characteristic free representation theory of the general linear group ii homological considerations in our first paper of this series cl we indicated how we were led to consider resolutions of schur and weyl modules of a particular form in order to prove the existence of these resolutions we were forced to enlarge the family of skew shapes to a class j containing new shapes whose corresponding schur and weyl modules had heretofore not been studied with these shapes in hand we proved in 1 the existence of some fundamental exact sequences and described how from these exact sequences we could use a mapping cone construction to build up the resolutions we were seeking for this mapping cone construction we needed maps and to provide maps we needed projectivity of tensor products of divided powers this led to the study of the schur algebra and its decomposition into orthogonal idempotents in sections 1 and 2 we review the information about the schur algebra that we need to carry out our program fortunately there is a very clear and detailed exposition of this subject in the notes of j a green s from which we borrowed very heavily in fact the main function of the first two sections is to condense and translate into our notation and terminology the relevant sections of green s notes in sections 3 and 4 we define the family of shapes j that we will study

resolutions of determinantal ideals the submaximal minors for many years there has been considerable interest in finding a resolution of the ideal generated by the minors of order p of a generic m x n matrix to put the problem more precisely suppose r is a commutative ring and x are variables with 1 i m and 1 j q n if we let r r x be the polynomial ring over r then we have the generic matrix x and we may form the ideal zp in r generated by the p x p minors of this matrix the problem then is to find an explicit free resolution of the ideal zp over the ring r it was proved by eagon and hochster lo that r i has a resolution of length m p l n p l but their proof consisted in showing that the ideal zp is perfect it did not provide a construction of the resolution in fact it is not known whether the betti numbers of the ideal zp depend on the characteristic of the ground ring r in 121 lascoux succeeded in giving an explicit resolution provided that the ground ring r contained the field q of rational numbers his construction rests heavily on the theory of schur functors and the fact that in characteristic zero the schur functors are the irreducible representations of the general linear group over the integers however the construction breaks down despite the fact that one can define the schur functors over an arbitrary commutative ring see 1 2 14 151 for various constructions of schur functors in analyzing the work of lascoux and its subsequent reworking by nielsen 131 some basic facts seemed to clamor for attention one was that within a resolution of r z there appeared to be two types of boundary maps one of degree 1 and one of degree p the maps of degree 1 were maps between sums of schur functors of fixed durfee square k see section 2 for definitions while the maps of degree p were from sums of schur functors of

on the arithmetic and homology of algebras of linear type three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras e g cohen macaulayness factoriality the main tools are certain fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing up rings

torsion freeness of symmetric powers of ideals let i be an ideal in a noetherian commutative ring r with unit let k 2 be an integer and let k i s k i i k be the canonical surjective r module homomorphism from the kth symmetric power of i to the kth power of i when pd r i 1 or when i is a perfect gorenstein ideal of grade 3 we provide a necessary and sufficient condition for k to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of i when i m is a maximal ideal of r we show that is an isomorphism if and only if r m is a regular local ring in all three cases for i our results yield that if is an isomorphism then at is also an isomorphism for each 1 t k

intertwining numbers the three rowed case a standard question in modular representation theory is what is the i zr p dimension of the zr p vector space ext k k the bars over a l m r the modules indicate the reduction of these modules modulo p the other notation is explained below these numbers are called intertwining numbers the attempts to determine these intertwining numbers are varied one approach is suggested by the following special case of a theorem found in w x ab1

sequences of linear type let a be a commutative ring 91 an ideal of a s 2l the symmetric algebra of 8 and r vl the recs algebra of vi there is a canonical surjection x s k which is homogeneous of degree zro and therefore gives canonical surjcctions r f 91 for every integer t 3 0 many recent papers 1 2 4 5 7 cc have investigated the problem of determining which ideals 91 have the property that a is an isomorphism such ideals have been dubbed ideals of lineur type s for example huneke showed that every ideal generated by a d sequence is of linear type 71 if ii is generated by n elements a a then for t 3 i there is an exact scquencc

a short proof of a theorem of m hashimoto abstract recently m hashimoto announced that the resolutions of determinantal ideals depend on the characteristic of the base field the purpose of this note is to give a short proof of this result one should stress that even though the method we use is different from hashimoto s his result provided us with the clue where the additional relation exists

bounds on the unstable eigenvalue for the asymmetric renormalization operator for period doubling we establish rigorous bounds for the unstable eigenvalue of the period doubling renormalization operator for asymmetric unimodal maps herglotz function techniques and cone invariance ideas are used our result generalizes an established result for conventional period doubling

schur de rham complex and its cohomology abstract we associate to a young diagram the schur de rham complex s we show that this complex is exact when the p core of is nontrivial and compute its cohomology when the p core is trivial and the p quotient of consists of a single diagram

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

acyclicity of schur complexes and torsion freeness of schur modules abstract let r be a noetherian commutative ring and m a r module with pd r m 1 that has rank necessary and sufficient conditions were provided in 1 for an exterior power k m to be torsion free when m is an ideal of r similar necessary and sufficient conditions were provided in 2 for a symmetric power s k m to be torsion free we extend these results to a broad class of schur modules l m en route for any map of finite free r modules f g we also study the general structure of the schur complexes l and provide necessary and sufficient conditions for the acyclicity of any given l by computing explicitly the radicals of the ideals of maximal minors of all its differentials

existence and properties of p tupling fixed points we prove the existence of fixed points of p tupling renormalization operators for interval and circle mappings having a critical point of arbitrary real degree r 1 some properties of the resulting maps are studied analyticity univalence behavior as r tends to infinity

on linear complete intersections or some of its localizations becomes a complete intersection is a major factor in section 1 we describe a jacobian formulation of serre s normality criterion for symmetric algebras it is used to provide several classes of examples of normal and non normal rees algebras of prime ideals in regular domains adding to the catalog of cohen macaulay domains that are symmetric algebras we examine almost complete intersections these bear a close relationship to certain height three ideals of

new proofs of the existence of the feigenbaum functions a new proof of the existence of analytic unimodal solutions of the cvitanovic feigenbaum functional equation g x g g x g x 1 const x r at 0 valid for all in 0 1 is given and the existence of the eckmann wittwer functions 8 is recovered the method also provides the existence of solutions for certain given values ofr and in particular forr 2 a proof requiring no computer

sphere covering measure concentration and source coding suppose a is a finite set let p be a discrete distribution on a and let m be an arbitrary mass function on a we give a precise characterization of the most efficient way in which a sup n can be almost covered using spheres of a fixed radius an almost covering is a subset c sub n of a sup n such that the union of the spheres centered at the points of c sub n has probability close to one with respect to the product distribution p sup n spheres are defined in terms of a single letter distortion measure on a sup n and an efficient covering is one with small mass m sup n c sub n in information theoretic terms the sets c sub n are rate distortion codebooks but instead of minimizing their size we seek to minimize their mass with different choices for m and the distortion measure on a our results give various corollaries as special cases including shannon s classical rate distortion theorem a version of stein s lemma in hypothesis testing and a new converse to some measure concentration inequalities on discrete spaces under mild conditions we generalize our results to abstract spaces and nonproduct measures

feigenbaum theory for unimodal maps with asymmetric critical point rigorous results we apply the methods of h epstein to prove the existence of a line of period 2 solutions of the feigenbaum period doubling renormalisation transformation these solutions govern the universal behaviour of families of unimodal maps with asymmetric critical points of degree d for which the d th derivative has differing left and right limits

analyticity properties of the feigenbaum function analyticity properties of the feigenbaum function a solution ofg x 1g g x withg 0 1 g 0 0 g 0 0 are investigated by studying its inverse function which turns out to be herglotz or anti herglotz on all its sheets it is found thatg is analytic and uniform in a domain with a natural boundary

permutation resolutions for specht modules of hecke algebras abstract in boltje and hartmann 2011 bh a chain complex was constructed in a combinatorial way which conjecturally is a resolution of the dual of the integral specht module for the symmetric group in terms of permutation modules in this paper we extend the definition of the chain complex to the integral iwahori hecke algebra and prove the same partial exactness results that were proven in the symmetric group case

a straightening algorithm for row convex tableaux abstract we produce a new basis for the schur and weyl modules associated to a row convex shape d the basis is indexed by a new class of straight tableaux which we introduce by weakening the usual requirements for standard tableaux spanning is proved via a new straightening algorithm for expanding elements of the representation into this basis for skew shapes this algorithm specializes to the classical straightening law the new straight basis is used to produce bases for flagged schur and weyl modules to provide groebner and sagbi bases for the homogeneous coordinate rings of some configuration varieties and to produce a flagged branching rule for row convex representations systematic use of supersymmetric letterplace techniques enables the representation theoretic results to be applied to representations of the general linear lie superalgebra as well as to the general linear group

a large deviation theory proof of the abstract alphabet source coding theorem a modified proof of t berger s 1971 abstract alphabet source coding with a fidelity criterion theorem is provided utilizing a variant of a result from large deviation theory this proof does not use an asymptotic equipartition property of any kind thus providing what appears to be a more direct approach to this key result

addendum specht modules for column convex diagrams characteristic free results for weyl modules abstract it is shown that the results in reiner and shimozono 12 have characteristic free analogues for weyl modules

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

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

on descent in dimension two and non split gorenstein modules in this paper we present a ring with a gorenstein module but no dualizing module the context of this example will be illustrated in the historical overview presented below in 21 sharp furnished the definition of a gorenstein module g of rank t over a local finite dimensional ring a g is finitely generated and for p spec a the bass number ll z g equals t if i hta fi and equals zero otherwise sharp went further to show that any ring admitting a gorenstein module must be cohen macaulay and also must have gorenstein formal fibres 21 we observe that a gorenstein module g of rank one over an n dimensional local ring a m k induces a natural equivalence of functors ext g hom e k on modules of finite length for this reason fossum foxby griffith and reiten assigned to gorenstein modules of rank one the name dualizing module s in the same paper it was obtained that any gorenstein a module is a direct sum of copies of the gorenstein a module of minimal rank of which up to isomorphism there is only one s it is clear that if a ring has a dualizing module then any of its gorenstein modules are direct sums of the dualizing module foxby s reiten 19 and sharp 21 showed that a local cohenmacaulay ring a admits a dualizing complex if and only if a has a dualizing module on the other hand grothendieck who initiated the notion of a dualizing complex provided the following result

on the divisor class group of rees algebras let r be a normal noetherian domain and let i be an ideal whose associated graded ring gr r is reduced barshay 2 showed that under these conditions the rees algebra s r it r 0 it 0 izt2 a e r t is normal a much simpler proof was given later by huneke 111 let p r it t be the extended rees algebra then pt e r t t 1 is normal and p tp e gr r is reduced it follows that p itself is normal this implies easily the normality of s as well we are interested in the class group cl s of s to obtain a neat result we strengthen our hypothesis and assume that gr r is a domain hochster 6 showed in this case that p is factorial if r is so the only tool needed to prove this is the following elementary but powerful theorem of nagata

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

on symplectic resolutions and factoriality of hamiltonian reductions recently herbig schwarz seaton have shown that 3 large representations of a reductive group g give rise to a large class of symplectic singularities via hamiltonian reduction we show that these singularities are always terminal we show that they are mathbb q factorial if and only if g has finite abelianization when g is connected and semi simple we show they are actually locally factorial as a consequence the symplectic singularities do not admit symplectic resolutions when g is semi simple we end with some open questions

schur modules weyl modules and capelli operators much work has been done in recent years towards presenting explicitly and with a minimum of machinery the irreducible representations of the general linear group gl n over a field k the representations obtained from the modules named after schur and weyl have undergone a tortuous refining process in the hands of notable authors akin buchsbaum and weyman 1 2 carter and lusztig 7 clausen 8 9 de concini eisenbud and procesi 10 green 16 james and kerber 19 towber 26 27 in the wake of the pioneering work of schur 23 and weyl 28 one objective of such simplifications has been that of obtaining information as to the reducibility of representations when the field k if of positive characteristic an objective which seems as yet out of reach the purpose of this work is that of organically presenting the main results on schur and weyl modules our techniques are at variance with most of those previously used and we should like to believe our proofs signify a return to the primordial combinatorial underpinning of the topic we hope thereby to have attained a maximum of characteristic freeness our two warhorses are 1 the well known straightening technique first used by doubilet rota and stein 15 and recently generalized to the supersymmetric case by grosshans rota and stein 17 an altogether different approach from which we have also benefitted has been developed by akin buchsbaum and weyman 1 both techniques has been skillfully displayed in the recent work of kouwenhoven 21 2 the use of polarization operators combined with the striking power of kostant s z form 20 in the guise of de sarme nien kung and rota s capelli operators 13 superalgebraic methods are used in obtaining two distinct standard basis theorems for the case of the exterior letterplace algebra such an algebra with its straightening possibilities was first considered by doubilet and rota 14 and then brought to fruish by akin buchsbaum and weyman 1 by these methods we derive what is perhaps the main novelty of the doi 10 1006 aima 1999 1872 available online at http www idealibrary com on

on the redundancy of lossy source coding with abstract alphabets the redundancy problem of lossy source coding with abstract source and reproduction alphabets is considered for coding at a fixed rate level it is shown that for any fixed rate r 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the rate r such that the distortion redundancy of c sub n defined as the difference between the performance of c sub n and the distortion rate function d p r of p is upper bounded by spl part d p r spl part r ln n 2n o ln n n for coding at a fixed distortion level it is demonstrated that for any d 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the fixed distortion d such that the rate redundancy of c sub n defined as the difference between the performance of c sub n and the rate distortion function r p d of p is upper bounded by 7ln n 6n o ln n n these results strengthen the traditional berger s 1968 1971 abstract alphabet source coding theorem and extend the positive redundancy results of zhang yang and wei see ibid vol 43 no 1 p 71 91 1997 and ibid vol 42 p 803 21 1996 on lossy source coding with finite alphabets and the redundancy result of wyner see ibid vol 43 p 1452 64 1997 on block coding of memoryless gaussian sources

cohen macaulay rees algebras and their specialization the rees algebra of an ideal j in a commutative ring r is by definition the graded algebra where t is an indeterminate in this paper we are concerned with proving that certain rees algebras are cohen macaulay and with answering under strong hypotheses the question if r s n is a factor ring of s and i jr when is i r a specialization of j s that is when is the natural epimorphism r a j s r i r an isomorphism these questions have interest partly because if s and s j s are cohen macaulay then so is gr s s j j j 16 1 and under these hypotheses if n is perfect and r s j s i r then r and 7 1 r are cohen macaulay too thus gr r is cohen macaulay and its torsion freeness and normality for exmple can be characterized in terms of analytic spreads as in 161 see section 3 for details we deal with the specialization question in section 1 it is answered for the case where as above s and 2 j s are cohen macaulay and n is 202

effective normality criteria for algebras of linear type abstract the algebras studied here are subalgebras of rings of polynomials generated by 1 forms so called rees algebras with coefficients in a noetherian ring given a normal domain r and a torsionfree module e with a free resolution f 2 f 1 f 0 e 0 we study the role of the matrices of syzygies in the normality of the rees algebra of e when the rees algebra r e and the symmetric algebra s e coincide the main results characterize normality in terms of the ideal ic s e and of the completeness of the first s symmetric powers of e where c rank and s rankf0 ranke it requires that r be a regular domain special results under broader conditions on r are still more effective

symbolic powers of perfect ideals of codimension 2 and birational maps abstract this work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field assuming that the entries of the corresponding presentation matrix are general linear forms the main contribution of the present approach is the use of the birational theory underlying the nature of the ideal and the details of a deep interlacing between generators of its symbolic powers and the inversion factors stemming from the inverse map to the birational map defined by the linear system spanned by the generators of this ideal a full description of the corresponding symbolic rees algebra is given in some cases one application is an affirmative solution of a conjecture of eisenbud mazur in 11 section 2

characteristic free resolutions of weyl and specht modules abstract we construct explicit resolutions of weyl modules by divided powers and of co specht modules by permutational modules we also prove a conjecture by boltje and hartmann 2010 7 on resolutions of co specht modules

on functional equations associated with the renormalization of non commuting circle mappings we investigate the functional equations associated with critical non commuting circle homeomorphisms using epstein s herglotz function methods we show that there are analytic solutions of a class of functional equations closely related to period two points of the circle map renormalization transformations for rotation number with continued fraction

rees algebras and almost linearly presented ideals abstract consider a grade 2 perfect ideal i in r k x 1 x d which is generated by forms of the same degree assume that the presentation matrix is almost linear that is all but the last column of consist of entries which are linear for such ideals we find explicit forms of the defining ideal of the rees algebra r i we also introduce the notion of iterated jacobian duals

an intersection number for the punctual hilbert scheme of a surface we compute the intersection number between two cycles a and b of complementary dimensions in the hilbert scheme h parameterizingr subschemes of given finite length n of a smooth projective surface s the n 1 cycle a corresponds to the set of finite closed subschemes the support of which has cardinality 1 the n 1 cycle b consists of the closed subschemes the support of which is one given point of the surface since b is contained in a indirect methods are needed the intersection number is a b 1 n 1n answering a question by h nakajima

the implicitization problem for ϕ pn p1 n 1 abstract we develop in this paper methods for studying the implicitization problem for a rational map p n p 1 n 1 defining a hypersurface in p 1 n 1 based on computing the determinant of a graded strand of a koszul complex we show that the classical study of macaulay resultants and koszul complexes coincides in this case with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions under suitable hypotheses these techniques enable us to obtain the implicit equation up to a power and up to some extra factor we give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of and what are the extra varieties that appear we also give some applications to the problem of computing sparse discriminants

complexes which arise from a matrix and a vector resolutions of divisors on certain varieties of complexes abstract consider the polynomial ring r r0 x y where r0 is a normal domain and x1 g and yg f are matrices of indeterminates the r ideal j i1 xy imin f g y defines a variety of complexes over r0 the divisor class group of r j is isomorphic to cl r0 z i where i is an ideal of r j generated by appropriately chosen lower order minors of y we produce the minimal r free resolution of i i for all integers i 1 if f is greater than or equal to g then j is a generic residual intersection of the generic grade g complete intersection i1 x the resolutions that we produce in this case are in many ways analogous to resolutions of divisors on generic residual intersections of grade two perfect ideals or grade three gorenstein ideals

the source coding theorem via sanov s theorem corresp a proof of shannon s source coding theorem is given using results from large deviation theory in particular sanov s theorem on convergence rates for empirical distributions is invoked to obtain the key large deviation result this result is used directly to prove the source coding theorem for discrete memoryless sources it is then shown how this theorem can be extended to ergodic polish space valued sources and continuous distortion measures

an implementable lossy version of the lempel ziv algorithm i optimality for memoryless sources a new lossy variant of the fixed database lempel ziv coding algorithm for encoding at a fixed distortion level is proposed and its asymptotic optimality and universality for memoryless sources with respect to bounded single letter distortion measures is demonstrated as the database size m increases to infinity the expected compression ratio approaches the rate distortion function the complexity and redundancy characteristics of the algorithm are comparable to those of its lossless counterpart a heuristic argument suggests that the redundancy is of order log log m log m and this is also confirmed experimentally simulation results are presented that agree well with this rate also the complexity of the algorithm is seen to be comparable to that of the corresponding lossless scheme we show that there is a tradeoff between compression performance and encoding complexity and we discuss how the relevant parameters can be chosen to balance this tradeoff in practice we also discuss the performance of the algorithm when applied to sources with memory and extensions to the cases of unbounded distortion measures and infinite reproduction alphabets

approximation complexes of blowing up rings ii this paper is a sequel to 111 where we studied the relationships that hold between the arithmetical properties of the rees ring 9 i and the symmetric algebra sym l of an ideal i and some of their fibers and the depth properties of the koszul homology modules h i r on a set of generators of i the connection between these objects is realized by certain differential graded algebras the so called approximation complexes that are built out of ordinary koszul complexes these compiexes show however different sensitivities being acyclic in situations much broader than the usual context of regular sequences it has been found that certain extensions thereof d sequences and proper sequences play here that role of acyclic sequence the interplay between sequences and approximation complexes will be a basic point of view here where we give

the jacobian module of a lie algebra there is a natural way to associate to the commuting variety c a of an algebra a a module over a polynomial ring it serves as a vehicle to study the arithmetical properties of c a particularly cohen macaulayness the focus here is on lie algebras and some of their representations

quantifying singularities with differential operators abstract the f signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties for example this invariant detects non singularity and strong f regularity however it is very difficult to compute motivated by different aspects of the f signature we define a numerical invariant for rings of characteristic zero or p 0 that exhibits many of the useful properties of the f signature we also compute many examples of this invariant including cases where the f signature is not known we also obtain a number of results on symbolic powers and bernstein sato polynomials

permanents doty coalgebras and dominant dimension of schur algebras abstract a hidden multiplication on a z n r the z dual of the integral schur algebra s z n r is explicitly constructed possibly without a unit the image of the multiplication map is shown to be spanned by bipermanents let k be any field of characteristic p 0 the image of the induced multiplication on a k n r a z n r z k turns out to coincide with the doty coalgebra d n r p of truncated symmetric powers combined with a new straightening formula for bipermanents it is proved that such a multiplication induces an isomorphism a k n r s k n r a k n r a k n r as s k n r bimodules if and only if r n p 1 if and only if d n r p a k n r as a result s k n r is a gendo symmetric algebra and its dominant dimension is at least two and admits a combinatorial characterization as long as r n p 1

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

criteria for equality of radicals of ideals abstract we provide criteria for equality of the radicals of ideals in terms of the grades of their specializations as an application we compute up to radicals the ideals of maximal minors for the differentials in two classes of complexes associated with a map of free modules