spectral collocation solutions to multiparameter mathieu s system our main aim is the accurate computation of a large number of specified eigenvalues and eigenvectors of mathieu s system as a multiparameter eigenvalue problem mep the reduced wave equation for small deflections is solved directly without approximations introduced by the classical mathieu functions we show how for moderate values of the cut off collocation parameter the qr algorithm and the arnoldi method may be applied successfully while for larger values the jacobi davidson method is the method of choice with respect to convergence accuracy and memory usage

a note on the backward errors for hermite eigenvalue problems we discuss the backward error analysis of the hermite eigenproblems three kinds of problems are included generalized eigenproblem matrix polynomial eigenproblem and multiparameter eigenproblem in the case of the approximate eigenvalue being real these problems have already been solved by higham and higham siam j matrix anal 20 1998 493 tisseur lin alg appl 309 2002 339 and hachstenbach and plestenjak lin alg appl 375 2003 63 this note as a complement of their results considers the complex case

spherical unitary highest weight representations in this paper we give an almost complete classification of the h spherical unitary highest weight representations of a hermitian lie group g where g h is a symmetric space of cayley type

on semi invariants and index for biparabolic seaweed algebras ii abstract in a continuation of i we study the semi invariant subalgebra sy q of s q for q a biparabolic subalgebra of a semisimple or of an affine lie algebra in most cases this is shown to be polynomial and we compute the degrees and weights of the generators we describe explicitly the canonical truncation q e of q for which sy q s q e q e let t denote the nilradical of q and q its derived algebra we show that the subalgebra s t q of s q q sy q is always polynomial and describe the degrees and weight of the generators finally a sum rule is given for the false degrees of s q q these are true degrees in most cases

on the harris knorr correspondence in p solvable groups abstract fix a prime p and let n be a normal subgroup of a finite p solvable group g suppose that b is a p block of n with abelian defect group q and b is a p block of g covering b let b be the brauer correspondent of b in n n q and let b be the unique p block of n g q that covers b and induces b we show that there exist height preserving bijections of irr b onto irr b and of ibr b onto ibr b such that the decomposition numbers d and d are equal for all irr b and all ibr b

the schur clifford subgroup of the brauer clifford group abstract we define a schur clifford subgroup of turull s brauer clifford group similar to the schur subgroup of the brauer group the schur clifford subgroup contains exactly the equivalence classes coming from the intended application to clifford theory of finite groups we show that the schur clifford subgroup is indeed a subgroup of the brauer clifford group as are certain naturally defined subsets we also show that this schur clifford subgroup behaves well with respect to restriction and corestriction maps between brauer clifford groups

characters of odd degree we prove the mckay conjecture on characters of odd degree a major step in the proof is the verication of the inductive mckay condition for groups of lie type and primes such that a sylow subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of harish chandra induction specics of characters of odd degree namely that most of them lie in the principal harish chandra series then allow us to deduce from it the mckay conjecture for the prime 2 hence for characters of odd degree

the mckay conjecture and galois automorphisms the main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group g can be computed locally the simplest of these conjectures is the mckay conjecture which asserts that the number of irreducible complex characters of g of degree not divisible by p is the same if computed in a p sylow normalizer of g in this paper we propose a much stronger version of this conjecture which deals with galois automorphisms in fact the same idea can be applied to the celebrated alperin and dade conjectures

extending irreducible representations of normal subgroups let h be a normal subgroup of some finite group g and let v be an irreducible right fh module where f is an arbitrary field it is one of the primary objectives of clifford theory to find necessary and sufhcient conditions so that v can be extended to an fg module an obvious necessary condition is that v has to be g invariant stable sufficiency relies upon various properties of the group extension h g w s where s g h and the module itself the theory is well developed in case v is absolutely irreducible recently isaacs 12 and dade 4 proved generalizing a theorem by gallagher that v can be always extended to an fg module provided h is a hall subgroup of g and v is g invariant the present paper resulted from an effort to understand this remarkable and somewhat surprising result assume in the sequel that v is g invariant let d end v be the opposite endomorphism ring of k basic clifford theory yields a group extension d 52 p p s such that there exists an fg module extending v if and only if 52 52 v splits over the multiplicative group d of the division algebra d see theorem 1 below in general d is noncommutative and so the usual cohomological methods seem to fail dade even remarked in 4 sceptically that none of the nice theorems about extensions of v such as those of gallagher which depend upon calculations in the cohomology groups of s with coefficients in a commutative group can be proved this way in the following situation however one can argue as usual

using g algebras for schur index computation abstract conditions for the existence of a bijective character correspondence that preserves schur indices are given based on the theory of central simple g algebras over fields of characteristic 0 it is then shown how this applies to schur index computation in the case of finite solvable groups

a reduction theorem for the mckay conjecture the mckay conjecture asserts that for every finite group g and every prime p the number of irreducible characters of g having p degree is equal to the number of such characters of the normalizer of a sylow p subgroup of g although this has been confirmed for large numbers of groups including for example all solvable groups and all symmetric groups no general proof has yet been found in this paper we reduce the mckay conjecture to a question about simple groups we give a list of conditions that we hope all simple groups will satisfy and we show that the mckay conjecture will hold for a finite group g if every simple group involved in g satisfies these conditions also we establish that our conditions are satisfied for the simple groups psl2 q for all prime powers q 4 and for the suzuki groups sz q and ree groups r q where q 2 e or q 3 e respectively and e 1 is odd since our conditions are also satisfied by the sporadic simple group j 1 it follows that the mckay conjecture holds for all primes p for every finite group having an abelian sylow 2 subgroup

lowest weight modules of sp2n r of minimal gelfand kirillov dimension abstract this paper is to classify genuine irreducible lowest weight modules of sp 2 n r of gelfand kirillov dimension n

calculating clifford classes for characters containing a linear character once abstract given a finite group g a normal subgroup n of g and an irreducible character of g the clifford class of describes the clifford theory including schur indices of with respect to the normal subgroup n assume there is a subgroup u of n and a linear character of u with multiplicity 1 in the restriction of to u we describe a method to calculate the clifford class of in certain cases this method helps in particular in the calculation of the schur indices of the irreducible characters of the groups contained between the general linear and the special linear groups

endoisomorphisms and character triple isomorphisms abstract this paper concerns aspects of clifford theory of finite groups in earlier papers turull proved that if two finite groups yielded the same element of the brauer clifford group then there was an endoisomorphism from one group to the other and furthermore that associated with each endoisomorphism there was an essentially unique correspondence of modules over many different fields from one group to the other the paper adapts the definition of character triple isomorphism so that it involves ordinary and brauer characters and it preserves fields of definition schur indices decomposition numbers and blocks it is proved that each endoisomorphism yields exactly one character triple isomorphism character triple isomorphisms can be composed restricted produced by direct sums extension of fields and these operations have their parallel for the endoisomorphisms one goal of the paper is to provide tools for the study of the character theory of finite groups in an accessible way suitable for applications

corestriction for algebras with group action abstract we define a corestriction map for equivariant brauer groups in the sense of frohlich and wall which contain as a special case the brauer clifford groups introduced by turull we show that this corestriction map has similar properties as the corestriction map in group cohomology especially galois cohomology in particular composing corestriction and restriction associated to a subgroup h g amounts to powering with the index g h

strengthening the mckay conjecture to include local fields and local schur indices abstract a central problem in the representation theory of finite groups is given a prime p the study of relationships between the representations of g and those of p local subgroups of g the simplest of these is perhaps the mckay conjecture which concerns the number of irreducible characters of degree prime to p even though this conjecture remains open in an effort to understand it a number of strengthenings of it have been proposed recently isaacs and navarro and later navarro have proposed interesting stronger conjectures indicating that congruences modulo p of the degree of the characters and the invariance under certain type of fixed galois automorphism could be taken into account as well we propose an even further strengthening of mckay s conjecture which takes into account in addition the p local field of definition of the irreducible characters as well as their p local schur indices

real polynomial diffeomorphisms with maximal entropy tangencies like the henon diffeomorphisms of r2 the quadratic maps of r have also played a central role in the field of dynamical systems these two families of dynamical systems fit together naturally which raises the question of the extent to which the dynamics is similar one difference is that our knowledge of these quadratic maps is much more thorough than our knowledge of these quadratic diffeomorphisms substantial progress in the theory of quadratic maps has led to a rather complete theoretical picture of their dynamics and an understanding of how the dynamics varies with the parameter despite significant recent progress in the theory of henon diffeomorphisms due to benedicks and carleson and many others there are still many phenomena that are not nearly so well understood in this two dimensional setting as they are for quadratic maps one phenomenon which illustrates the difference in the extent of our knowledge in dimensions one and two is the dependence of the complexity of the system on parameters in one dimension the nature of this dependence is understood and the answer is summarized by the principle of monotonicity loosely stated this is the assertion that the complexity of fa does not

clifford classes for some overgroups of the special linear groups abstract in a turull j algebra 235 2001 275 314 we calculated the schur index of each of the irreducible characters of the finite special linear groups in the present paper we calculate the schur index of all the irreducible characters of some overgroups of the special linear groups the overgroups in question are the special linear groups extended by diagonal automorphisms and the subgroups of the general linear group that contain the special linear group to each conjugacy class of irreducible characters of the special linear group in each overgroup is associated a clifford class the clifford class controls all the irreducible characters of the overgroup and intermediate subgroups that are related to the given irreducible by clifford theory knowing only the clifford class we can parametrize all the irreducible characters of the intermediate subgroups and compute for each parametrized irreducible character its field of character values as well as its schur index over each field we explicitly compute the clifford class in each case and deduce from it the information on the schur index of all the irreducible characters of the overgroups

the brauer clifford group and rational forms of finite groups abstract let k be a perfect field and g be a finite subgroup of gl n k the aim of this paper is to study the following question is it possible to find a subgroup of gl n k conjugate to g with a set of fundamental polynomial invariants whose coefficients lie in the field k using the brauer clifford group introduced by turull this question is split into two natural parts the first part is to recognize elements in the brauer clifford group and the second part is to decide a field theoretic question involving galois cohomology for the first part a subgroup of the brauer clifford group which can be identified with a subdirect product of the brauer group and a second cohomology group is introduced the main question is answered completely if g is an absolutely irreducible subgroup of gl n q and isomorphic to psl 2 f q furthermore if k is the real field or a finite field then a satisfactory treatment is possible for any finite group

resolutions and hilbert series of determinantal varieties and unitary highest weight modules abstract let g k be a hermitian symmetric pair and let g and k denote the corresponding complexified lie algebras let g k p p be the usual decomposition of g as a k module there is a natural correspondence between k c orbits in p and a distinguished family of unitarizable highest weight modules for g called the wallach representations we denote by yk the closure of the k c orbit in p that is associated to the kth wallach representation in this article we give explicit formulas for the numerator polynomials of the hilbert series of the varieties yk by using bgg resolutions of unitarizable highest weight modules a preliminary result gives a new branching formula for a certain two parameter family of finite dimensional representations of the even orthogonal groups our work is an extension of previous work by enright and willenbring

the brauer clifford group abstract clifford theory provides well behaved character correspondences between different groups which have isomorphic quotients given one such quotient group we define the brauer clifford group we show that each character of the original groups gives rise to a specific element of the brauer clifford group when two characters of different groups yield the same element of the brauer clifford group we obtain a very well behaved character correspondence between the characters of the different groups which preserves not only induction restriction multiplicities but also fields of values for the corresponding characters and schur indices we also show that the brauer clifford group has a natural homomorphism into a brauer group the brauer clifford group can be thought of as a refinement of the previously introduced clifford classes

on clifford theory with galois action abstract let g be a finite group n a normal subgroup of g and irr n let f be a subfield of the complex numbers and assume that the galois orbit of over f is invariant in g we show that there is another triple g 1 n 1 1 of the same form such that the character theories of g over and of g 1 over 1 are essentially the same over the field f and such that the following holds g 1 has a cyclic normal subgroup c contained in n 1 such that 1 n 1 for some linear character of c and such that n 1 c is isomorphic to the abelian galois group of the field extension f f 1 more precisely having the same character theory means that both triples yield the same element of the brauer clifford group brcliff g f defined by a turull

eigenvalues pseudospectrum and structured perturbations we investigate the behavior of eigenvalues under structured perturbations we show that for many common structures such as complex symmetric toeplitz symmetric toeplitz circulant and others the structured condition number is equal to the unstructured condition number for normwise perturbations and prove similar results for real perturbations an exception are complex skewsymmetric matrices we also investigate componentwise complex and real perturbations here hermitian and skew hermitian matrices are exceptional for real perturbations furthermore we characterize the structured complex and real pseudospectrum for a number of structures and show that often there is little or no significant difference to the usual unstructured pseudospectrum

endoisomorphisms yield module and character correspondences abstract in an earlier paper turull 2012 12 the author introduced the concept of endoisomorphisms and showed its natural connection with clifford theory of finite groups associated with clifford theory are module and character correspondences in the present paper the author shows that each endoisomorphism produces a unique module and character correspondence with excellent compatibility results

the strengthened alperin weight conjecture for p solvable groups abstract the alperin weight conjecture is a well known conjecture and it is central to the modern representation theory of finite groups it is known to be true for many types of finite groups but it remains open in particular the conjecture is known to be true for finite p solvable groups in this paper we prove that a very strong form of the conjecture holds for all finite p solvable groups it follows that the strengthening of the alperin weight conjecture that includes galois automorphisms over the p adic numbers holds for all finite p solvable groups this strengthening of the alperin weight conjecture was suggested in an earlier work of navarro on the mckay conjecture the present paper provides the first direct evidence beyond the calculation of individual cases and the case of groups of odd order for this strengthening

character correspondences in solvable groups let g be a finite solvable group p be some prime let p be a sylow p subgroup of g and let n be its normalizer in g it is known that the number of irreducible characters of g of degree prime to p equals the number of irreducible characters of n of degree prime to p let f be any field of characteristic zero such that if g n is even then f contains qp q 2 the intersection of the field of p adic numbers with the field of rational numbers extended by all roots of unity whose order is a power to two in this paper we show that there exists a bijection from the set of all irreducible characters of g of degree prime to p to the set of all the irreducible characters of degree prime to p of n such that it preserves the degrees modulo p all the field of values over f and the schur index over every field containing f

hermitian symmetric spaces and their unitary highest weight modules abstract the purpose of this article is to determine the set of unitarizable highest weight modules corresponding to hermitian symmetric spaces of the noncompact type the major step is that of proving unitarity at the last possible place with this established the description of the full set of unitarizable highest weight modules follows by a straightforward tensor product argument combined with the main ingredients of the proof of the key theorem bernstein gelfand gelfand and a diagramatic representation of the set of positive noncompact roots

structured backward error for palindromic polynomial eigenvalue problems a detailed structured backward error analysis for four kinds of palindromic polynomial eigenvalue problems ppep left sum ell 0 d a ell lambda ell right x 0 quad a d ell varepsilon a ell star quad rm for ell 0 1 ldots lfloor d 2 rfloor is one of the two actions transpose and conjugate transpose and varepsilon in pm 1 each of them has its application background with the case star taking transpose and e 1 attracting a great deal of attention lately because of its application in the fast train modeling computable formulas and bounds for the structured backward errors are obtained the analysis reveals distinctive features of ppep from general polynomial eigenvalue problems pep investigated by tisseur linear algebra appl 309 339 361 2000 and by liu and wang appl math comput 165 405 417 2005

perturbation theory for homogeneous polynomial eigenvalue problems we consider polynomial eigenvalue problems p a alpha beta x 0 in which the matrix polynomial is homogeneous in the eigenvalue alpha beta uc2 in this framework infinite eigenvalues are on the same footing as finite eigenvalues we view the problem in projective spaces to avoid normalization of the eigenpairs we show that a polynomial eigenvalue problem is well posed when its eigenvalues are simple we define the condition numbers of a simple eigenvalue alpha beta and a corresponding eigenvector x and show that the distance to the nearest ill posed problem is equal to the reciprocal of the condition number of the eigenvector x we describe a bihomogeneous newton method for the solution of the homogeneous polynomial eigenvalue problem homogeneous pep

above the glauberman correspondence let s be a finite solvable group and suppose s acts on the finite group n and they have coprime orders then the celebrated glauberman correspondence provides a natural bijection from the set irrs n of irreducible characters of n which are invariant under the action of s to the set irr cn s of all irreducible characters of the centralizer of s in n suppose further that the semidirect product sn is a normal subgroup of a finite group g let irrs n and let irr cn s be its glauberman correspondent we prove that there is a bijection with good compatibility properties from the set irr g of the irreducible characters of g above to irr ng s such that in the case when s is a p group for some prime p it preserves fields of values and schur indices over qp the field of p adic numbers using this result we also prove a strengthening of the mckay conjecture for all p solvable groups

brauer clifford equivalence of full matrix algebras abstract the brauer clifford group was introduced to describe the clifford theory for finite groups it was proved that it has a natural homomorphism into a brauer group and the kernel of this homomorphism is the set of all equivalence classes of g algebras which are full matrix algebras in this paper we prove that this kernel is isomorphic to a second cohomology group in the clifford theory for finite groups situation we characterize families of characters which yield elements in the full matrix subgroup of the brauer clifford group as those where an appropriate character has schur index one we also show in this case how to compute the element of the second cohomology group associated with this family of characters

schur indices and schur groups w x schur groups were introduced in 8 as a certain kind of smallest groups admitting characters with nontrivial schur index indeed every finite group g having an irreducible character x with m x 1 involves such schur groups they occur as terminal steps by repeated application of the brauer witt theorem over local fields of characteristic zero there is a connection with the schur subgroup s k of the brauer group br k of such a field k this s k has been studied in detail by w x w x yamada 12 and quite recently by adem and milgram 1 from the character theoretic point of view it appears to be of interest to give explicit generators for s k in terms of schur groups see section 6 in the course w x of this investigation the authors noticed that there must be a gap in 8 concerning the exceptional dyadic schur groups yielding nontrivial 2 lo cal index it turned out that the definition of the schur groups of type qd q was too restrictive the correct definition will be given in section 5 below we take the opportunity to discuss these terminal q reduc2 tions once again and the class of 2 groups involved theorem 4 and 5 the schur groups admit faithful irreducible characters which are induced from linear characters of cyclic normal subgroups for the sake of clarity we begin our present work by making some general comments on schur indices in group extensions this should be compared with turull w x 11 our approach is perhaps more elementary and we just demonstrate w x how to use the results on stable clifford theory 7 more details may be w x found in the recent thesis 5 of the first named author

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

clifford theory and endoisomorphisms this paper describes the relation between clifford theory of finite groups over any field and the brauer clifford group let g and g be finite groups let g g be a surjective group homomorphism with kernel h and let f be any field let k be any extension field of f and let s be an irreducible kh module we show that to s is associated in a natural way a specific element s f of a brauer clifford group defined over and f as a tool to prove the existence of this association and to study its properties we use endoisomorphisms these are simply certain isomorphisms of related endomorphism algebras as g algebras over f we show that two modules for two different finite groups have the same in an appropriate sense element of the brauer clifford group if and only if there exists an appropriate endoisomorphism

numerical computation of an analytic singular value decomposition of a matrix valued function this paper extends the singular value decomposition to a path of matricese t an analytic singular value decomposition of a path of matricese t is an analytic path of factorizationse t x t s t y t t wherex t andy t are orthogonal ands t is diagonal to maintain differentiability the diagonal entries ofs t are allowed to be either positive or negative and to appear in any order this paper investigates existence and uniqueness of analytic svd s and develops an algorithm for computing them we show that a real analytic pathe t always admits a real analytic svd a full rank smooth pathe t with distinct singular values admits a smooth svd we derive a differential equation for the left factor develop euler like and extrapolated euler like numerical methods for approximating an analytic svd and prove that the euler like method converges

hilbert series howe duality and branching for classical groups an extension of the littlewood restriction rule is given that covers all pertinent parameters and simplifies to the original under littlewood s hypotheses two formulas are derived for the gelfand kirillov dimension of any unitary highest weight representation occurring in a dual pair setting one in terms of the dual pair index and the other in terms of the highest weight for a fixed dual pair setting all the irreducible highest weight representations which occur have the same gelfand kirillov dimension we define a class of unitary highest weight representations and show that each of these representations l has a hilbert series hl q of the form

backward error condition numbers and pseudospectra for the multiparameter eigenvalue problem we define and evaluate the normwise backward error and condition numbers for the multiparameter eigenvalue problem mep the pseudospectrum for the mep is defined and characterized we show that the distance from a right definite mep to the closest non right definite mep is related to the smallest unbounded pseudospectrum some numerical results are given

finite dimensional multiparameter spectral theory the nonderogatory case abstract a completeness theorem for nonderogatory eigenvalues of multiparameter systems is proved in a finite dimensional setting also a basis for the second root subspace of a simple eigenvalue is given weakly elliptic multiparameter systems are introduced it is shown that simple real eigenvalues of such systems are nonderogatory

extending irreducible modules an important part of clifford s theory does not work properly for arbitrary fields r to see why we must recall the main steps of this theory we are interested in extensions to rh modules of a fixed h invariant irreducible rn module a where n is a normal subgroup of a finite group h we first pass to the factor algebra u rh 3 modulo the ideal 3 3 rh rh3 generated by the annihilator 3 of 8 in rn if sth is given its natural grading by the factor group g h n see 6 2 below then 3 is a graded ideal so that u is a strongly g graded ring in the sense of 3 or section 1 below whose l component u rn d is a finite dimensional simple ralgebra with i as its irreducible module it follows that the extensions of 8 to rh modules are just the pull backs of the extensions of 2i to 2l modules via the natural epimorphism of rh onto u when 52 is algebraically closed or more generally when i is split over r then 2i is the kronecker product

clifford theory and galois theory i abstract we combine galois theory with crossed products by letting grading groups act on ground fields we show that clifford theory for suitable normal subgroups preserves such actions

the brauer clifford group of g rings abstract in previous work the author introduced the brauer clifford group of certain g algebras this group is useful because to every irreducible character of a normal subgroup of a finite group one can associate a unique element of a specific brauer clifford group and this element controls the clifford theory of this character in its ambient group in the present paper we define the brauer clifford group of g rings this new definition only requires us to discuss tensor products over the underlying g ring and it is simpler than the earlier one we prove that the new definition yields a group which is canonically isomorphic to the brauer clifford group of a corresponding suitable g algebra

clifford theory of simple modules clifford theory primarily is concerned with the representations of normal subgroups and quotient groups of finite groups the basic contribution being due to a h clifford 3 modern developments founded on the concepts of group graded rings and crossed products have shown close relations with commutative and non commutative galois theory this is most conspicuous in stable clifford theory of simple modules suppose n w h g is an extension of finite groups and v is a simple f n module where f is any field since passage from the inertia group to the whole group usually is easy to handle we may and do assume that v is g invariant or h invariant stable then the annihilator i ann v of v generates a homogeneous ideal z of f h and a f h z is a fully g graded algebra with simple l component r f n z general clifford theory is concerned with such g graded algebras a and their modules cf dade 7 lo assume the l component r a is simple artinian as above then the centre k z r has in a natural way the structure of a g field the main object of study is the endomorphism ring e end v of the a module induced from the simple r module v this is a fully g graded algebra as well generalizing wedderburn s theorem we show that a z m e as g graded algebras where d is the dimension of v over the division algebra d end v theorem 2 1 in particular k z d has the same g structure with respect to e and a and e are morita equivalent in some strong sense now assume that dim r d2m2 is finite and that the schur index m m v is relatively prime to the order of the automorphism group g c k of k induced by g then the graded structure of a e can be described by some distinguished cohomology class w v e h g k more precisely w oa v determines a crossed product sz of k with g