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

good filtrations and strong f regularity of the ring of u p invariants abstract let k be an algebraically closed field of positive characteristic g a reductive group over k and v a finite dimensional g module let p be a parabolic subgroup of g and u p its unipotent radical we prove that if s sym v has a good filtration then s u p is strongly f regular

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

multiplicity free subgroups of reductive algebraic groups abstract we introduce the notion of a multiplicity free subgroup of a reductive algebraic group in arbitrary characteristic this concept already exists in the work of kramer for compact connected lie groups we give a classification of reductive multiplicity free subgroups and as a consequence obtain a simple proof of a theorem of kleshchev

hermitian matrices with a bounded number of eigenvalues abstract conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex hermitian matrices with a bounded number of distinct eigenvalues a minimal generating system of the vanishing ideal of degenerate three by three hermitian matrices is given and the structure of the corresponding coordinate ring as a module over the special unitary group is determined the method applies also for degenerate real symmetric three by three matrices for arbitrary n partial information on the minimal degree component of the vanishing ideal of the variety of n n hermitian matrices with a bounded number of eigenvalues is obtained and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex hermitian matrices

ordered sets with the standardizing property and straightening laws for algebras of invariants abstract in math z 176 1981 359 374 i explicitly determined the invariants of a certain class of unipotent group actions and obtained a positive partial answer to hilbert s 14th problem for nonreductive groups the class of groups for which the method worked remained quite obscure theorem 4 2 of the present paper gives a precise description of the cases where the algebras of invariants are spanned by standard bitableaux hence have a straightening law the unipotent groups in question radizielle untergruppen of gln correspond up to conjugation to finite partially ordered sets the promised description is done by properties of the ordered sets that are easy to test this is another example where combinatorial methods are important for the theory of invariants

higher order numerical schemes for affinely controlled nonlinear systems a systematic method for the derivation of high order schemes for affinely controlled nonlinear systems is developed using an adaptation of the stochastic taylor expansion for control systems we construct taylor schemes of arbitrary high order and indicate how derivative free runge kutta type schemes can be obtained furthermore an approximation technique for the multiple control integrals appearing in the schemes is proposed

skew modules for reductive groups skew modules for symmetric groups have been investigated by james and peel 13 farahat and peel 9 and clausen s more recently skew modules for general linear groups have been investigated by akin buchsbaum weyman 3 l 2 and kouwenhoven 161 our aim in this paper is to set up in a rather general context a theory of skew modules for reductive algebraic groups for a partition 1 of it james peel 13 obtain a filtration of the specht module sj viewed as a module for the young subgroup corresponding to a decomposition n a b of n the successive quotients in the filtration have the form s h sj p kronecker product where p runs over the partitions of a such that p is a skew partition and s denotes the skew specht module the analogue of this result is proved for general linear groups by akin buchsbaum and weyman 3 theorem 11 4 1 l and kouwenhoven 16 proposition ii 2 2a the analogue for reductive groups is more or less our definition of skew modules this filtration property is in fact a characterisation of skew modules see 1 3 4 and our definition for general linear groups therefore agrees with that of akin buchsbaum and weyman 3 and kouwenhoven 161 it becomes apparent rather quickly from our point of view that the skew modules have a particular kind of filtration see 2 1 1 which in the case of the general linear group is a filtration by schur modules this is a result of kouwenhoven 16 proposition 11 2 2b a for general linear groups kouwenhoven is able to give an explicit description of such a filtration one may then obtain as in kouwenhoven 16 proposition ii 2 2b b the result of james and peel 131 on the existence of specht series for skew modules for the symmetric groups by specializing to a certain weight space i e applying the schur functor as in ll chap 63 we are also able to prove see 2 3 3 the existence of a certain kind of resolution considered by akin and buchsbaum see 1 sects 83 for skew modules for general linear groups our main point of reference is the representation theory of reductive 465 0021 8693 88 3 00

regularized nonparametric volterra kernel estimation abstract in this paper the regularization approach introduced recently for nonparametric estimation of linear systems is extended to the estimation of nonlinear systems modeled as volterra series the kernels of order higher than one representing higher dimensional impulse responses in the series are considered to be realizations of multidimensional gaussian processes based on this prior information about the structure of the volterra kernel is introduced via an appropriate penalization term in the least squares cost function it is shown that the proposed method is able to deliver accurate estimates of the volterra kernels even in the case of a small amount of data points

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

uber bewertungen welche unter einer reduktiven gruppe invariant sind letg be a reductive group defined over an algebraically closed fieldk and letx be ag variety in this paper we studyg invariant valuationsv of the fieldk of rational functions onx these objects are fundamental for the theory of equivariant completions ofx letb be a borel subgroup andu the unipotent radical ofb it is proved thatv is uniquely determined by its restriction toku then we study the set of invariant valuations having some fixed restrictionv0 tokb ifv0 is geometric i e induced by a prime divisor then this set is a polyhedron in some vector space in characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupwx thus the classification of invariant valuations is almost reduced to the classification of valuations ofkb

groups quasi isometric to complex hyperbolic space we show that any finitely generated group quasi isometric to complex hyperbolic space is a finite extension of a properly discontinuous cocompact subgroup of the isometry group

minimal and maximal operator spaces and operator systems in entanglement theory abstract we examine k minimal and k maximal operator spaces and operator systems and investigate their relationships with the separability problem in quantum information theory we show that the matrix norms that define the k minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k positive linear maps and bound entanglement similarly we investigate the k super minimal and k super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k block positive operators and unnormalized states with schmidt number no greater than k respectively we characterize a class of norms on the k super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the schmidt number of a quantum state that generalizes the recently developed separability criterion based on trace contractive maps

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

quantum gate fidelity in terms of choi matrices we provide new results for computing and comparing the quantum gate fidelity of quantum channels via their choi matrices we extend recent work that showed that there exist non dual pairs of quantum channels with equal gate fidelity by providing an explicit characterization of all such channels we use our characterization to show that when the dimension is 2 or 3 under slightly stronger hypotheses the gate fidelity of two channels is equal if and only if their difference equals the difference of some unital map and its dual a fact that has been shown to be false when the dimension is 4 or larger we also present a formula for the minimum gate fidelity of a channel in terms of a well studied norm on a compression of its choi matrix as a consequence several new ways of bounding and approximating the minimum gate fidelity follow including a simple semidefinite program to compute it for qubit channels

reduction and reconstruction for symmetric ordinary differential equations we discuss the reduction and reconstruction problem for ordinary differential equations that admit a linear symmetry group the goal is to prove that modulo reduction there remain only linear differential equations and to construct these explicitly extending previous work on one parameter groups we show this for certain unipotent and solvable groups and for all semisimple groups some applications to relative equilibria are given

cones of positive maps and their duality relations the structure of cones of positive and k positive maps acting on a finite dimensional hilbert space is investigated special emphasis is given to their duality relations to the sets of superpositive and k superpositive maps we characterize k positive and k superpositive maps with regard to their properties under taking compositions a number of results obtained for maps are also rephrased for the corresponding cones of block positive k block positive separable and k entangled operators due to the jamiolkowski choi isomorphism generalizations to a situation where no such simple isomorphism is available are also made employing the idea of mapping cones as a side result to our discussion we show that extreme entanglement witnesses which are optimal should be of special interest in entanglement studies

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

a family of norms with applications in quantum information theory we consider a family of vector and operator norms defined by the schmidt decomposition theorem for quantum states we use these norms to tackle two fundamental problems in quantum information theory the classification problem for k positive linear maps and entanglement witnesses and the existence problem for nonpositive partial transpose bound entangled states we begin with an analysis of the norms showing that the vector norms can be explicitly calculated and we derive several inequalities in order to bound the operator norms and compute them in special cases we then use the norms to establish what appears to be the most general spectral test for k positivity currently available showing how it implies several other known tests as well as some new ones building on this work we frame the nonpositive partial transpose bound entangled problem as a concrete problem on a specific limit specifically that a particular entangled werner state is bound entangled if and only if a certain norm inequality ho

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

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

disguising quantum channels by mixing and channel distance trade off we consider the reverse problem of the distinguishability of two quantum channels which we call the disguising problem given two quantum channels the goal here is to make the two channels identical by mixing with some other channels with minimal mixing probabilities this quantifies how much one channel can disguise as the other in addition the possibility to trade off between the two mixing probabilities allows one channel to be more preserved less mixed at the expense of the other we derive lower and upper bounds of the trade off curve and apply them to a few example channels optimal trade off is obtained in one example we relate the disguising problem and the distinguishability problem by showing that the former can lower and upper bound the diamond norm we also show that the disguising problem gives an upper bound on the key generation rate in quantum cryptography

choi matrices norms and entanglement associated with positive maps on matrix algebras abstract we study positive maps of b k into b h for finite dimensional hilbert spaces k and h our main emphasis is on how choi matrices and estimates of their norms with respect to mapping cones reflect various properties of the maps special attention will be given to entanglement properties and k positive maps

spectral conditions for positive maps we provide partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions this construction generalizes the celebrated choi example of a map which is positive but not completely positive it is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements such maps provide basic tools to study quantum entanglement in multipartite systems

a cohomological characterization of parabolic subgroups of reductive algebraic groups let g be an algebraic group defined over an algebraically closed field k and rat g the category of rational g modules for any closed subgroup h of g and any rational h module v let v ig denote the rational g module induced from i the induction functor ig rat h rat g is left exact and we denote its derived functors by l j for n o 1 2 of course these derived functors are 0 for n 2 1 whenever 1 g is exact and by a result of cline parshall and scott 3 this happens precisely when g h is an afftne variety so the case of an afftne quotient is characterized by the vanishing of the higher derived functors in a like vein in this paper we will attempt to find a property of these functors which characterizes the case of a projective quotient since by definition g h is projective iff h p is a parabolic subgroup of g we are in effect looking for a cohomological characterization of parabolic subgroups of g the property of the functors l j we consider is a certain finiteness condition explained below the author was originally led to consider this question in 14 where induction from certain nonparabolic subgroups is studied and is shown to have this finiteness property it is well known that for every rational h module v there is an induced bundle l on g h with rank equal to dim i such that l j v identifies naturally with the sheaf cohomology group h g h l see 4 for details thus we may apply standard results from algebraic geometry to obtain some basic facts for example lg g 0 for n dim g h and serre s theorem gives the easy half of the result of cline parshall and scott men

an algorithm to explore entanglement in small systems a quantum state s entanglement across a bipartite cut can be quantified with entanglement entropy or more generally schmidt norms using only schmidt decompositions we present a simple iterative algorithm to maximize schmidt norms depending on the choice of norm the optimizing states maximize or minimize entanglement possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace recognizing that convergence but not success is certain we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels

the inverse eigenvalue problem for entanglement witnesses abstract we consider the inverse eigenvalue problem for entanglement witnesses which asks for a characterization of their possible spectra or equivalently of the possible spectra resulting from positive linear maps of matrices we completely solve this problem in the two qubit case and we derive a large family of new necessary conditions on the spectra in arbitrary dimensions we also establish a natural duality relationship with the set of absolutely separable states and we completely characterize witnesses i e separating hyperplanes of that set when one of the local dimensions is 2

how often is a random quantum state k entangled the set of trace preserving positive maps acting on density matrices of size d forms a convex body we investigate its nested subsets consisting of kpositive maps where k 2 d working with the measure induced by the hilbert schmidt distance we derive asymptotically tight bounds for the volumes of these sets our results strongly suggest that the inner set of k 1 positive maps forms a small fraction of the outer set of k positive maps these results are related to analogous bounds for the relative volume of the sets of k entangled states describing a bipartite d d system

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

on conjugating representations and adjoint representations of semisimple groups where ea is a direct sum of irreducible g modules of highest weight 2 here we consider the conjugating representation in arbitrary characteristic and prove the appropriate version of richardson s theorem under small and almost certainly unnecessary characteristic root system restrictions one can no longer decompose a g into homogeneous components but instead we prove the existence of an ascending

nonlinear system identification for multivariable control via discrete time chen fliess series abstract the nonlinear system identification problem is solved for a multivariable nonlinear input output system that can be represented in terms of a chen fliess functional expansion the problem is formulated in terms of a regression which is linear in the parameters and has a nonlinear regressor an inductive implementation of the nonlinear regressor is developed using the underlying noncommutative algebraic and combinatorial structures the method is demonstrated in an adaptive control application involving a two input two output lotka volterra dynamical system

on the dimension of subspaces with bounded schmidt rank we consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states this is motivated in part by results of hayden et al e print arxiv quant ph 0407049 commun math phys 265 95 2006 which show that in large d d dimensional systems there exist random subspaces of dimension almost d2 all of whose states have entropy of entanglement at least logd o 1 it is also a generalization of results on the dimension of completely entangled subspaces which have connections with the construction of unextendible product bases here we take as entanglement measure the schmidt rank and determine for every pair of local dimensions da and db and every r the largest dimension of a subspace consisting only of entangled states of schmidt rank r or larger this exact answer is a significant improvement on the best bounds that can be obtained using the random subspace techniques in hayden et al we also determine the converse the l

discrete time approximations of fliess operators a convenient way to represent a nonlinear input output system in control theory is via a chen fliess functional expansion or fliess operator the general goal of this paper is to describe how to approximate fliess operators with iterated sums and to provide accurate error estimates for two different scenarios one where the series coefficients are growing at a local convergence rate and the other where they are growing at a global convergence rate in each case it is shown that the error estimates are asymptotically achievable for certain worst case inputs the paper then focuses on the special case where the operators are rational i e they have rational generating series and thus are realizable in terms of bilinear ordinary differential state equations in particular it is shown that a discretization of the state equation via a kind of euler approximation coincides exactly with the discrete time fliess operator approximator of the continuous time rational operator

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

operator spaces with few completely bounded maps we construct several examples of hilbertian operator spaces with few completely bounded maps in particular we give an example of a separable 1 hilbertian operator space x 0 such that whenever x is an infinite dimensional quotient of x 0 x is a subspace of x and t x rightarrow x is a completely bounded map then t i x s where s is compact hilbert schmidt and s 2 16 s cb s 2 moreover every infinite dimensional quotient of a subspace of x 0 fails the operator approximation property we also show that every banach space can be equipped with an operator space structure without the operator approximation property

fixed points of commutative super operators in this paper we study fixed points of commutative super operators first under some circumstances we discuss the question of necessary and sufficient conditions for the set of fixed points of a super operator to be non trivial then we establish a structure lemma to completely characterize this set using this lemma we show that if a commutative super operator is also a generalized luders operator then its fixed points are precisely those operators intertwining the two sequences of operation elements associated with it specifically the main results of weihua and junde 2010 j phys a math theor 43 395206 are special cases of our results

fliess operators on lp spaces convergence and continuity fliess operators as input output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory in the classical analysis of these operators certain growth conditions on the coe3cients in their series representations insure uniform and absolute convergence provided every input is uniformly bounded by some 4xed upperbound in some emerging applications however it is more natural to consider other classes of inputs in this paper lp function spaces are considered in particular it is shown that the classic growth conditions also provide su3cient conditions for convergence and continuity when the admissible inputs are from a ball in lp t0 t 0 t where t is bounded and p 1 in addition stronger global growth conditions are given that apply even for the case where t is unbounded when the coe3cients of a fliess operator have a state space representation it is shown that the state space model will always locally realize the correspondinginput output map on lp t0 t 0 t for su3ciently small t 0 if certain well posedness conditions are satis4ed then the state space model will globally realized the input output mapping for unbounded t when the coe3cients satisfy the global growth condition c 2002 elsevier science b v all rights reserved

a tensor network kalman filter with an application in recursive mimo volterra system identification abstract this article introduces a tensor network kalman filter which can estimate state vectors that are exponentially large without ever having to explicitly construct them the tensor network kalman filter also easily accommodates the case where several different state vectors need to be estimated simultaneously the key lies in rewriting the standard kalman equations as tensor equations and then implementing them using tensor networks which effectively transforms the exponential storage cost and computational complexity into a linear one we showcase the power of the proposed framework through an application in recursive nonlinear system identification of high order discrete time multiple input multiple output mimo volterra systems the identification problem is transformed into a linear state estimation problem wherein the state vector contains all volterra kernel coefficients and is estimated using the tensor network kalman filter the accuracy and robustness of the scheme are demonstrated via numerical experiments which show that updating the kalman filter estimate of a state vector of length 10 9 and its covariance matrix takes about 0 007 s on a standard desktop computer in matlab

algebraic hamiltonian actions in this paper we deal with a hamiltonian action of a reductive algebraic group g on an irreducible normal affine poisson variety x we study the quotient morphism mu g x g x g rightarrow mathfrak g g of the moment map mu g x x rightarrow mathfrak g we prove that for a wide class of hamiltonian actions including for example actions on generically symplectic varieties all fibers of the morphism g x g have the same dimension we also study the stein factorization of g x g namely let c g x denote the spectrum of the integral closure of mu g x mathbb k mathfrak g g in mathbb k x g we investigate the structure of the mathfrak g g scheme c g x our results partially generalize those obtained by f knop for the actions on cotangent bundles and symplectic vector spaces

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