on projective resolutions of simple modules over the borel subalgebra s n r of the schur algebra s n r for n 3 we construct projective resolutions for simple modules over the borel subalgebras s 2 r and s 3 r of the schur algebras s 2 r and s 3 r over a field of positive characteristic in the case s 2 r we get a minimal projective resolution we also find projective resolutions of minimal possible length for weyl modules over the schur algebra s 2 r corresponding to the regular weights

finite resolutions of modules for reductive algebraic groups of a suitable g module m by modules which are required to be direct sums of tensor products of exterior powers of the natural representation in particular in z 31 for partitions 1 and p resolutions are constructed for the modules l f and l f the schur functor corresponding to 1 and the skew schur functor corresponding to a p evaluated at f where f is a free module over a field or the integers the purpose of this paper is to characterise the gl modules which admit such a resolution as those modules which have a filtration in which each successive quotient has the form l f for some partition p by contrast with 2 3 we do not produce resolutions explicitly our methods are independent of those of akin and buchsbaum we give a new proof of their result on the existence of a resolution for l f and are algebraic group theoretic in nature we have therefore cast our main result the theorem of section 1 as a statement about resolutions for reductive algebraic groups over an algebraically closed field one may frequently wish to work over a more general coefficient ring see l 3 and so we treat in some detail chevalley group schemes and general linear group schemes over a principal ideal domain 2 6 and 2 7 giving the appropriate versions of our main result and relating it to the work of akin buchsbaum and weyman

explicit solvability of a class of maliuzhinets equations with matrix coefficients a coupled system of maliuzhinets functional equations for a pair of unknown functions is studied it is shown that the system can be solved in an explicit form for a specific but non trivial class of its matrix coefficients two examples arising in electromagnetic diffraction theory in a wedge shaped region are briefly discussed

schur subalgebras and an application to the symmetric group abstract let k be an infinite field of prime characteristic p and let d r be positive integers of the same parity satisfying a certain congruence condition we prove that the schur algebra s 2 d is isomorphic to a subalgebra of the form es 2 r e where e is a certain idempotent of s 2 r translating this result via ringel duality to the symmetric groups d and r we obtain lattice isomorphisms between specht modules between young modules and between permutation modules here modules labelled by the partitions r k k correspond to modules labelled by d k k this provides a representation theoretical interpretation for part of the fractal structures observed for the decomposition numbers of the symmetric groups corresponding to two part partitions

rayleigh wave scattering by two adhering elastic wedges we consider the problem of wave propagation in a composite bi material elastic wedge with traction free faces the wedge is composed of two distinct isotropic elastic solids which are perfectly bonded along their common face the angles of each wedge are essentially arbitrary satisfying 0 up to 2 2 1 2 2 up to 360 in particular we solve the problem of reflection transmission and diffraction of an incoming rayleigh wave on one of the wedge faces the problem is treated by the method of sommerfeld integrals whose spectral functions are defined through singular integral equations contour integration and analytic continuation we first follow the maliuzhinetz method and derive functional equations for the spectral functions we convert the maliuzhinetz functional equations into well posed singular integral equations which are shown to be suitable for effective inversion the solution of the integral equations explicitly determines the sommerfeld amplitudes inside some strips of the complex plane then explicit recursive formulas analytically continue the amplitudes to a larger region of the complex plane in a finite number of steps which correspond to multiple reflections from the boundaries the integral equations admit clear physical interpretation in terms of fermat9s reflection principle and their right hand sides are directly related to the boundary values of incident waves it is shown how our previous solutions on homogeneous wedges are recovered from the bi material wedge case we also show how several degenerate cases such as a liquid solid wedge and a liquid liquid wedge can be treated in appropriate limits finally the integral equations are solved numerically for particular material combinations and wedge angles to show the usefulness of the solution the results are plotted graphically and shown to be in general agreement with relevant special cases where prior results are available

linear fourier and stochastic analysis to extend the traditional fourier theory of stationary processes some new boundedness notions for processes and for random measures are introduced this leads for these processes and measures to plancherel and hausdorff young type formulae and to a decomposition theory via dilations and multiplications various applications of our methods are also presented

diffraction at skew incidence by an anisotropic impedance wedge in electromagnetism theory a new class of canonical cases an original mathematical approach for the diffraction of an electromagnetic skew incident wave by a wedge allows us to reduce such a problem to new elementary functional equations using this method we present an exact analytical solution with closed form expressions for a class of wedge of any angle with a certain type of anisotropic boundary condition which does not require any field component to vanish on the faces we thereby consider the important case where the principal axes of anisotropy are along and normal to the edge the relative impedance matrix attached to each face being diagonal with its determinant equal to unity

q schur algebras asymptotic forms and quantum sln abstract we studied asymptotic methods for q schur algebras and related quantum groups and constructed a kind of asymptotic form for q schur algebras and quantum special linear groups the structure and representations for these forms were also discussed

the global dimension of schur algebras for gl2 and gl3 abstract we first define the notion of good filtration dimension and weyl filtration dimension in a quasi hereditary algebra we calculate these dimensions explicitly for all irreducible modules in sl 2 and sl 3 we use these to show that the global dimension of a schur algebra for gl 2 and gl 3 is twice the good filtration dimension to do this for sl 3 we give an explicit filtration of the modules by modules of the form f l where is a dominant weight and is p restricted

extensions of symmetric tensors by alternating tensors on demontre que les groupes d extension des tenseurs symetriques par des tenseurs alternes sont determines par les composantes homogenes des modules d homologie de l hyperalgebre du groupe additif g a

constructing quantized enveloping algebras via inverse limits of finite dimensional algebras abstract it is known that a generalized q schur algebra may be constructed as a quotient of a quantized enveloping algebra u or its modified form u on the other hand we show here that both u and u may be constructed within an inverse limit of a certain inverse system of generalized q schur algebras working within the inverse limit u clarifies the relation between u and u this inverse limit is a q analogue of the linear dual r g of the coordinate algebra of a corresponding linear algebraic group g

presenting generalized schur algebras in types b c d abstract we give explicit presentations by generators and relations of certain generalized schur algebras associated with tensor powers of the natural representation in types b c d this extends previous results in type a obtained by two of the authors the presentation is compatible with the serre presentation of the corresponding universal enveloping algebra in types c d this gives a presentation of the corresponding classical schur algebra the image of the representation on a tensor power since the classical schur algebra coincides with the generalized schur algebra in those types this coincidence between the generalized and classical schur algebra fails in type b in general

on vertices of simple modules for symmetric groups of small degrees abstract we present some algorithmic methods for the computation of vertices of indecomposable and simple modules over group algebras in prime characteristic furthermore we apply these to the simple modules of the symmetric groups and determine the vertices of all simple modules of the symmetric groups of degree at most 14 and 15 in characteristic 2 and 3 respectively with one exception

wiener hopf analysis of the em diffraction by an impedance discontinuity in a planar surface and by an impedance half plane the wiener hopf technique is used to solve two canonical problems the first problem considered is the electromagnetic em diffraction by a planar surface with an impedance discontinuity two part surface of an arbitrarily polarized plane wave obliquely incident to the axis of the two dimensional objects the second problem considers the em diffraction by a half plane with equal impedances on both sides the solutions obtained are cast in a matrix notation which is useful for diffraction problems the exact formal solutions are expressed in terms of integrals which can be asymptotically evaluated uniform asymptotic expressions are obtained where the presence of the geometrical optics go poles as well as the surface wave poles near the saddle point are fully taken into account several numerical examples are presented and it is shown that the solutions are continuous across the shadow boundaries of the go and surface wave fields

harmonizability v boundedness 2 p boundedness of stochastic processes some new classes of discrete time non stationary processes related to the harmonizable andv bounded classes are introduced a few characterizations are obtained which in turn unify thev bounded theory our main results depend on a special form of grothendieck s inequality

homological properties of quantised borel schur algebras and resolutions of quantised weyl modules abstract we continue the development of the homological theory of quantum general linear groups previously considered by the first author the development is used to transfer information to the representation theory of quantised schur algebras the acyclicity of induction from some rank one modules for quantised borel schur subalgebras is deduced this is used to prove the exactness of the complexes recently constructed by boltje and maisch giving resolutions of the co specht modules for hecke algebras

on the ext groups between weyl modules for gln abstract this paper studies extension groups between certain weyl modules for the algebraic group gl n over the integers main results include 1 a complete determination of ext groups between weyl modules whose highest weights differ by a single root and 2 determination of ext 1 between an exterior power of the defining representation and any weyl module the significance of these results for modular representation theory of gl n is discussed in several remarks notably the first result leads to a calculation of ext groups between neighboring weyl modules for gl n and also recovers the gl n case of a recent result of andersen some generalities about ext groups between weyl modules and a brief overview of known results about these groups are also included

characteristic free representation theory of the general linear group on etudie systematiquement les z formes des representations rationnelles du groupe lineaire generale

on schur algebras and related algebras vi some remarks on rational and classical schur algebras abstract in a recent paper dipper and doty 4 introduced certain finite dimensional algebras associated with the natural module of the general linear group and its dual which they call rational schur algebras we give a proof via tilting modules that these algebras are in fact generalised schur algebras using the same technique we show that certain finite dimensional algebras associated with classical groups introduced by doty 20 are quasi hereditary algebras a generalised schur algebras may be viewed as a quotient of the algebra of distributions of a reductive group by a certain ideal we give generators for this ideal

a generic algebra associated to certain hecke algebras abstract we initiate the systematic study of endomorphism algebras of permutation modules and show they are obtainable by a descent from a certain generic algebra infinite dimensional in general coming from the universal enveloping algebra of sl n or gl n the endomorphism algebras and the generic algebras are cellular in the latter case of profinite type in the sense of r m green we give several equivalent descriptions of these algebras find a number of explicit bases for them and describe indexing sets for their irreducible representations moreover we show that the generic algebra embeds densely in an endomorphism algebra of a certain infinite dimensional induced module

representation type of schur algebras we give a complete classification of the classical schur algebras and the infinitesimal schur algebras which have tame representation type in combination with earlier work of some of the authors on semisimplicity and finiteness this completes the classification of representation type of all classical and infinitesimal schur algebras in all characteristics

on the defining relations for generalized q schur algebras abstract we show that the defining relations needed to describe a generalized q schur algebra as a quotient of a quantized enveloping algebra are determined completely by the defining ideal of a certain finite affine variety the points of which correspond bijectively to the set of weights this explains unifies and extends previous results

composition factors of tensor products of truncated symmetric powers let g be the general linear group of degree n over an algebraically closed field k of characteristic p 0 we study the m fold tensor product s e m of the truncated symmetric algebra s e of the symmetric algebra s e of the natural module e for g we are particularly interested in the set of partitions occurring as the highest weight of a composition factor of s e m we explain how the determination of these composition factors is related to the determination of the set of composition factors of the m fold tensor product s e m of the symmetric algebra we give a complete description of the composition factors of s e m in terms of distinguished partitions our main interest is in the classical case but since the quantised version is essentially no more difficult we express our results in the general context throughout

representation type of the blocks of category os abstract in this paper the authors investigate the representation type of the blocks of the relative parabolic category o s for complex semisimple lie algebras a complete classification of the blocks corresponding to regular weights is given the main results of the paper provide a classification of the blocks in the mixed case when the simple roots corresponding to the singular set and s do not meet

a proof of the mullineux conjecture a partition of a positive integer n is a sequence 1 2 m 0 of integers such that i n for a positive integer p a partition 1 2 m or its young diagram is called p regular if it does not have p or more equal parts i e if there does not exist t m p 1 with t t 1 t p 1 let f be a field of characteristic p 0 it is well known that irreducible representations of the symmetric group sn over f are naturally parametrized by pregular partitions of n cf for example 9 12 if is such a partition we denote the corresponding irreducible module by d let sgnn be the one dimensional sign representation of sn over f i e sgnn f as a vector space and g f sgn g f for any g sn f f here sgn g is just the sign of the permutation g it is clear that for any irreducible d the tensor product d sgnn is also irreducible the problem usually called the problem of mullineux is to find the p regular partition such that d sgnn d put d sgnn dn in this way a bijection bn on the set pn of p regular partitions of n is defined for each positive integer n and the problem is

the varieties for some specht modules abstract j carlson introduced the cohomological and rank variety for a module over a finite group algebra we give a general form for the largest component of the variety for the specht module for the partition p p of p 2 restricted to a maximal elementary abelian p subgroup of rank p we determine the varieties of a large class of specht modules corresponding to p regular partitions to any partition of np of not more than p parts with empty p core we associate a unique partition of np where the rank variety of the restricted specht module s e n to a maximal elementary abelian p subgroup e n of rank n is v e n k if and only if v e n s v e n k in some cases where is a 2 part partition we show that the rank variety v e n s is v e n k in particular the complexity of the specht module s is n

some remarks on tilting modules double centralisers and lie modules abstract in a recent paper 14 s doty and j m douglass consider the degree r component l r e of the free lie algebra l e on an n dimensional vector space e over a suitable field k of characteristic not dividing r the authors discuss amongst other things the question of when this space regarded as a module for the general linear group of degree n equivalently the schur algebra s n r has the double centraliser property some general classes are described but the question in general is left open here we show that at most one block component of l r e can fail to have the double centraliser property and give some examples of the failure of the property for that block and therefore for the lie module

on the blocks of semisimple algebraic groups and associated generalized schur algebras in this paper we give a new proof for the description of the blocks of any semisimple simply connected algebraic group when the characteristic of the field is greater than 5 the first proof was given by donkin and works in arbitrary characteristic our new proof has two advantages first we obtain a bound on the length of a minimum chain linking two weights in the same block second we obtain a sufficient condition on saturated subsets of the set of dominant weights which ensures that the blocks of the associated generalized schur algebra are simply the intersection of the blocks of the algebraic group with the set however we show that this is not the case in general for the symplectic schur algebras disproving a conjecture of renner

schur weyl duality for orthogonal groups we prove schur weyl duality between the brauer algebra bn m and the orthogonal group om k over an arbitrary infinite field k of odd characteristic if m is even we show that each connected component of the orthogonal monoid is a normal variety this implies that the orthogonal schur algebra associated to the identity component is a generalized schur algebra as an application of the main result an explicit and characteristic free description of the annihilator of n tensor space v n in the brauer algebra bn m is also given

the grothendieck inequality for bilinear forms on c algebras abstract the following generalization of grothendieck s inequality is proved for any bounded bilinear form v on a pair of c algebras a b there exist two states 1 2 on a and two states 1 2 on b such that v x y v 1 x x 2 xx 2 1 2 1 y y 2 yy 2 1 2 for all x a and all y b an inequality of this type was proved a few years ago by pisier in the case where one of the c algebras has the bounded approximation property it follows from the above inequality that any bounded linear map t of a c algebra into the dual of a c algebra has a factorization t r s through a hilbert space such that r s 2 t

schur algebras of classical groups abstract throughout this paper the base field will be c by doty s definition s doty polynomial representations algebraic monoids and schur algebras of classical type j pure appl algebra 123 1998 165 199 a schur algebra of a classical group g is the image of the representation map c g end c c n r where c n is the natural representation and r any natural number these schur algebras are semisimple over c firstly we determine when the schur algebras are generalized schur algebras in donkin s sense see s donkin on schur algebras and related algebras i j algebra 104 1986 310 328 the main step is to decompose the tensor space c n r using path model by littelmann p littelmann a littlewood richardson rule for symmetrizable kac moody algebras invent math 116 1994 329 346 secondly we relate schur algebras with different parameters and form inverse systems from schur algebras in the same type we find the inverse limit naturally contains the universal enveloping algebra of the corresponding lie algebra

wiener hopf splitting of a 2 x 2 matrix explicit expressions for the wiener hopf factors of a 2 x 2 matrix are determined in terms of a single scalar examples of the application to various matrices are given

a note on decomposition numbers for reductive algebraic groups in a recent paper 7 james proves two theorems on the decomposition numbers for the general linear groups and working within the context of green s monograph 5 derives properties of the decomposition numbers for the symmetric groups this note grew out of an attempt to formulate for arbitrary reductive groups these theorems of james we prove a natural result on decomposition numbers for reductive groups from which james s theorems are deduced the result the theorem of section 2 is the analogue of jantzen s satz 1 18 of 8 on decomposition numbers for representations of complex semisimple lie algebras also proved independently by deodhar very little direct use of 5 is made though some of the ideas are cribbed from that source as one may see by comparing the properties of the functor fa in section 2 with green s treatment see section 6 2 of s of ese theory s a ring e an idempotent

diffraction by a highly contrast transparent wedge the problem of a plane wave diffraction by a highly contrast transparent wedge of an arbitrary opening is studied with the aid of the sommerfeld integrals it is reduced to a system of coupled maliuzhinets equations by use of the theory of s integrals the system of functional equations is transformed to a system of linear equations in a banach space in the case of the high contrast of a material inside the wedge in comparison with that in the wedge s exterior the linear equations are solved by means of perturbation theory convergence of the corresponding neumann series is proved singularities of the integrands in the sommerfeld integrals are investigated application of the steepest descent method leads to the determination of the reflected transmitted and diffracted waves expressions for the diffraction coefficients are also represented

on one approach to an electromagnetic diffraction problem in a wedge shaped region an electromagnetic diffraction problem in a wedge shaped region is reduced to a system of coupled functional equations by means of sommerfeld integrals anisotropic impedance boundary conditions are satisfied on the wedge s faces this system of functional equations is solved by regular perturbation method it is shown that for weak anisotropy the solution is presented by converging series which are neumann recurrent series for linear equations with contracting operators in a general case the problem is reduced to linear equations with compact operators the wave field asymptotic is computed for the region outside the vicinity of the edge of the wedge

on the good filtration dimension of weyl modules for a linear algebraic group let g be a linear algebraic group over an algebraically closed field of characteristic p whose corresponding root system is irreducible in this paper we calculate the weyl filtration di mension of the induced g modules and the simple g modules l fora regular weight we use this to calculate some ext groups of the form ext extl l and ext where are regular and is the weyl module of highest weight we then deduce the projective dimensions and injective dimensions for l and for a regular weight in associated generalised schur algebras we also deduce the global dimension of the schur algebras for gln s n r when p n and for s mp p with m an integer

on schur algebras and related algebras ii this paper is a continuation of 7 the purpose of 7 was to set up in the context of reductive algebraic groups a theory of generalized schur algebras a special case of our construction gives the schur algebras s n r studied extensively in lo we concentrate here on the schur algebras themselves and show that our construction yields new results in their representation theory by means of the schur functor see 10 chap 6 we obtain new results in the representation theory of the symmetric groups the main results are as follows the schur algebras s n r defined over a principal ideal domain r have finite global dimension see also 1 i over a field or complete valuation ring r every finitely generated projective s n r module has a weyl module filtration we prove the irreducibility of the socle of certain weyl modules see also 15 we describe a connection between weights of projective indecomposable modules for the schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation we show that over a complete valuation ring or a field every young module for a symmetric group has a specht series in particular the projective indecomposables have specht series finally we show that for r n and a complete valuation ring or field r the blocks of the schur algebra s n r are given by the rule embodied in nakayama s conjecture the notations and conventions are as in the first part of this work 7

representation theory of reductive normal algebraic monoids new results in the representation theory of semisimple algebraic monoids are obtained based on renner s monoid version of chevalley s big cell the semisimple algebraic monoids have been classified by renner the rational representations of such a monoid are the same thing as polynomial representations of the associated reductive group of units in the monoid and this representation category splits into a direct sum of subcategories by homogeneous degree we show that each of these homogeneous subcategories is a highest weight category in the sense of cline parshall and scott and so equivalent with the module category of a certain finite dimensional quasihereditary algebra which we show is a generalized schur algebra in s donkin s sense

straightening rule for an m truncated polynomial ring abstract we consider a certain quotient of a polynomial ring categorified by both the isomorphic green rings of the symmetric groups and schur algebras generated by the signed young permutation modules and mixed powers respectively they have bases parametrised by pairs of partitions whose second partitions are multiples of the odd prime p the characteristic of the underlying field we provide an explicit formula rewriting a signed young permutation module respectively mixed power in terms of signed young permutation modules respectively mixed powers labelled by those pairs of partitions as a result for each partition we discovered the number of compositions such that can be rearranged to and whose partial sums of are not divisible by p

skew weyl modules for gln and degree reduction for schur algebras abstract we prove a strong characteristic free analogue of the classical adjoint formula s s f s f in the ring of symmetric functions this is done by showing that the representative of a suitably chosen functor involving a tensor product is the skew weyl module by strong we mean that this representative preserves not only hom groups but higher ext groups also a fact which can be used to compute some homological invariants of weyl modules for gln via recursion on degree we use the following main tools existence of weyl filtrations in tensor products of weyl modules the akin buchsbaum weyman constructions of weyl modules and certain vanishing properties of ext groups

stratifying endomorphism algebras associated to hecke algebras abstract let g be a finite group of lie type and let k be a field of characteristic distinct from the defining characteristic of g in studying the non describing representation theory of g the endomorphism algebra s g k end kg j ind g p j k plays an increasingly important role in type a by work of dipper and james s g k identifies with a q schur algebra and so serves as a link between the representation theories of the finite general linear groups and certain quantum groups this paper presents the first systematic study of the structure and homological algebra of these algebras for g of arbitrary type because s g k has a reinterpretation as a hecke endomorphism algebra it may be analyzed using the theory of hecke algebras its structure turns out to involve new applications of kazhdan lusztig cell theory in the course of this work we prove two stratification conjectures about coxeter group representations made by e cline b parshall and l scott mem amer math soc 591 1996 and we formulate a new conjecture about the structure of s g k we verify this conjecture here in all rank 2 examples

brauer algebras symplectic schur algebras and schur weyl duality in this paper we prove the schur weyl duality between the symplectic group and the brauer algebra over an arbitrary infinite field k we show that the natural homomorphism from the brauer algebra b n 2m to the endomorphism algebra of the tensor space k 2m n as a module over the symplectic similitude group gsp 2m k or equivalently as a module over the symplectic group sp 2m k is always surjective another surjectivity that of the natural homomorphism from the group algebra for gsp 2m k to the endomorphism algebra of k 2m n as a module over b n 2m is derived as an easy consequence of s oehms s results s oehms j algebra 1 244 2001 19 44

quantum weyl reciprocity for cohomology we study the relationship between the cohomology of q schur algebras and hecke algebras in type a a key tool which has independent interest involves a specific hecke algebra resolution introduced in a special case by deodhar this resolution is somewhat akin to the koszul resolution for lie algebra cohomology in that it has an explicit description yet applies for all modules an application of the properties of this resolution establishes a quantum weyl reciprocity connecting in a range of degrees the cohomology of the q schur algebra with that of the hecke algebra this work also provides independently of the resolution cohomological inequalities which apply in all degrees and which sometimes can be related to kazhdan lusztig polynomials finally we extend to cohomology a decomposition number method of erdmann and illustrate its usefulness in computations this method enables us to exhibit some properties of symmetric group cohomology which if true would imply the sl n version of lusztig s characteristic p character conjecture

a utd solution for the scattering by a wedge with anisotropic impedance faces skew incidence case asymptotic expressions for the fields scattered by an anisotropic impedance wedge at oblique incidence are derived in the context of the uniform geometrical theory of diffraction utd they are obtained by resorting to a perturbative approach considering the normal incidence case as the imperturbed configuration we observe that the limits of applicability of this approximate analytical solution extend far beyond those of standard perturbative approaches allowing us to account for deviations from the normal incidence case of 20 spl deg to 30 spl deg

on central idempotents in the brauer algebra abstract we provide a method for constructing central idempotents in the brauer algebra using the splitting of short exact sequences of bimodules from this we determine certain primitive central idempotents by working over a suitable integral ring we hence demonstrate an efficient method of constructing pieces of the representation theory of the brauer algebra over artinian rings from the integral case

on some trivial source specht modules abstract the paper presented here focuses on the classification of trivial source specht modules we completely classify the trivial source specht modules labelled by hook partitions we also classify the trivial source specht modules labelled by two part partitions in the odd characteristic case moreover in the even characteristic case we prove a result for the classification of the trivial source specht modules labelled by partitions with 2 weight 2 which justifies a conjecture of 16

on the continuous wavelet transform of second order random processes some second order properties of random processes such as periodic correlation stationarity harmonizability self similarity are characterized via corresponding properties of their wavelet transform any one of these properties of the wavelet transform characterizes the corresponding property of the increments of the random process of order equal to the order of regularity of the analyzing wavelet these results are then specialized to fractional brownian motion and other self similar processes

on the almost sure convergence of series of stationary and related nonstationary variables let x n be for example a weakly stationary sequence or a lacunary system with finite pth moment 1 p 2 and let a n be a sequence of scalars we obtain here conditions which ensure the almost sure convergence of the series a n x n when x n is an orthonormal sequence the classical rademacher menchov theorem is recovered this is then applied to study the strong consistency of least squares estimates in multiple regression models

modular representations of the supergroup q n i the representation theory of the algebraic supergroup q n has been studied quite intensively over the complex numbers in recent years especially by penkov and serganova 18 19 20 culminating in their solution 21 22 of the problem of computing the characters of all irreducible finite dimensional representations ofq n the characters of one important family of irreducible representations the so called polynomial representations had been determined earlier by sergeev 24 exploiting an analogue of schur weyl duality connecting polynomial representations of q n to the representation theory of the double covers n of the symmetric groups in 2 we used sergeev s ideas to classify for the first time the irreducible representations of n over fields of positive characteristic p 2 in the present article and its sequel we begin a systematic study of the representation theory of q n in positive characteristic motivated by its close relationship to n let us briefly summarize the main facts proved in this article by purely algebraic techniques let g q n defined over an algebraically closed field k of characteristic p 6 2 see 2 3 for the precise definition in 4 we construct the superalgebra dist g of distributions on g by reduction modulo p from a kostant z form for the enveloping superalgebra of the lie superalgebra q n c this provides one of the main tools in the remainder of the paper there is an explicit equivalence between the category of representations of g and the category of integrable dist g supermodules see corollary 5 7 in 6 we classify the irreducible representations of g by highest weight theory they turn out to be parametrized by the set x p n 1 n z 1 n with i i 1 only if p i for x p n the corresponding irreducible representation is denoted l and is constructed naturally as the simple socle of an induced representation h indbu where b is a borel subgroup of g and u is a certain irreducible representation of b of dimension a power of 2 the main difficulty here is to show that h0 6 0 for x p n which we prove by exploiting the main result of 2 classifying the irreducible polynomial representations of g so ultimately the proof that h0 6 0 depends on a counting argument involving p regular conjugacy classes in n

schur algebras of brauer algebras i a classical problem of invariant theory and of lie theory is to determine endomorphism rings of representations of classical groups for instance of tensor powers of the natural module schur weyl duality or of full direct sums of tensor products of exterior powers ringel duality in this article the endomorphism rings of full direct sums of tensor products of symmetric powers over symplectic and orthogonal groups are determined these are shown to be isomorphic to schur algebras of brauer algebras as defined in henke and koenig math z 272 3 4 729 759 2012 this implies structural properties of the endomorphism rings such as double centraliser properties quasi hereditary and a universal property as well as a classification of simple modules

affine strict polynomial functors and formality abstract we introduce the notion of an affine strict polynomial functor we show how this concept helps to understand homological behavior of the operation of frobenius twist in the category of strict polynomial functors over a field of positive characteristic we also point out for an analogy between our category and the category of representations of the group of algebraic loops on g l n

a criterion for quasi hereditary and an abstract straightening formula a criterion is given to show that a k algebra a is quasi hereditary if it can be defined over an integral domain r and if there is a certain commutative semisimple subalgebra satisfying a technical but easily verified condition which roughly states that over the field of fractions k of r the formal characters of the semisimple k algebra generated by the r algebra defining a satisfy an ordering condition this applies in particular to schur algebras where various proofs of quasi hereditary are known by de concini eisenbud and procesi by donkin by parshall and by j a green generalized schur algebras covering a result of donkin q schur algebras dipper and james parshall and wang and temperley lieb algebras westbury

rational cohomology and cohomological stability in generic representation theory with f q a finite field of characteristic p let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i q be the category whose objects are functors from finite dimensional f q vector spaces to inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i vector spaces friedlander and suslin have introduced a category inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i of strict polynomial functors which has the same relationship to inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i q that the category of rational gl m modules has to the category of gl m f q modules our main theorem says that for all finite objects f g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i and all s the natural restriction map from inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i f k g k to inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i f g is an isomorphism for all large enough k and q here f k denotes f twisted by the frobenius k times this combines with an analogous theorem of cline parshall scott and van der kallen to show that for all finite f g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 08i and all s evaluation on an m dimensional vector space v m induces an isomorphism from inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 09i f g to ext s gl m f q f v m f v m for all large enough m and q thus group cohomology of the finite general linear groups has often been identified with maclane or topological hochschild cohomology

stable processes moving averages versus fourier transforms we study the relationship between fourier transforms and moving averages of stable processes the two classes have a large intersection however as soon as the noise is required to be bounded they become disjoint

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal

the brauer algebra and the symplectic schur algebra let k be an algebraically closed field of characteristic p 0 let m r be integers with m 1 r 0 and m r and let s0 2m r be the symplectic schur algebra over k as introduced by the first author we introduce the symplectic schur functor derive some basic properties of it and relate this to work of hartmann and paget we do the same for the orthogonal schur algebra we give a modified jantzen sum formula and a block result for the symplectic schur algebra under the assumption that r and the residue of 2m mod p are small relative to p from this we deduce a block result for the orthogonal schur algebra under similar assumptions we also show that in general the block relations of the brauer algebra and the symplectic and orthogonal schur algebra are the same finally we deduce from the previous results a new proof of the description of the blocks of the brauer algebra in characteristic 0 as obtained by cox de visscher and martin

diffraction of an electric polarized wave by a dielectric wedge the problems of diffraction of an e polarized plane wave by an inhomogeneous imperfectly conducting dielectric wedge or a homogeneous dielectric wedge with vanishing conductivity are formulated as integral equations conditions for existence and uniqueness of the solutions of the integral equations are found the integral equations are solved by iterative methods an explicit expression for the far field is found the series expansion of the far field near the edge of a homogeneous wedge converges very fast for relative permittivities in the interval from one to ten the physical optical approximation of the far field is poor