitrary abelian 2 groups we also show that a block with defect groups of the form c2m c2m for m 2 has one of two morita equivalence types and hence is morita equivalent to the brauer correspondent block of the normaliser of a defect group this completes the analysis of the morita equivalence types of 2 blocks with abelian defect groups of rank 2 from which we conclude that donovan s conjecture holds for such 2 groups a further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16 the proof uses the classification of finite simple groups

on the zelmanovian classification of prime jb triples complex jb triples are complex banach spaces endowed with a triple product subjected to suitable conditions of algebraic and analytic nature see section 1 for the definition they were introduced by w kaup in order to provide an algebraic setting for the study of bounded symmetric domains in complex banach spaces the open unit ball of every complex jb triple is a bounded symmetric domain ka1 and every bounded symmetric domain in any complex banach space is bi holomorphically equivalent to the open unit ball of a suitable complex jb triple ka3 fundamental examples of complex jb triples are provided by complex c algebras with triple product defined by xyz x 1 2 xy z zy x 1

blocks with transitive fusion systems suppose that all nontrivial subsections of a p block b are conjugate where p is a prime by using the classification of the finite simple groups we prove that the defect groups of b are either extraspecial of order p 3 with p in 3 5 or elementary abelian

composition factors from the table of marks let b g be the burnside ring for a finite group g and let t g be the table of marks of g it is known that in general neither b g nor t g determines the finite group g that is one can easily construct pairs of non isomorphic finite groups such that their table of marks is the same hence their burnside rings are isomorphic in this note we show that the composition factors of g are determined by the table of marks t g

on the brauer feit bound for abelian defect groups we improve the brauer feit bound on the number of irreducible characters in a p block for abelian defect groups by making use of halasi and podoski every coprime linear group admits a base of size two http arxiv org abs 1212 0199v1 7 and kessar and malle ann math 178 2 321 384 11 we also prove brauer s k b conjecture for 2 blocks with abelian defect groups of rank at most 5 and 3 blocks and 5 blocks with abelian defect groups of rank at most 3

equivariance and extendibility in finite reductive groups with connected center we show that several character correspondences for finite reductive groups g are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to g has connected center the correspondences we consider are the so called jordan decomposition of characters introduced by lusztig and the generalized harish chandra theory of unipotent characters due to broue malle michel in addition we consider a correspondence giving character extensions due to the second author in order to verify the inductive mckay condition from isaacs malle navarro for the non abelian finite simple groups of lie types 3 mathsf d 4 mathsf e 8 mathsf f 4 2 mathsf f 4 and mathsf g 2

the structure of strongly prime quadratic jordan algebras abstract we extend the structure theory for strongly prime jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2 we show that such j are either 1 hermitian forms h 0 a j h q a lying between an ample subspace of hermitian elements in a prime associative algebra a and those in its martindale ring of quotients q a 2 clifford forms with a scalar extension j j which is an algebra j j q 1 of a nondegenerate quadratic from q with basepoint 1 or 3 albert forms with a scalar extension j j which is a 27 dimensional split albert algebra j h 3 k of 3 3 hermitian matrices over an 8 dimensional split octonion algebra k as a consequence the simple jordan algebras of arbitrary dimension are either hermitian h 0 a for simple a or clifford forms in j q 1 or are albert algebras in particular our methods give an idempotent free classification of all simple finite dimensional algebras which are i special the key to this characterization is the existence of zel manov polynomials in the free special jordan algebra which are both hermitian polynomials whose values always look like hermitian elements and at the same time clifford polynomials nontrivial on 3 3 hermitian matrices hence on special algebras with 3 interconnected idempotents if a zel manov polynomial does not vanish on j then j is of hermitian type and if it does vanish on j then j is of clifford type one example of such a polynomial is q48 p16 x1 y1 z1 w1 p16 x2 y2 z2 w2 p16 x3 y3 z3 w3 for p16 d2x y z 2 dx y w dx y w dx y z x y z thus the structure of jordan algebras depends very directly on polynomial identities and commutators

sp6 2a is good for the mckay alperin weight and related local global conjectures abstract the so called local global conjectures in the representation theory of finite groups relate the representation theory of g to that of certain proper subgroups such as the normalizers of particular p groups recent results by several authors reduce some of these conjectures to showing that a certain collection of stronger conditions holds for all finite simple groups here we show that g sp 6 2 a is good for these reductions for the mckay conjecture the alperin weight conjecture and their blockwise versions

herstein s theorems and simplicity of hermitian jordan systems abstract in this paper we extend herstein s first construction relating associative and jordan ideals to pairs and triple systems as a consequence we show that an associative pair or triple system is simple if and only if its jordan symmetrization is simple we also generalize herstein s second construction to ample subsystems of associative algebras pairs and triple systems which provides information on their simplicity when the associative structure is simple

zel manov polynomials in quadratic jordan triple systems abstract zel manov polynomials are elements of a free special jordan system which are both hermitian their values look like hermitian elements and clifford they do not vanish on systems t containing h 3 h m 3 t these polynomials decide the classification of strongly prime special jordan systems t is either clifford or hermitian according as some zel manov polynomial does or does not vanish on t the existence of such polynomials has been established by mccrimmon and zel manov for quadratic jordan algebras and by zel manov for linear jordan triple systems and pairs in this paper we carry out the construction for quadratic jordan triple systems and pairs

associative systems of left quotients abstract in this paper we construct the maximal left quotient system of every pair of modules in some generalized matrix rings these rings can be seen as 3 graded algebras or as superalgebras we show the relation among the three different notions of left quotients

polynomial identities and nonidentities of split jordan pairs abstract we show that split jordan pairs over rings without 2 torsion can be distinguished by polynomial identities with integer coefficients in particular this holds for simple finite dimensional jordan pairs over algebraically closed fields of characteristic not 2 we also generalize results of drensky and racine and of rached and racine on polynomial identities of respectively jordan algebras and jordan triple systems

confirmation for wielandt s conjecture abstract let be a set of primes by h wielandt s definition sylow theorem holds for a finite group g if all maximal subgroups of g are conjugate in the paper the following statement is proven assume that is a union of disjoint subsets and and a finite group g possesses a hall subgroup which is a direct product of a subgroup and a subgroup furthermore assume that both the sylow theorem and theorem hold for g then the sylow theorem holds for g this result confirms a conjecture posed by h wielandt in 1959

rationality of finite group representations and local subgroups abstract we show that local fields of values and local schur indices of irreducible characters in a block of a finite group are controlled by local subgroups a method for computing certain schur indices and an application for the prime 2 are given

on restriction of characters to defect groups abstract we put forward a blockwise version of a recent conjecture of 6 on finite groups let b be a p block of a finite group g with defect group d let irr b be a character with positive height in this note we conjecture that the number of distinct linear constituents of the restriction d is 0 or at least p we prove that this is indeed the case for various classes of finite groups and p blocks

primitivity in jordan systems is ubiquitous abstract the main result of the paper asserts that if a jordan pair v v v is primitive at some 0 b v then it is primitive at any 0 b v also if a jordan triple system t is primitive at some 0 b t then it is primitive at any 0 b t as a tool similar results concerning one sided primitivity and primitivity of associative pairs and triple systems are established equivalences between our definitions of primitivity and some other appearing in the literature are also obtained as a consequence

computational proof of the mackey formula for q 2 let g be a connected reductive group defined over a finite field with q elements we prove that the mackey formula for the lusztig induction and restriction holds in g whenever q 2 or g does not have a component of type e

on the morita frobenius numbers of blocks of finite reductive groups abstract we show that the morita frobenius number of the blocks of the alternating groups the finite groups of lie type in defining characteristic and the ree and suzuki groups is 1 we also show that the morita frobenius number of almost all of the unipotent blocks of the finite groups of lie type in non defining characteristic is 1 and that in the remaining cases it is at most 2

a characterization of saturated fusion systems over abelian 2 groups abstract given a saturated fusion system f over a 2 group s we prove that s is abelian provided any element of s is f conjugate to an element of z s this generalizes a theorem of camina herzog leading to a significant simplification of its proof more importantly it follows that any 2 block b of a finite group has abelian defect groups if all b subsections are major furthermore every 2 block with a symmetric stable center has abelian defect groups

numerical solution of the sine gordon equation two new difference schemes are proposed for the sine gordon equation they have the advantage that there is a discrete energy which is conserved their stability and convergence is proved also we show some numerical computations using one of these schemes

large varvec p prime orbits for varvec p p nilpotent linear groups let g be a p nilpotent linear group on a finite vector space v of characteristic p suppose that g v is odd let p be a sylow p subgroup of g we show that there exist vectors v 1 and v 2 in v such that c g v 1 cap c g v 2 p a striking conjecture of malle and navarro offers a simple global criterion for the nilpotence in the sense of broue and puig of a p block of a finite group our result implies that this conjecture holds for groups of odd order

on unipotent blocks and their ordinary characters there is a great deal of evidence suggesting that block theory for finite reductive groups should lead to an enrichment of the harish chandra theory for those groups see s 5 1 fs2 p2 3 4 h 6 1 10 an e generalized harish chandra theory for e a positive integer is introduced in bmm i a the context is as follows let g be a connected reductive group defined over lfq with associated frobenius endomorphism f let e 1 an integer when l is an f stable levi subgroup of g one denotes by r and re the lusztig functor and its adjoint a unipotent e pair l l is defined by an e split levi subgroup l of g and a unipotent character z d l 1 one writes li z1 l2 j 2 if and only if l1 c l 2 and zl l2 l 1 2 l 4 0 see 3 l the notion of unipotent e cuspidal pairs and the resulting partition of g g v 1 are defined as in the classical case corresponding to e 1 if f is a prime not dividing q and e is the order of q mod f it seems natural in connection with blocks to ask if the partition of the unipotent characters 8 g v l resulting from e theory coincides with f blocks the blocks of g v which are defined by unipotent characters are called unipoteut f blocks we prove

2 blocks whose defect group is homocyclic and whose inertial quotient contains a singer cycle abstract we consider a block b of a finite group with defect group d c 2 m n and inertial quotient e containing a singer cycle an element of order 2 n 1 this implies e e f where e c 2 n 1 f c n and e acts transitively on the elements in d of order 2 we classify the basic morita equivalence classes of b over a complete discrete valuation ring o when m 1 b is basic morita equivalent to the principal block of one of s l 2 2 n f d e or j 1 where j 1 occurs only when n 3 when m 1 b is basic morita equivalent to d e