characteristic free decomposition of skew schur functors this note is a generalization of our thesis brandeis university 1984 whose material can be found in 2 our aim is to show that by the methods presented there one can produce a characteristic free filtration of the skew schur functor l f whose associated graded module is the direct sum of schur modules l f one would expect from the characteristic zero theory as in 2 notations and facts are freely borrowed from l

the decomposition of lie powers let g be a group f a field of prime characteristic p and v a finite dimensional fg module let l v denote the free lie algebra on v regarded as an fg submodule of the free associative algebra or tensor algebra t v for each positive integer r let l r v and t r v be the r th homogeneous components of l v and t v respectively here l r v is called the r th lie power of v our main result is that there are submodules b 1 b 2 of l v such that for all r b r is a direct summand of t r v and whenever m geqslant 0 and k is not divisible by p the module l p mk v is the direct sum of l p m b k l p m 1 b pk l 1 b p mk thus every lie power is a direct sum of lie powers of p power degree the approach builds on an analysis of t r v as a bimodule for g and the solomon descent algebra

linking first occurrence polynomials over f2 by steenrod operations abstract it is proved that for every irreducible representation l of the full matrix semigroup mn f 2 the first occurrence of l as a composition factor in the polynomial algebra p f 2 x1 xn is linked by a steenrod operation to the first occurrence of l as a submodule in p this steenrod operation is given explicitly as the image of an admissible monomial in the steenrod squares sqr under the canonical anti automorphism of the mod 2 steenrod algebra a the first occurrences of both kinds are also linked to higher degree occurrences of l by elements of the milnor basis of a

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

grobner methods for representations of combinatorial categories given a category c of a combinatorial nature we study the following fundamental question how does the combinatorial behavior of c affect the algebraic behavior of representations of c we prove two general results the first gives a combinatorial criterion for representations of c to admit a theory of grobner bases from this we obtain a criterion for noetherianity of representations the second gives a combinatorial criterion for a general rationality result for hilbert series of representations of c this criterion connects to the theory of formal languages and makes essential use of results on the generating functions of languages such as the transfer matrix method and the chomsky schutzenberger theorem our work is motivated by recent work in the literature on representations of various specific categories our general criteria recover many of the results on these categories that had been proved by ad hoc means and often yield cleaner proofs and stronger statements for example we give a new more robust proof that fi modules originally introduced by church ellenberg farb and a family of natural generalizations are noetherian we give an easy proof of a generalization of the lannes schwartz artinian conjecture from the study of generic representation theory of finite fields we significantly improve the theory of modules introduced by snowden in connection to syzygies of segre embeddings and we establish fundamental properties of twisted commutative algebras in positive characteristic

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

lie powers of modules for groups of prime order let l v be the free lie algebra on a finite dimensional vector space v over a field k with homogeneous components l n v for n geq 1 if g is a group and v is a kg module the action of g extends naturally to l v and the l n v become finite dimensional kg modules called the lie powers of v in the decomposition problem the aim is to identify the isomorphism types of indecomposable kg modules with their multiplicities in unrefinable direct decompositions of the lie powers this paper is concerned with the case where g has prime order p and k has characteristic p as is well known there are p indecomposables denoted here by j 1 dots j p where j r has dimension r a theory is developed which provides information about the overall module structure of l v and gives a recursive method for finding the multiplicities of j 1 dots j p in the lie powers l n v for example the theory yields decompositions of l v as a direct sum of modules isomorphic either to j 1 or to an infinite sum of the form j r oplus j p 1 oplus j p 1 oplus ldots with r geq 2 closed formulae are obtained for the multiplicities of j 1 dots j p in l n j p and l n j p 1 for r p 1 the indecomposables which occur with non zero multiplicity in l n j r are identified for all sufficiently large n

on the cohomology of specht modules we investigate the cohomology of the specht module s for the symmetric group d we show if 0 i p 2 then hi d s is isomorphic to hs i b w0 where s d d 1 2 b is the borel subgroup of the algebraic group gld k and 1d is the weight of the determinant representation we obtain similar isomorphisms of ext di s m with b cohomology which in turn yield isomorphisms of cohomology for borel subgroups of gln k for varying n d in the case i 0 and the case i 1 for certain we apply our result and known symmetric group results of james and erdmann to obtain new information about b cohomology finally we show that specht module cohomology is closely related to cohomology for the frobenius kernel b1 for small primes

the composition factors of fp x1 x2 x3 as a gl 3 p module let gl n p denote the general linear group over the prime field fp and let sd denote the right gl n p module consisting of all homogeneous polynomials of degree d in n variables x1 x2 x over f we wish to determine the simple composition factors of sd unfortunately satisfactory models for all the simple gl n p modules themselves are not known there are p p 1 of these simple modules up to isomorphism and all of them are known to occur in the symmetric algebra s cf sd the general problem is therefore a very difficult one in our present state of knowledge the purpose of this paper is to offer a solution to this problem in the case n 3 this extends the work of david carlisle who calculated in his manchester thesis 2 the occurrences of all the l dimensional modules and developed a method for dealing with the general case let us begin with some general remarks about the problem the case n 1 is trivial as s d detd the dth power of the determinant representation the case n 2 is also easily disposed of for instance by glover 9 the simple modules are the tensor products of the sd for d 1 and d p the module s is not simple for example the pth power sf of the natural module is a proper submodule of 411 0021 8693192 3 00

the onset and end of the gunn effect in extrinsic semiconductors a hopf bifurcation analysis of the spontaneous current oscillation in direct current dc voltage biased extrinsic semiconductors is given for the classical model of the gunn effect in n gaas for semiconductor lengths l larger than a certain minimal value the steady state is linearly unstable for voltages in an interval phi alpha phi omega as l increases the branch of time periodic solutions bifurcating at simple eigenvalues when phi phi alpha turns from subcritical to supercritical and then back to subcritical again for very long semiconductors a quasi continuum of oscillatory modes bifurcates from the steady state at the onset of the instability the bifurcating branch is then described by a scalar reaction diffusion equation with cubic nonlinearity subject to antiperiodic boundary conditions on a subinterval of 0 l for the electron velocity curve we have considered the bifurcation is subcritical which may agree with experimental observations in n gaas an extension o

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

on projective and injective polynomial modules let k denote a field for each n 0 we have a category pol n of finite dimensional polynomial modules and a category rat n of rational modules over k for r 0 we write pol n r for the full subcategory of pol n whose objects are the polynomial modules of degree r each v pol n has a unique decomposition v r 0 v r where v r is polynomial of degree r furthermore the category pol n r is naturally equivalent to the category of modules for the finite dimensional associative schur algebra s n r it follows that every polynomial module has a projective cover and an injective hull we here attempt to describe those polynomial modules which are both projective and injective we give a complete description in the cases n 2 3 and make a conjecture which if true gives a complete description in the general case a finite dimensional polynomial module which is injective and projective must be a tilting module and thanks to a natural duality on the category of polynomial modules our problem is equivalent to that of determining which

bifurcation behavior of a superlattice model we present a complete description of the stationary and dynamical behavior of semiconductor superlattices in the framework of a discrete drift model by means of numerical continuation singular perturbation analysis and bifurcation techniques the control parameters are the applied dc voltage phi and the doping nu in nondimensional units we show that the organizing centers for the long time dynamics are takens bogdanov bifurcation points in a broad range of parameters and we cast our results in a phi nu phase diagram for small values of the doping the system has only one uniform solution where all the variables are almost equal for high doping we find multistability corresponding to domain solutions and the stationary solutions may exhibit chaotic spatial behavior in the intermediate regime of nu the solution can be time periodic depending on the bias the oscillatory regions are related to the appearance and disappearance of hopf bifurcation tongues which can be sub or supercri

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

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

new approximations for minimum weighted dominating sets and minimum weighted connected dominating sets on unit disk graphs given a node weighted graph the minimum weighted dominating set mwds problem is to find a minimum weighted vertex subset such that for any vertex it is contained in this subset or it has a neighbor contained in this set and the minimum weighted connected dominating set mwcds problem is to find a mwds such that the graph induced by this subset is connected in this paper we study these two problems on a unit disk graph a 4 e approximation algorithm for an mwds based on a dynamic programming algorithm for a min weight chromatic disk cover is presented meanwhile we also propose a 1 e approximation algorithm for the connecting part by showing a polynomial time approximation scheme for a node weighted steiner tree problem when the given terminal set is c local and thus obtain a 5 e approximation algorithm for an mwcds

invariant subspaces of the ring of functions on a vector space over a finite field abstract iffqis the finite field of characteristicpand orderq ps let f q be the category whose objects are functors from finite dimensionalfq vector spaces tofq vector spaces and with morphisms the natural transformations between such functors a fundamental object in f q is the injectiveifqdefined by i f q v f q v s v x q x we determine the lattice of subobjects ofifq it is the distributive lattice associated to a certain combinatorially defined poset i p s whoseqconnected components are all infinite with one trivial exception an analysis of i p s reveals that every proper subobject of an indecomposable summand ofifqis finite thusifqis artinian filteringifqand i p s in various ways yields various finite posets and we recover the main results of papers by doty kovacs and krop on the structure ofs v xq overfq ands v over f p

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

subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebras let a t be the subalgebra of the steenrod algebra generated by sqi i 2t we determine which a module indecomposable summands of f2 x1 xn are free over a t by first showing this to be equivalent to the group theoretic question of determining which irreducible modular representations of mn f2 or gln f2 occur as composition factors in the truncated polynomial algebra f2 x1 xn x12 xn2 the second question is then answered

cohomology and generic cohomology of specht modules for the symmetric group cohomology of specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the borel subgroup b of the general linear group gld k but this has never been exploited to prove new symmetric group results using work of doty on the submodule structure of symmetric powers of the natural gld k module together with work of andersen on cohomology for b and its frobenius kernels we prove new results about hi d s we recover work of james in the case i 0 then we prove two stability theorems one of which is a generic cohomology result for specht modules equating cohomology of sp with sp2 this is the first theorem we know relating specht modules s and sp the second result equates cohomology of s with s pa for large a

approximation schemes for covering and packing problems in image processing and vlsi a unified and powerful approach is presented for devising polynomial approximation schemes for many strongly np complete problems such schemes consist of families of approximation algorithms for each desired performance bound on the relative error e o with running time that is polynomial when e is fixed though the polynomiality of these algorithms depends on the degree of approximation e being fixed they cannot be improved owing to a negative result stating that there are no fully polynomial approximation schemes for strongly np complete problems unless np p the unified technique that is introduced here referred to as the shifting strategy is applicable to numerous geometric covering and packing problems the method of using the technique and how it varies with problem parameters are illustrated a similar technique independently devised by b s baker was shown to be applicable for covering and packing problems on planar graphs

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

first degree cohomology of specht modules and extensions of symmetric powers abstract let d denote the symmetric group of degree d and let k be a field of positive characteristic p for p 2 we give an explicit description of the first cohomology group h 1 d sp of the specht module sp over k labelled by a partition of d we also give a sufficient condition for the cohomology to be non zero for p 2 and we find a lower bound for the dimension the cohomology of specht modules has been considered in many papers including 10 12 15 and 21 our method is to proceed by comparison with the cohomology for the general linear group g gl n k and then to reduce to the calculation of ext b 1 s d e k where b is a borel subgroup of g where s d e denotes the dth symmetric power of the natural module e for g and k denotes the one dimensional b module with weight the main new input is the description of module extensions by extensions sequences coherent triples of extension sequences and coherent multi sequences of extension sequences and the detailed calculation of the possibilities for such sequences these sequences arise from the action of divided powers elements in the negative part of the hyperalgebra of g our methods are valid also in the quantised context and we aim to treat this in a separate paper

modular branching rules and the mullineux map for hecke algebras of type a in a series of papers 21 22 23 kleshchev has recently proved a branching rule for modular representations of the symmetric group s r on r symbols fix a field f of arbitrary characteristic p then kleshchev s branching rule gives a precise description of the socle of the restriction of an irreducible fs r module to fs r 1 in 24 kleshchev applies this branching rule to give a purely combinatorial description of the mullineux map which describes the irreducible fs r module obtained by tensoring an arbitrary irreducible fs r module with the 1 dimensional sign representation in 13 see also 3 ford and kleshchev show with some technical combinatorics that kleshchev s description of the mullineux map is equivalent to a quite different algorithm conjectured by mullineux 27 in 1979 thus proving the so called mullineux conjecture the main purpose of this paper is to prove the quantum analogues of these results replacing the group algebra fs r with the corresponding hecke algebra h r over f at an arbitrary root of unity q f our main result is the hecke algebra analogue of kleshchev s modular branching rule it turns out that the branching rule for h r only depends on the integer where is the smallest natural number such that 1 q q 1 0 in f if q 1 is precisely the characteristic p of f and our branching rule specializes to the classical case considered by kleshchev as a consequence of this branching rule combining kleshchev s argument in 24 with the known block structure of h r from 9 we are able to deduce the quantum analogue of kleshchev s description of the corresponding mullineux map for h r the proof by ford and kleshchev in 13 that this is equivalent to mullineux s original algorithm does not depend on p being prime so our result is equivalent by 13 to a quantum version of the mullineux conjecture replacing p by we remark that there is now a quite different proof that kleshchev s algorithm gives the mullineux map using crystal bases this was first observed for fields of characteristic 0 in 25 modulo a conjecture which has now been proved 17 2 relating crystal bases for a certain affine quantum group to decomposition matrices of h r over fields of characteristic 0 the result in arbitrary characteristic can be deduced from the characteristic 0 case by an

a case study for homoclinic chaos in an autonomous electronic circuit a trip taken from takens bogdanov to hopf sbil nikov abstract the dynamics of an autonomous electronic oscillator is analysed both theoretical study with methods of local bifurcation theory and numerical simulation by using continuation methods to detect global behaviours are carried out comment is made on the presence of homoclinic connections which organise the periodic and nonperiodic behaviour the richness and complexity of the periodic oscillations are described and the presence of isolated branches of odd period orbits are emphasized further attention is paid to the chaotic attractor coexisting with sil nikov homoclinicity even in the presence of a hopf degeneracy the hopf sil nikov singularity

polynomially and infinitesimally injective modules abstract the injective polynomial modules for a general linear group g of degree n are labelled by the partitions with at most n parts working over an algebraically closed field of characteristic p we consider the question of which partitions correspond to polynomially injective modules that are also injective as modules for the restricted enveloping algebra of the lie algebra of g the question is related to the index of divisibility of a polynomial module in general and an explicit answer is given for n 2

metabelian lie powers of the natural module for a general linear group abstract consider a free metabelian lie algebra m of finite rank r over an infinite field k of prime characteristic p given a free generating set m acquires a grading its group of graded automorphisms is the general linear group g l r k so each homogeneous component m d is a finite dimensional g l r k module the homogeneous component m 1 of degree 1 is the natural module and the other m d are the metabelian lie powers of the title this paper investigates the submodule structure of the m d in the main result a composition series is constructed in each m d and the isomorphism types of the composition factors are identified both in terms of highest weights and in terms of steinberg s twisted tensor product theorem their dimensions are also given it turns out that the composition factors are pairwise non isomorphic from which it follows that the submodule lattice is finite and distributive by the birkhoff representation theorem any such lattice is explicitly recognizable from the poset of its join irreducible elements the poset relevant for m d is then determined by exploiting a 1975 paper of yu a bakhturin on identical relations in metabelian lie algebras

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

approximations for steiner trees with minimum number of steiner points abstract given n terminals in the euclidean plane and a positive constant find a steiner tree interconnecting all terminals with the minimum number of steiner points such that the euclidean length of each edge is no more than the given positive constant this problem is np hard with applications in vlsi design wdm optical networks and wireless communications in this paper we show that a the steiner ratio is 1 4 that is the minimum spanning tree yields a polynomial time approximation with performance ratio exactly 4 b there exists a polynomial time approximation with performance ratio 3 and c there exists a polynomial time approximation scheme under certain conditions

composition factors of tensor products of symmetric powers abstract we determine the composition factors of the tensor product s e s e of two copies of the symmetric algebra of the natural module e of a general linear group over an algebraically closed field of positive characteristic our main result may be regarded as a substantial generalisation of the tensor product theorem of krop 6 and sullivan 8 on composition factors of s e we earlier answered the question of which polynomially injective modules are infinitesimally injective in terms of the divisibility index we are now able to give an explicit description of the divisibility index for polynomial modules for general linear groups of degree at most 3

dynamical approach study of spurious steady state numerical solutions of nonlinear differential equations i the dynamics of time discretization and its implications for algorithm development in computational fluid dynamics abstract the goal of this paper is to utilize the theory of nonlinear dynamics approach to investigate the possible sources of errors and slow convergence and nonconvergence of steady state numerical solutions when using the time dependent approach for nonlinear hyperbolic and parabolic partial differential equations terms this interdisciplinary research belongs to a subset of a new field of study in numerical analysis sometimes referred to as the dynamics of numerics and the numerics of dynamics at the present time this new interdisciplinary topic is still the property of an isolated discipline with all too little effort spent in pointing out an underlying generality that could make it adaptable to diverse fields of applications this is the first of a series of research papers under the same topic our hope is to reach researchers in the fields of computational fluid dynamics cfd and in particular hypersonic and combustion related cfd by simple examples in which the exact solutions of the governing equations are known the application of the apparently straightforward numerical technique to genuinely nonlinear problems can be shown to lead to incorrect or misleading results one striking phenomenon is that with the same initial data the continuum and its discretized counterpart can asymptotically approach different stable solutions this behavior is especially important for employing a time dependent approach to the steady state since the initial data are usually not known and a freestream condition or an intelligent guess for the initial conditions is often used with the unique property of the different dependence of the solution on initial data for the partial differential equation and the discretized counterpart it is not easy to delineate the true physics from numerical artifacts when numerical methods are the sole source of solution procedure for the continuum part i concentrates on the dynamical behavior of time discretization for scalar nonlinear ordinary differential equations in order to motivate this new yet unconventional approach to algorithm development in cfd and to serve as an introduction for parts 11 and iii of the same series of research papers

bgg algebras and the bgg reciprocity principle the bgg reciprocity principle is a famous result in the representation theary of complex semisimple lie algebras proved by bernstein gelfand and gelfand in bgg 21 it may be viewed as an analogue for the category 6 of the brauer reciprocity principle in the modular representation theory of finite groups and of humphreys reciprocity principle in the representation theory of restricted lie algebras indeed the result of humphreys served as motivation for the bgg result similar results have since been proved in a number of other settings by various approaches for all these results one works in a suitable category of modules in which each simple module l has a projective cover p in addition there is a family of intermediate modules mi such that each p has a filtration with m s as quotients then the appropriate reciprocity principle states that m occurs as a filtration quotient in p as often as l occurs as a composition factor in mi in the case of the category 0 the simple modules are the simple highest weight modules the category is introduced as a setting for projective covers to exist and the intermediate modules are the verma modules the proof of bgg reciprocity in bgg 21 is fairly straightforward once the basic facts about the category 0 are developed the crucial point is the construction of enough projective modules in 0 once these are constructed the existence of projective covers the existence of suitable filtrations on these and the reciprocity principle are ail proved with reference to the construction however another proof of reciprocity exists which is more natural in the sense that it makes no further reference to the construction of projective modules this proof can be used in some of the other settings as well although in any given setting no drastic simplifications result i believe it is still worthwhile to draw attention to this more conceptual approach the properties of the category 0 which are needed for the proof may be

the module structure of a group action on a polynomial ring abstract for any representation of a p group g on a vector space of dimension 3 over a finite field k of characteristic p we show how the symmetric algebra regarded as a kg module can be expressed as a direct sum of kg modules each one of which is isomorphic to a summand in low degree it follows that for any group g only a finite number of isomorphism classes of summands can occur

modular representations of gl 3 fp symmetric squares and mod p cohomology of gl 3 z abstract we prove certain cases of nonabelian reciprocity between mod p galois representations and hecke eigenclasses in the mod p cohomology of gl 3 z using the symmetric square liftings from gl 2 and congruences on hecke eigenvalues we need to develop some modular representation theory for gl 3 z p in some cases we also need to invoke certain computer calculations to finish the proof

double centralizer properties dominant dimension and tilting modules double centralizer properties play a central role in many parts of algebraic lie theory soergel s double centralizer theorem relates the principal block of the bernstein gelfand gelfand category o of a semisimple complex lie algebra with the coinvariant algebra i e the cohomology algebra of the corresponding flag manifold schur weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic or via the quantized version in nondefining characteristic in this paper we exhibit algebraic structures behind these double centralizer properties we show that the finite dimensional algebras relevant in this context have dominant dimension at least two with respect to some projective injective or tilting modules general arguments which combine methods from ring theory qf 3 rings and dominant dimension with tools from representation theory approximations tilting modules then imply the validity of these double centralizer properties as well as new ones in contrast to the traditional proofs e g by the fundamental theorems of invariant theory no computations are necessary

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

an analogue of serre s conjecture for galois representations and hecke eigenclasses in the mod p cohomology of gl n ℤ

the permutation representation of sp 2m fp acting on the vectors of its standard module abstract this paper studies the permutation representation of a finite symplectic group over a prime field of odd characteristic on the vectors of its standard module the submodule lattice of this permutation module is determined the results yield additive formulae for the p ranks of various incidence matrices arising from the finite symplectic spaces

the morava k theories of some classifying spaces let p be a finite abelian p group with classifying space bp we compute in representation theoretic terms the morava k theories of the stable wedge summands of bp in particular we obtain a simple and purely group theoretic description of the rank of k s bg for any finite group g with an abelian p sylow subgroup a minimal amount of topology quickly reduces the problem to an algebraic one of analyzing truncated polynomial algebras as modular representations of the semigroup m 1 z p 1 main results for a fixed prime p there exists a sequence of cohomology theories k 1 k 2 with k s periodic with period 2 ps 1 these are the morava k theories 13 and generalize ordinary complex k theory in the sense that k 1 is one of the p 1 isomorphic summands of k z p recently it has been becoming clear that they play a central role in homotopy theory see for example the work of m hopkins and j smith 6 k s has further computational virtues k s x is always a free module over the coefficient ring k s and there is a kunneth isomorphism a major outstanding problem is to find good models for the spaces representing these theories related to this is the question of finding for finite groups g a group theoretic description for the rings k s bg analogous to atiyah s isomorphism 2 r g k bg the author together with j harris recently analyzed stable wedge decompositions of classifying spaces of abelian p groups 5 in this paper we compute in representation theoretic terms the morava k theories of the resulting wedge summands as a consequence we obtain a very simple and purely group theoretic description of the rank of k s bg for any finite group g with an abelian p sylow subgroup to state our results we establish some notation it is convenient to let k o be rational cohomology we let k s x always denote the reduced sth morava k theory of x if x is a space or spectrum we let k5 x equal the rank of k s x as a k s module and by abuse of notation we let ks g ks bg for a finite group g bg is the union of bg with a disjoint basepoint received bv the editors may 27 1986 1980 marhemat7es subject cl lssific ltion primary 55n22 secondary 20g05 research partially supported by the sloan foundation and the nsf t1987 american mathematical society 0002 9947 87 1 00 25 per page

galois representations with conjectural connections to arithmetic cohomology in this paper we extend a conjecture of a ash and w sinnott relating niveau 1 galois representations to the mod p cohomology of congruence subgroups ofsln z to include galois representations of higher niveau we then present computational evidence for our conjecture in the case n 3 in the form of three dimensional galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture our examples include galois representations with nontrivial weight and level as well as irreducible three dimensional representations that are in no obvious way related to lower dimensional representations in addition we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture

higher extensions between modules for abstract we calculate ext sl 2 k ext sl 2 k l ext sl 2 k l ext sl 2 k l l where is the weyl module of highest weight l is the simple sl 2 k module of highest weight and our field k is algebraically closed of positive characteristic we also get analogous results for the dipper donkin quantisation to do thus we construct the lyndon hochschild serre spectral sequence in a new way and find a new condition for the e 2 page of any spectral sequence to be the same as the e page

algorithms for connected set cover problem and fault tolerant connected set cover problem given a set v of elements s a family of subsets of v and g a connected graph on vertex set s a connected set cover csc is a subfamily r of s such that every element in v is covered by at least one set of r and the subgraph g r of g induced by r is connected if furthermore g r is k connected and every element in v is covered by at least m sets in r then r is a k m csc in this paper we present two approximation algorithms for the minimum csc problem and one approximation algorithm for the minimum 2 m csc problem performance ratios are analyzed these are the first approximation algorithms for csc problems in general graphs with guaranteed performance ratios

decompositions of some specht modules i abstract we give a decomposition as a direct sum of indecomposable modules of several types of specht modules in characteristic 2 these include the specht modules labelled by hooks whose decomposability was considered by murphy 15 since the main arguments are essentially no more difficult for hecke algebras at parameter q 1 we proceed in this generality

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

the permutation action of finite symplectic groups of odd characteristic on their standard modules abstract we study the space of functions on a finite dimensional vector space over a field of odd order as a module for a symplectic group we construct a basis of this module with the following special properties each submodule generated by a single basis element under the symplectic group action is spanned as a vector space by a subset of the basis and has a unique maximal submodule from these properties the dimension and composition factors of the submodule generated by any subset of the basis can be determined these results apply to incidence geometry of the symplectic polar space yielding the symplectic analogue of hamada s additive formula for the p ranks of the incidence matrices between points and flats a special case leads to a closed formula for the p rank of the incidence matrix between the points and lines of the symplectic generalized quadrangle over a field of odd order together with earlier results on the 2 ranks this result completes the determination of the p ranks for these quadrangles

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

on the representations of the full matrix semigroup on homogeneous polynomials this paper is the successor of 11 in the closing theorem there we determined the composition factors of w for a regular i 1 6 11 here we are going to describe the structure of wa in the singular case the second theme of this paper is the criterion for distributivity of the m modules wcd since the necessary conditions for that established in 1 8 51 limit d to d p p 1 we just have to sort out the latter case this will be done in the third section of this paper for the convenience of the reader we recall some notation terms and definitions used in 11

invariants of specht modules abstract in 14 hemmer conjectures that the module of fixed points for the symmetric group m of a specht module for n with n m over a field of positive characteristic p has a specht series when viewed as a n m module we provide a counterexample for each prime p the examples have the same form for p 5 and we treat the cases p 3 and p 2 separately

two and higher dimensional pattern matching in optimal expected time algorithms with optimal expected running time are presented for searching the occurrences of a two dimensional m x m pattern p in a two dimensional n x n text t over an alphabet of size c the algorithms are based on placing in the text a static grid of test points determined only by n m and c not dynamically by earlier test results using test strings read from the test points the algorithms eliminate as many potential occurrences of p as possible the remaining potential occurrences are separately checked for actual occurrences a suitable choice of the test point set leads to algorithms with expected running time o n2logc m2 m2 using the uniform bernoulli model of randomness this is shown to be optimal by a generalization of a one dimensional lower bound result by yao experimental results show that the algorithms are efficient in practice too the method is also generalized for the k mismatches problem the resulting algorithm has expected running time o kn2logc m2 m2 provided that k leq m lfloor m lceil log c m 2 rceil rfloor 1 2 all algorithms need preprocessing of p which takes time and space o m2 the text processing can be done on line using a rather small window the algorithms easily generalize to d dimensional matching for any d

a note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup abstract let g be a connected semisimple simply connected algebraic group over an algebraically closed field k of positive characteristic the action of g on itself by conjugation induces on the coordinate algebra k g and hence on the coordinate algebra k g r of the infinitesimal subgroup g r the structure of a rational g module in the recent paper 3 by bendel nakano pillen and sobaje the authors ask whether the tensor product of k g r with the rth steinberg module st r has a good filtration here we show that this is the case provided that the characteristic of k is good for g we do so by relating k g r to a certain quotient of the coordinate algebra k g of the lie algebra g of g and using a koszul resolution we note that the argument may also be applied to certain polynomial modules occurring in the context of doty s conjecture 17 4 2 11