clifford theory for cyclic quotient groups abstract we describe in detail the clifford theory for cyclic quotient groups we pay particular attention to the field of definition of the characters involved as well as their schur indices

on the maximum entropy method for interval covariance sequences given an interval covariance sequence the existence of a maximum entropy spectral estimate is considered it is shown that a maximum entropy spectral estimate does exist and moreover it is unique

graded multiplicities in the exterior algebra abstract this paper deals with the graded multiplicities of the smallest irreducible representations of a simple lie algebra in its exterior algebra an explicit formula for the graded multiplicity of the adjoint representation in terms of the weyl group exponents was conjectured by a joseph a proof of this conjecture based on the properties of macdonald polynomials is given in the present paper the same method allows us to calculate the multiplicity of the simple module with highest weight equal to the short dominant root

the schur indices of the irreducible characters of the special linear groups abstract we compute the schur indices of each irreducible character of sl n q the special linear group for all n 1 and for all q a power of a prime

maximum entropy power spectrum estimation with uncertainty in correlation measurements the purpose of this paper is to present a multidimensional mem algorithm valid for nonuniformly sampled arrays which satisfies a correlation approximating constraint to this end the correlation matching equality constraints of the usual mem are replaced by a single inequality constraint whose form is based on a measure of the noise in the given autocovariance function acf in this way one can incorporate into the model knowledge of the noisy nature of the given acf since the given acf is usually estimated from the samples of the wavefield specifically the covariance matrix of the correlation estimates is used in a quadratic form that weights the difference between the given acf and the one matched by the power spectrum the maximization of entropy under this inequality constraint leads ultimately to a steepest descent algorithm the algorithm has been tested with 1 d synthetic data representing multiple sinusoids buried in additive white noise the performance of this modified mem algorithm is compared to a traditional mem algorithm for extendible acf s and for different snr s examples of the mem spectrum are given for the case of nonextendible acf s

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

power spectrum estimation with randomly spaced correlation samples three algorithms are developed to reproduce uniformly spaced correlation measurements for power spectrum estimation some general relations between the randomly spaced correlation samples and the power spectrum are derived the details of the three algorithms that can be used to reproduce uniformly spaced correlation measurements are included simulation examples using sinusoidal signals are given to examine the adverse effects due to the correlation randomness as well as to demonstrate improvements achieved by the suggested algorithms

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

refinements of dade s projective conjecture for p solvable groups dade s projective conjecture is known to be true for finite p solvable groups thanks to work of g r robinson but remains open in general work of isaacs and navarro suggested to uno and boltje refinements of this conjecture these refinements were studied for finite p solvable groups by glesser in the present paper inspired by earlier work of turull we propose further refinements of the conjecture that take into account the schur indices and the elements of the brauer group we prove that all these refinements of dade s projective conjecture hold for all finite p solvable groups in particular we obtain that the version of dade s projective conjecture which involves character degree residues modulo p fields of definition and schur indices as well as the full strength of boltje s conjecture both hold for all finite p solvable groups the proof develops a clifford theory for normalizers of chains of p subgroups which allows one to reduce the calculation of the relevant sums to simpler groups

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

the mckay conjecture is true for the sporadic simple groups abstract the mckay conjecture states that the number of irreducible complex characters of a group g that have degree prime to p is equal to the same number for the sylow p normalizer in g we verify this conjecture for the 26 sporadic simple groups

odd character correspondences in solvable groups abstract 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 assume that n has odd order then we prove 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 degree modulo p the field of values and the schur index over every field of characteristic zero this strengthens a more general recent result a turull character correspondences in solvable groups j algebra 295 2006 157 178 but only for the case under consideration here in addition we prove some other strong character correspondences that have very good rationality properties as one consequence we prove that a solvable group g has a non trivial rational irreducible character with degree prime to p if and only if the order of the normalizer of a sylow p subgroup of g has even order

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

clifford algebra analogue of the hopf koszul samelson theorem theρ decompositionc g end vρ c p and the g module structure of g

schur indices and abelian defect groups abstract we show that the irreducible characters lying in 2 blocks of finite solvable groups with abelian defect groups have trivial 2 local schur indices

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

exterior powers of the adjoint representation and the weyl ring of e8 abstract i derive explicitly all polynomial relations in the character ring of e 8 of the form k e 8 p k 1 8 0 where k e 8 is an arbitrary exterior power of the adjoint representation and i is the ith fundamental character this has simultaneous implications for the theory of relativistic integrable systems seiberg witten theory quantum topology orbifold gromov witten theory and the arithmetic of elliptic curves the solution is obtained by reducing the problem to a large but finite dimensional linear problem which is amenable to an efficient solution via distributed computation

a minimum free energy hybrid algorithm for 2 d spectral estimation a separable quasi 2 d spectral estimation algorithm is presented this hybrid algorithm combines the 1 d minimum free energy method with the 1 d periodogram the algorithm reduces to the corresponding maximum entropy hybrid spectral estimator in some cases the minimum free energy hybrid algorithm is preferable to the maximum entropy version when the estimated autocorrelation sequence is not extendable or when the signal to noise ratio is low a simulation helps illustrate this point

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

a formula for calculating some schur indices abstract we describe a method for calculating the schur indices of certain characters of finite groups this method helps in particular in the calculation of the schur indices of the irreducible characters of the special linear groups

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

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

character correspondences in solvable groups our concern in this paper is with the ordinary complex number character theory of solvable groups if the lengths of some of the earlier papers on this subject l 3 8 are a reliable indication this is a surprisingly complicated and nontrivial theory nevertheless some of the important results are considerably less deep than one would be led to believe by the number of pages involved this paper contains no new theorems and few new ideas it is addressed to the reader who while familiar with character theory has not mastered the literature on characters of solvable groups its purpose is to give such a reader a feeling for the subject and to provide him with accessible proofs for a few of the main results although we certainly do not attempt to prove the strongest or most general results possible or known nevertheless the paper does contain a few items which have application outside of solvable groups in particular we have included in section 4 a brief exposition of tensor induction although our main result could have been proved using an ad hoc argument in place of this general technique the author is now convinced after considerable initial resistance that tensor induction is valuable and deserves further exposure naturally a key idea in the study of characters of solvable groups is to consider the relationships between characters and chief sections in particular suppose that k l is an abelian chief section of g and let 9 e irr k be invariant in g then according to the going down theorem theorem 6 18 of 9 one of the following must occur

height zero characters and normal subgroups in p solvable groups abstract fix a prime number p and let n be a normal subgroup of a finite p solvable group g let b be a p block of n and suppose b is a p block of g covering b let d be a defect group for the fong reynolds correspondent of b with respect to b and let b be the unique p block of n n g d having defect group d and inducing b suppose further that irr b and let irr 0 b be the set of irreducible characters in b of height zero that lie over we show that the number of characters in irr 0 b is equal to the number of characters in t irr 0 b t where t runs through the inertial group t of b in g this result generalizes a theorem of t okuyama and m wajima which confirms the alperin mckay conjecture for p solvable groups

first layer formulas for characters of sl n c some problems concerning the decomposition of certain characters of sl n c are studied from a combinatorial point of view the specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the euler characteristic of hanlon s so called macdonald complex a general recursion is given for computing the irreducible decomposition of these characters the recursion is explicitly solved for the first layer representations which are the irreducible representations corresponding to partitions of n in the case of the exterior algebra this settles a conjecture of gupta and hanlon a further application of the recursion is used to give a family of formal laurent series identities that generalize the equal parameter q dyson theorem

the schur index of projective characters of symmetric and alternating groups the present paper answers a natural question that arises when one sets side by side two major contributions of issai schur to the theory of group representations in a monumental paper 13 schur described the projective characters of the symmetric group sn and the alternating group an for all n he calculated the representation groups of sn and an for all n and calculated the character table of these representation groups thereby describing the projective characters of sn and an in another seminal paper 12 schur defined and studied what is now called the schur index if 4f is an irreducible character of some finite group g and k is a field of characteristic zero the schur index mk g of qf with respect to k is the least positive multiplicity of q in a character afforded by a kg module the question is what are the schur indices of the characters of the representation groups of sn and an we give here a rule that describes the schur index of every irreducible character of the representation groups of sn and an for all n a substantial amount of research has been devoted to the schur index efforts have been directed both to finding its general properties and to calculating it for important classes of examples for example benard 1 showed that the schur index of every character of the weyl groups of type e6 e7 and e8 is one in 2 feit gives short proofs for the calculation of the schur index for many specific groups including benard s weyl groups of type e6 e7 and e8 and the representation groups of the sporadic simple groups other authors have studied the schur index for groups of lie type see for example 4 or 10 various methods are available to give upper bounds for the schur indices but it is usually harder to find their value when they are larger than one hence for the groups of lie type one can show that their schur indices are at most two in many cases and at most one in some cases but the question of exactly for what characters it is one and for which it is two has not been consistently answered

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

e8 spectral curves i provide an explicit construction of spectral curves for the affine mathrm e 8 relativistic toda chain their closed form expression is obtained by determining the full set of character relations in the representation ring of mathrm e 8 for the exterior algebra of the adjoint representation this is in turn employed to provide an explicit construction of both integrals of motion and the action angle map for the resulting integrable system i consider two main areas of applications of these constructions on the one hand i consider the resulting family of spectral curves in the context of the correspondences between toda systems 5d seiberg witten theory gromov witten theory of orbifolds of the resolved conifold and chern simons theory to establish a version of the b model gopakumar vafa correspondence for the mathrm sl n l e murakami ohtsuki invariant of the poincar e integral homology sphere to all orders in 1 n on the other i consider a degenerate version of the spectral curves and prove a 1 dimensional landau ginzburg mirror theorem for the frobenius manifold structure on the space of orbits of the extended affine weyl group of type mathrm e 8 introduced by dubrovin zhang equivalently the orbifold quantum cohomology of the type mathrm e 8 polynomial mathbb c p 1 orbifold this leads to closed form expressions for the flat co ordinates of the saito metric the prepotential and a higher genus mirror theorem based on the chekhov eynard orantin recursion i will also show how the constructions of the paper lead to a generalisation of a conjecture of norbury scott to ade mathbb p 1 orbifolds and a mirror of the dubrovin zhang construction for all weyl groups and choices of marked roots

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

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

geometric monodromy semisimplicity and maximality let x be a connected scheme smooth and separated over an alge braically closed field k of characteristic p 0 let f y x be a smooth proper morphism and x a geometric point on x we prove that the tensor invariants of bounded length d of 1 x x acting on the etale cohomology groups h yx fl are the reduction modulo l of those of 1 x x acting on h yx l for l greater than a constant depending only on f y x d we apply this result to show that the geometric variant with fl coefficients of the grothendieck serre semisimplicity conjecture namely that 1 x x acts semisimply on h yx fl for l 0 is equivalent to the condition that the image of 1 x x acting on h yx ql is almost maximal in a precise sense what we call almost hyperspecial with respect to the group of ql points of its zariski closure ultimately we prove the geometric variant with fl coefficients of the grothendieck serre semisimplicity conjecture

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

on the calculation of local invariants of irreducible characters abstract let k be a finite extension of the field q p of p adic numbers let be an irreducible character of a finite group whose values are in k associated with and k is an element of the brauer group of k and therefore by standard results a local invariant in q z the paper defines some classes of finite groups that the paper calls p basic groups and gives formulas to calculate the local invariant of their characters it also shows how one can calculate the local invariant of any irreducible character of any finite group that has values in k by reducing the problem to the case where the groups are p basic and using the explicit formulas given in the paper

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

constructing representations of the finite symplectic group sp 4 q abstract let g be a finite group and be an irreducible character we say that a subgroup h is a subgroup if the restriction h of to h has at least one linear constituent of multiplicity 1 not every pair g has a subgroup but subgroups can be found in many cases the existence of such subgroups is of interest for several reasons one being that knowledge of a subgroup enables us to give a simple construction of a matrix representation of g affording in this paper we show that when g sp 4 q where q is a power of an odd prime p and h is a sylow p subgroup of g then h is a subgroup for every irreducible character with one exception we also find a p subgroup which is a subgroup for the exceptional character

degree divisibility in character correspondences abstract recent results of the author prove the existence of bijections of certain sets of irreducible characters which satisfy many nice properties in particular these correspondences imply a strengthened version of the mckay conjecture for solvable finite groups which takes into account fields of values and schur indices in the present paper we prove that these correspondences have also the additional property that the degree of the corresponding character divides the degree of the original character and we say something about the quotient as well

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

the adjoint representation inside the exterior algebra of a simple lie algebra abstract for a simple complex lie algebra g we study the space of invariants a g g g which describes the isotypic component of type g in g as a module over the algebra of invariants g g as main result we prove that a is a free module of rank twice the rank of g over the exterior algebra generated by all primitive invariants in g g with the exception of the one of highest degree

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

spectral estimation for sensor arrays the array processing problem is briefly discussed and an abstract spectral estimation problem is formulated this problem involves the estimation of a multidimensional frequency wave vector power spectrum from measurements of the correlation function and knowledge of the spectral support the investigation of correlation matching spectral estimates leads to the extendibility question does there exist any positive spectrum on the spectral support that exactly matches a given set of correlation samples in answering this question a mathematical framework is developed in which to analyze and design spectral estimation algorithms pisarenko s method of spectral estimation which models the spectrum as a sum of impulses plus a noise component is extended from the time series case to the more general array processing case pisarenko s estimate is obtained as the solution of a linear optimization problem which can be solved using a linear programming algorithm such as the simplex method