brauer characters and rationality a finite group g has no non trivial rational valued irreducible p brauer characters if and only if g has no non trivial rational elements of order not divisible by p

symplectic groups symplectic spreads codes and unimodular lattices abstract it is known that the symplectic group sp 2 n p has two complex conjugate irreducible representations of degree p n 1 2 realized over q p provided that p 3 mod 4 in the paper we give an explicit construction of an odd unimodular sp 2 n p 2 invariant lattice p n in dimension p n 1 for any p n 3 mod 4 such a lattice has been constructed by r bacher and b b venkov in the case p n 27 a second main result says that these lattices are essentially unique we show that for n 3 the minimum of p n is at least p 1 2 and at most p n 1 2 the interrelation between these lattices symplectic spreads of f p 2 n and self dual codes over f p is also investigated in particular using new results of u dempwolff and l bader w m kantor and g lunardon we come to three extremal self dual ternary codes of length 28

generators and relations for the unitary group of a skew hermitian form over a local ring abstract let s be an involutive local ring and let u 2 m s be the unitary group associated to a nondegenerate skew hermitian form defined on a free s module of rank 2m a presentation of u 2 m s is given in terms of bruhat generators and their relations this presentation is used to construct an explicit weil representation of the symplectic group s p 2 m r when s r is commutative and is the identity when s is commutative but is arbitrary with fixed ring r an elementary proof that the special unitary group s u 2 m s is generated by unitary transvections is given this is used to prove that the reduction homomorphisms s u 2 m s s u 2 m s and u 2 m s u 2 m s are surjective for any factor ring s of s the corresponding results for the symplectic group s p 2 m r are obtained as corollaries when is the identity

hermitian and skew hermitian forms over local rings abstract we investigate the structure of possibly degenerate e hermitian forms over local rings we prove classification theorems in the cases where the ring is complete and either the form is nondegenerate or the ring is a discrete valuation ring in the latter case we describe a complete set of invariants for such forms based on a generalisation of the classical notion of the radical of the form

quasiprimitive linear groups with quadratic elements abstract a semisimple complex linear transformation with exactly two distinct eigenvalues is called a quadratic element in this paper the finite irreducible linear groups generated by quadratic elements of order 3 for which the multiplicities of the eigenvalues are distinct are determined this is used to obtain bounds on the degree of an irreducible primitive linear group containing such elements there are no irreducible primitive linear groups containing quadratic elements of order 4 with eigenvalues 1 i with distinct multiplicities where i is a primitive fourth root of 1

a relaxed deteriorated pss preconditioner for nonsymmetric saddle point problems from the steady navier stokes equation for nonsymmetric saddle point problems arising from the steady navier stokes equations pan ng and bai presented a deteriorated positive definite and skew hermitian splitting dpss preconditioner pan et al 2006 to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the unconditional convergence property of the dpss iteration method is proved and a relaxed dpss rdpss preconditioner is proposed the rdpss preconditioner is much closer to the coefficient matrix than the dpss preconditioner in certain norm which straightforwardly results in an rdpss iteration method the convergence conditions of the rdpss iteration are analyzed and the optimal parameter which minimizes the spectral radius of the rdpss iteration matrix is derived using the rdpss preconditioner to accelerate some krylov subspace methods like gmres is also studied the eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the rdpss preconditioner

a relaxed dimensional factorization preconditioner for the incompressible navier stokes equations in this paper we introduce a relaxed dimensional factorization rdf preconditioner for saddle point problems properties of the preconditioned matrix are analyzed and compared with those of the closely related dimensional splitting ds preconditioner recently introduced by benzi and guo 7 numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the rdf preconditioner with respect to both mesh size and viscosity

the structure of rank 3 permutation modules for o2n 2 and um 2 acting on nonsingular points abstract we study the odd characteristic structure of permutation modules for the rank 3 natural actions of o 2 n 2 n 3 and u m 2 m 4 on nonsingular points of their standard modules

a reducible weil representation of sp 4 realized by differential operators in the space of smooth functions on h2 s1 this work presents a novel way to obtain the associated romanovski functions rn m x with n m in the three separate regions in terms of n and m we obtain the raising and lowering relations with respect to the both indices simultaneously in the three regions then a reducible weil representation of the real lie algebra sp 4 is realized in the space of complex valued smooth functions on h2 s1 by differential forms for the cartan weyl basis its invariant subspace is the second rare instance of the highest weight irreducible representation of sp 4 all whose weight spaces are one dimensional

cross characteristic representations of odd characteristic symplectic groups and unitary groups in ls landazuri and seitz gave lower bounds for irreducible representations of chevalley groups in non defining characteristic when referring to irreducible representations for quasisimple groups g we will assume that the modules are nontrivial on f g see also sz gpps ho for some improvements on these bounds these results have proved to be useful in many applications in particular they have been used to classify the maximal subgroups of classical groups containing an element of prime order acting irreducibly on a subspace of large dimension cf gpps and to show that low dimensional modules in characteristic p for groups with no normal p subgroup are semisimple see gu it is also important to identify the modules which have dimension close to the smallest possible dimension and to prove that there are no irreducible modules with dimension in a certain range above it this was done in gpps and gt for sln q further improvements were obtained by brundan and kleshchev brk hiss and malle hm have obtained results similar to gt for unitary groups in this paper we consider the groups g sp2n q with n 2 and q p odd and g sun q with n 3 landazuri and seitz ls had already shown that the minimal dimension d of any irreducible module in the nondefining characteristic has dimension exactly q 1 2 for the symplectic case and q 1 q 1 for the unitary case it was proved in gpps that aside from some small exceptions every irreducible kg module in a nondefining characteristic has dimension d d 1 or at least dimension 2d in characteristic 0 tiep and zalesskii tz1 using deligne lusztig

some characterizations of the weil representations of the symplectic and unitary groups abstract let g be a finite symplectic or unitary group we characterize the weil representations of g via their restriction to a standard subgroup then we complete the determination of complex representations of g with specific minimal polynomials of certain elements by showing that they coincide with the weil representations

hall higman type theorems for semisimple elements of finite classical groups minimum polynomials of semisimple elements of prime power order p a of finite classical groups in nontrivial irreducible cross characteristic representations are studied in particular an analogue of the hall higman theorem is established which shows that the degree of such a polynomial is at least p a 1 p 1 with a few explicit exceptions

a modified dimensional split preconditioner for generalized saddle point problems abstract for large and sparse saddle point linear systems arising from 2d linearized navier stokes equations benzi and guo recently studied a dimensional split ds preconditioner appl numer math 61 2011 66 76 by further applying it to generalized saddle point problems in this paper we present a modified dimensional split mds preconditioner this new preconditioner is based on a splitting of the generalized saddle point matrix resulting in an unconditional convergent fixed point iteration the basic iteration is accelerated by a krylov subspace method like restarted gmres the implementation of the mds preconditioner is discussed and a similar case is also analyzed finally numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the mds preconditioner

weil representations via abstract data and heisenberg groups a comparison abstract let b be a ring not necessarily commutative having an involution and let u 2 m b be the unitary group of rank 2m associated to a hermitian or skew hermitian form relative to when b is finite we construct a weil representation of u 2 m b via heisenberg groups and find its explicit matrix form on the bruhat elements as a consequence we derive information on generalized gauss sums on the other hand there is an axiomatic method to define a weil representation of u 2 m b and we compare the two weil representations thus obtained under fairly general hypotheses when b is local not necessarily finite we compute the index of the subgroup of u 2 m b generated by its bruhat elements besides the independent interest this subgroup and index are involved in the foregoing comparison of weil representations

the spectral properties of the hermitian and skew hermitian splitting preconditioner for generalized saddle point problems in this paper we consider the hermitian and skew hermitian splitting hss preconditioner for generalized saddle point problems with nonzero 2 2 blocks the spectral property of the preconditioned matrix is studied in detail under certain conditions all eigenvalues of the preconditioned matrix with the original system being non hermitian will form two tight clusters one is near 0 0 and the other is near 2 0 as the iteration parameter approaches to zero from above so do all eigenvalues of the preconditioned matrix with the original system being hermitian numerical experiments are given to demonstrate the results

a relaxed positive definite and skew hermitian splitting preconditioner for saddle point problems for saddle point linear system whose coefficient matrix is non hermitian of positive definite hermitian part pan et al 2006 proposed a positive definite and skew hermitian splitting pss preconditioner in this paper based on the pss preconditioner we present a relaxed positive definite and skew hermitian splitting rpss preconditioner we analyze the spectral properties of the new preconditioned matrix and conclude that the rpss preconditioner has much better properties than pss numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the new preconditioner

the weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring we find all irreducible constituents of the weil representation of a unitary group um a of rank m associated to a ramified quadratic extension a of a finite commutative local and principal ring r of odd characteristic we show that this weil representation is multiplicity free with monomial ir reducible constituents we also find the number of these constituents and describe them in terms of clifford theory with respect to a congruence sub group we find all character degrees in the special case when r is a field

a practical formula for computing optimal parameters in the hss iteration methods in the hss iteration methods proposed by bai golub and ng z z bai g h golub m k ng hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems siam j matrix anal appl 24 2003 603 626 the determination of the optimal parameter is a tough task when solving a non hermitian positive definite linear system in this paper a new and simple strategy for obtaining the optimal parameter is proposed which computes the optimal parameter by solving a cubic polynomial equation the coefficients of this polynomial are determined by several traces of some matrices related to the symmetric and skew symmetric parts of the coefficient matrix of the real linear system numerical experiments show that our new strategy is very effective for approximating the optimal parameter in the hss iteration methods as it leads to fast convergence of the method

a class of constraint preconditioners for nonsymmetric saddle point matrices we consider the use of a class of constraint preconditioners for the application of the krylov subspace iterative method to the solution of large nonsymmetric indefinite linear systems the eigensolution distribution of the preconditioned matrix is determined and the convergence behavior of a krylov subspace method such as gmres is described the choices of the parameter matrices and the implementation of the preconditioning step are discussed numerical experiments are presented

surjective word maps and burnside s p aq b theorem we prove surjectivity of certain word maps on finite non abelian simple groups more precisely we prove the following if n is a product of two prime powers then the word map x y mapsto x ny n is surjective on every finite non abelian simple group if n is an odd integer then the word map x y z mapsto x ny nz n is surjective on every finite quasisimple group these generalize classical theorems of burnside and feit thompson we also prove asymptotic results about the surjectivity of the word map x y mapsto x ny n that depend on the number of prime factors of the integer n

even unimodular lattices associated with the weil representation of the finite symplectic group soit q une puissance d un nombre premier impair p et soit g le groupe symplectique sp 2n q quand n est pair et q p est un nombre premier avec p 3 mod 4 il existe un treillis integral rationnel g invariant l dans m qui supporte une forme unimodulaire definie positive symetrique paire m est un qg module et q le corps des rationnels

a relaxed upper and lower triangular splitting preconditioner for the linearized navier stokes equation abstract based on the block triangular splitting of the original coefficient matrix a relaxed upper and lower triangular splitting rult preconditioner for the linearized incompressible navier stokes equations is considered the convergence analysis of the rult iteration method is presented the spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed respectively then the quasi optimal parameters of the rult preconditioner are derived finally some numerical results are carried out to show the effectiveness of the rult preconditioner and verify that the rult preconditioner outperforms the mds grs mrs and rbts preconditioners for the gmres method for solving the linearized incompressible navier stokes equation in terms of the number of iterations and computation times

a positivity preserving inexact noda iteration for computing the smallest eigenpair of a large irreducible m m matrix in this paper based on the noda iteration we present inexact noda iterations ini to find the smallest eigenvalue and the associated positive eigenvector of a large irreducible nonsingular m m matrix the positivity of approximations is critical in applications and if the approximations lose the positivity then they may be meaningless and could not be interpreted we propose two different inner tolerance strategies for solving the inner linear systems involved and prove that the convergence of resulting ini algorithms is globally linear and superlinear with the convergence order frac 1 sqrt 5 2 1 52 respectively the proposed ini algorithms are structure preserving and maintains the positivity of approximate eigenvectors we also revisit the exact noda iteration and establish a new quadratic convergence result all the above is first done for the problem of computing the perron root and the positive perron vector of an irreducible nonnegative matrix and is then adapted to computing the smallest eigenpair of the irreducible nonsingular m m matrix numerical examples illustrate that the proposed ini algorithms are practical and they always preserve the positivity of approximate eigenvectors we compare them with the jacobi davidson method the implicitly restarted arnoldi method and the explicitly restarted krylov schur method all of which cannot guarantee the positivity of approximate eigenvectors and illustrate that the overall efficiency of the ini algorithms is competitive with and can be considerably higher than the latter three methods

character degrees and random walks in finite groups of lie type for a finite group h let irr h denote the set of irreducible characters of h and define the zeta function zeta h t sum chi in irr h chi 1 t for real t 0 we study the asymptotic behaviour of zeta h t for finite simple groups h of lie type and also of a corresponding zeta function defined in terms of conjugacy classes applications are given to the study of random walks on simple groups and on base sizes of primitive permutation groups

group representations and lattices this paper began as an attempt to understand the euclidean lattices that were recently constructed using the theory of elliptic curves by elkies e and shioda shl sh2 from the point of view of group representations the main idea appears in a note of thompson th2 if one makes strong irreducibility hypotheses on a rational representation v of a finite group g then the g stable euclidean lattices in v are severely restricted unfortunately these hypotheses are rarely satisfied when v is absolutely irreducible over q but there are many examples where the ring endg v is an imaginary quadratic field or a definite quaternion algebra these representations allow us to construct some of the mordell weil lattices considered by elkies and shioda as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves in 1 we discuss lattices and hermitian forms on y7 and in 2 4 the strong irreducibility hypotheses we wish to make in 5 we show how our hypotheses imply the existence of a finite number up to isomorphism of euclidean z g lattices l in v with endg l a maximal order in endg v we give some examples with dim v 8 in 6 and in 7 9 discuss the invariants of l such as the dual lattice and theta function the rest of the paper is devoted to examples in most g is a finite group of lie type and v is obtained as an irreducible summand of the weil representation of g some of the representation theoretic problems left open by this paper are to find all examples of pairs g v satisfying the strong irreducibility hypotheses of 2 4 and to determine the invariants shortest nonzero vector theta function thompson series of the g lattices l so effortlessly constructed inside v

degrees of rational characters of finite groups abstract a classical theorem of john thompson on character degrees states that if the degree of any complex irreducible character of a finite group g is 1 or divisible by a prime p then g has a normal p complement in this paper we consider fields of values of characters and prove some improvements of this result

on parameterized block triangular preconditioners for generalized saddle point problems in this paper we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems these preconditioners generalize the common block diagonal and triangular preconditioners we will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues numerical experiments of a model stokes problem are presented

preconditioned conjugate gradient method for generalized least squares problems abstract a variant of the preconditioned conjugate gradient method to solve generalized least squares problems is presented if the problem is min ax b t w 1 ax b with a r m n and w r m m symmetric and positive definite the method needs only a preconditioner a 1 r n n but not the inverse of matrix w or of any of its submatrices freund s comparison result for regular least squares problems is extended to generalized least squares problems an error bound is also given

motivations and realizations of krylov subspace methods for large sparse linear systems we briefly introduce typical and important direct and iterative methods for solving systems of linear equations concretely describe their fundamental characteristics in viewpoints of both theory and applications and clearly clarify the substantial differences among these methods in particular the motivations of searching the solution of a linear system in a krylov subspace are described and the algorithmic realizations of the generalized minimal residual gmres method are shown and several classes of state of the art algebraic preconditioners are briefly reviewed all this is useful for correctly deeply and completely understanding the application scopes theoretical properties and numerical behaviors of these methods and is also helpful in designing new methods for solving systems of linear equations

two new variants of the hss preconditioner for regularized saddle point problems two new preconditioners which can be viewed as improved variants of the hermitian and skew hermitian splitting hss preconditioner are presented for regularized saddle point problems the unconditionally convergent properties of the corresponding iteration methods are deduced and the selection of optimal iteration parameters resulting in fast convergence for the two iteration methods are discussed moreover eigenvalue bounds of the preconditioned matrices and upper bounds of the degree of their minimal polynomials are obtained compared with the hss preconditioner they are much better approximations to the coefficient matrix of the saddle point problem and they have better convergence properties and spectrum distributions numerical experiments from the stokes problem are presented which illustrates the advantages of the two preconditioners over the hss preconditioner and some other preconditioners to accelerate the convergence rate of gmres

sign preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second order elliptic operators this paper is concerned with the p1 finite element approximation of the eigenvalue problem of second order elliptic operators subject to the dirichlet boundary condition the focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem it is shown that when the stiffness matrix is an irreducible m matrix the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain mesh conditions leading to such a stiffness matrix are also studied a sufficient condition is that the mesh is simplicial interiorly connected and acute when measured in the metric specified by the inverse of the diffusion matrix the acute requirement can be replaced by the delaunay condition in two dimensions numerical results show that when the stiffness matrix is not an m matrix a finite element approximation can be structurally different from the continuous eigenvalue problem the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue in modulus can even be complex for the case with nonsymmetric operators

inexact generalized noda iterations for generalized eigenproblems abstract in this paper we present an inexact generalized noda iteration igni for finding the perron pair of the generalized eigenproblem arising from economic models we prove that the approximate eigenvalue sequence generated by igni converges globally linearly furthermore we also propose an enhanced inexact generalized noda iteration eigni and prove that the approximate eigenvector sequence generated by eigni algorithms converges superlinearly numerical examples illustrate that the proposed igni and eigni algorithms are efficient

a generalized relaxed positive definite and skew hermitian splitting preconditioner for non hermitian saddle point problems for non hermitian saddle point problems with the non hermitian positive definite 1 1 block zhang et al 2014 presented a relaxed positive definite and skew hermitian splitting rpss preconditioner to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the convergence property of the grpss iteration method is proved and a generalized rpss grpss preconditioner is proposed the grpss preconditioner is much closer to the coefficient matrix than the rpss preconditioner in certain norm which straightforwardly results in an grpss iteration method we employing the grpss preconditioner to accelerate some krylov subspace methods like gmres the spectral distribution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the grpss preconditioner