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

on the blocks of the walled brauer algebra we determine the blocks of the walled brauer algebra in characteristic zero these can be described in terms of orbits of the action of a weyl group of type a on a certain set of weights in positive characteristic we give a linkage principle in terms of orbits of the corresponding affine weyl group we also classify the semisimple walled brauer algebras in all characteristics

new and improved methods to analyze and compute double scalar multiplications we address several algorithms to perform a double scalar multiplication on an elliptic curve all the methods investigated are related to the double base number system dbns and extend previous work of doche et al 25 we refine and rigorously prove the complexity analysis of the joint binary ternary jbt algorithm experiments are in line with the theory and show that the jbt requires approximately 6 percent less field multiplications than the standard joint sparse form jsf method to compute n p m q we also introduce a randomized version of the jbt called jbt rand that gives total control of the number of triplings in the expansion that is produced so it becomes possible with the jbt rand to adapt and tune the number of triplings to the coordinate system and bit length that are used to further decrease the cost of a double scalar multiplication then we focus on koblitz curves for extension degrees enjoying an optimal normal basis of type ii we discuss a joint dbns approach that reduces the number of field multiplications by at least 35 percent over the traditional jsf for other extension degrees represented in polynomial basis the joint dbns is still relevant provided that appropriate bases conversion methods are used in this situation tests show that the speedup over the jsf is then larger than 20 percent finally when the use of the dbns becomes unrealistic for instance because of the lack of an efficient normal basis or the lack of memory to allow an efficient conversion we adapt the joint binary ternary algorithm to koblitz curves giving rise to the joint method whose complexity is analyzed and proved the joint induces a speedup of about 10 percent over the jsf

the double base number system and its application to elliptic curve cryptography we describe an algorithm for point multiplication on generic elliptic curves based on a representation of the scalar as a sum of mixed powers of 2 and 3 the sparseness of this so called double base number system combined with some efficient point tripling formulae lead to efficient point multiplication algorithms for curves defined over both prime and binary fields side channel resistance is provided thanks to side channel atomicity

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

the malliavin calculus a functional analytic approach in his groundbreaking articles s 91 malliavin introduced a technique for obtaining elliptic regularity results using function space calculus in his formulation the function space calculus on which his theory rests is the martingale calculus coming from the ornstein uhlenbeck process on wiener space cf 121 for a detailed discussion of malliavin s theory from this point of view although this formulation has a great deal to recommend it and is particularly pleasing to afficionados of the so called brownian sheet most analysts including those who are reasonably facile with the machinery of probability theory are unlikely to find the technical difficulties inherent in this approach outweighed by the eventual rewards shigekawa 111 discovered that there is an alternative formulation of malliavin s theory in shigekawa s formulation no mention need be made of the ornstein uhlenbeck process instead what he relies on is a sobolev type extension of the frechet dervative with wiener measure playing the role played by lebesgue measure in the finite dimensional context a third formulation of malliavin s theory was recently provided by bismut l bismut s idea is to exploit the quasi invariance of wiener measure as it is manifested in a beautiful relation due to haussmann 4 the present article has two aims in the first place section 1 2 and 3 are devoted to yet another formulation of malliavin s calculus although the end results are precisely the same as those obtained in 121 no reference is made to the ornstein uhlenbeck process on which the original theory depended instead the machinery is entirely that of elementary functional analysis and is devoid of any sophisticated martingale calculus it has been my intent that these three sections should be readily accessible to any analyst who has had some acquintance with wiener measure

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

multiple wiener ito integrals possessing a continuous extension letf w be a wiener functional defined byf w in f wherein f denotes the multiple wiener ito integral of ordern of the symmetricl2 0 1 n kernelf we show that a necessary and sufficient condition for the existence of a continuous extension off i e the existence of a function o from the continuous functions on 0 1 which are zero at zero to which is continuous in the supremum norms and for which o w f w a s is that there exists a multimeasure dt1 dt n on 0 1 n such thatf t1 t n t1 1 t n 1 a e lebesgue on 0 1 n recall that a multimeasure a1 a n is for every fixedi and every fixedai ai 1 ai 1 an a signed measure ina i and there exists multimeasures which are not measures it is furthermore shown that iff t1 t2 t n t1 1 t n 1 then all the tracesf k k leqq left frac n 2 right off exist eachf k induces ann 2k multimeasure denoted by k the following relation holds i n f sum limits k 0 n 2 left frac 1 2 right k frac n k n 2k int limits 0 1 n 2k w t 1 cdot cdot cdot w t n 2k cdot mu k dt 1 dt n 2k and each of the integrals in the above expression equals the multiple stratonovich or ogawa type integral of the tracef k namely int limits 0 1 n 2k w t 1 cdot cdot cdot w t n 2k mu k dt 1 dt n 2k i n 2k circ f k

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

blocks of affine quantum schur algebras abstract the affine quantum schur algebra is a certain important infinite dimensional algebra whose representation theory is closely related to that of quantum affine gl n finite dimensional irreducible modules for the affine quantum schur algebra s n r v were classified in 15 where v c is not a root of unity we will classify blocks of the affine quantum schur algebra s n r v in this paper

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

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

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

conversion algorithms and implementations for koblitz curve cryptography in this paper we discuss conversions between integers and tau adic expansions and we provide efficient algorithms and hardware architectures for these conversions the results have significance in elliptic curve cryptography using koblitz curves a family of elliptic curves offering faster computation than general elliptic curves however in order to enable these faster computations scalars need to be reduced and represented using a special base tau expansion hence efficient conversion algorithms and implementations are necessary existing conversion algorithms require several complicated operations such as multiprecision multiplications and computations with large rationals resulting in slow and large implementations in hardware and microcontrollers with limited instruction sets our algorithms are designed to utilize only simple operations such as additions and shifts which are easily implementable on practically all platforms we demonstrate the practicability of the new algorithms by implementing them on altera stratix ii fpgas the implementations considerably improve both computation speed and required area compared to the existing solutions

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

generalized stochastic integrals and the malliavin calculus the paper first reviews the skorohod generalized stochastic integral with respect to the wiener process over some general parameter space t and it s relation to the malliavin calculus as the adjoint of the malliavin derivative some new results are derived and it is shown that every sufficiently smooth process ut t t can be decomposed into the sum of a malliavin derivative of a wiener functional and a process whose generalized integral over t vanishes using the results on the generalized integral the bismut approach to the malliavin calculus is generalized by allowing non adapted variations of the wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions

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

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

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

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

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

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

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

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

malliavin derivatives and derivatives of functionals of the wiener process with respect to a scale parameter let f c 0w be a functional of the wiener process with variance parameter c 2 0 and let f cw be an extension of f c 0w to f cw c in 0 c 0 relations are derived between the malliavin derivatives between the derivatives with respect to the scale parameter partial f rho cw partial rho p 1 and noncoherent derivatives such as de f cw sqrt varepsilon c tilde w mid w d varepsilon varepsilon 0 where tilde w is another wiener process independent of w and between the generator of the nontime homogeneous ornstein uhlenbeck process

fixed point functors for symmetric groups and schur algebras abstract let d be the symmetric group for 1 f m be the functor which takes a d module u to the space of fixed points u m which is naturally a module for d m this functor was previously used by the author to study cohomology of the symmetric group but little is known about it this paper initiates a study of f m first we relate it to james work on row and column removal and decomposition numbers for the schur algebra next we determine the image of dual specht modules permutation and twisted permutation modules and some young and twisted young modules under f m in particular f m acts as first row removal on dual specht modules s with 1 m and as first column removal on twisted young and twisted permutation modules corresponding to partitions with m parts finally we prove that determining f m on the young modules is equivalent to determining the decomposition numbers for the schur algebra

a change of variables formula for stratonovich integrals and existence of solutions for two point stochastic boundary value problems existence results for a class of two point non adaptive stochastic boundary value problems are derived by combining a new change of variables formula with optimal control techniques the class of equations treated is significantly wider than that for which previously known existence results apply

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

a fast hardware architecture for integer to taunaf conversion for koblitz curves scalar multiplication in elliptic curve cryptography is the most computational intensive operation efficiency of this operation can be significantly improved in hardware implementations by using frobenius endomorphisms which require integer to adic nonadjacent form conversion because conversion is one of the limiting factors in some of koblitz curve based cryptosystems it has become an interesting problem in this paper we propose two algorithms and a novel hardware architecture to double the speed of integer to adic nonadjacent form conversion

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

markov field property of stochastic differential equations the purpose of this paper is to prove a characterization of the conditional independence of two independent random variables given a particular functional of them in terms of a factorization property as an application we discuss the markov field property for solutions of stochastic differential equations with a boundary condition involving the values of the process at times t 0 and t 1

applications of the degree theorem to absolute continuity on wiener space let h p be an abstract wiener space and define a shift on byt f wheref is anh valued random variable we study the absolute continuity of the measurespot 1and f p ot 1 with respect top using the techniques of the degree theory of wiener maps where f det2 1 f exp f 1 2 f 2

boundary value processes estimation and identification recent results obtained for boundary value processes and the associated smoothing and identification problems are presented in this paper both lumped and distributed parameter models are considered some open problems are discussed and the fundamental mathematical difficulties that arise in studying nonlinear extensions of the proposed models are mentioned

on the relation between the stratonovich and ogawa integrals l auteur montre que l existence de l integrale d ogawa implique l existence de l integrale de stratonovich et que les deux sont egales sous condition d une definition convenable de l integrale stochastique de stratonovich non adaptee

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

on the 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

homomorphisms between weyl modules for sl3 k we classify all homomorphisms between weyl modules for sl3 k when k is an algebraically closed field of characteristic at least three and show that the hom spaces are all at most one dimensional as a corollary we obtain all homomorphisms between specht modules for the symmetric group when the labelling partitions have at most three parts and the prime is at least three we conclude by showing how a result of fayers and lyle on hom spaces for specht modules is related to earlier work of donkin for algebraic groups let g be a reductive algebraic group over an algebraically closed field of characteristic p 0 an important class of modules for such a group are the weyl modules labelled by dominant weights these can be constructed relatively explicitly and their heads provide a full set of simple modules for g equivalently one can study the duals of these modules denoted which have the advantage of being induced from one dimensional modules for a borel subgroup in determining the structure of such modules or indeed their cohomology the calculation of hom spaces between them is a useful tool relatively little is known in general about such hom spaces in type a when andare related by a suitable single reflection explicit non zero homomorphisms from to were constructed with some restrictions by carter and lusztig 3 and more generally by carter and payne 4 the corresponding cases in other types were considered by franklin 14 while it is clear that there should be a hierarchy of families of homomorphisms corresponding to different powers of p the only case where the above results provide a complete classification is when g is sl2 where it is relatively easy to determine all hom spaces exactly 6 the only other general results in this area by andersen 1 and koppinen 18 concern homo morphisms between modules labelled by weights which are close together typically such results show that certain hom spaces are non zero or in some cases one dimensional for weights which are far apart and not related by a single reflection almost nothing is known in this paper we will determine all homomorphisms between weyl modules for sl3 when p 3 and provide a recursive procedure for determining the composition factors arising in the image or kernel of such maps in most cases from these results we will also classify all homomorphisms between specht modules for the symmetric groups corresponding to three part partitions when p 3 after a section of preliminaries we review the sl3 data concerning p filtrations that we will need from 21 this describes certain filtrations of induced modules which will allow us to proceed by induction together with the set of p good homomorphisms which will be fundamental in our later constructions we also recall a theorem of carter and payne 4 on the existence of certain homomorphisms these will be the two key sets of data which we need to determine all possible homomorphisms with the notation developed up to that point in place in section 4 we can give the strategy to be followed in the remainder of the paper and in particular the translation functor arguments

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

convergence to spdes in stratonovich form we consider the perturbation of parabolic operators of the form t p x d by large amplitude highly oscillatory spatially dependent potentials modeled as gaussian random fields the amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order we show that when the dimension is smaller than the order of the elliptic pseudo differential operator p x d the perturbed parabolic equation admits a solution given by a duhamel expansion moreover as the correlation length of the potential vanishes we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the stratonovich sense the theory of mild solutions for such stochastic partial differential equations is developed

approximation solution of nonlinear stratonovich volterra integral equations by applying modification of hat functions in this paper we studied the hyers ulam rassias stability for nonlinear stratonovich volterra integral equations also we obtained stochastic operational matrix of modification of hat functions mhfs to numerically solve the nonlinear stratonovich volterra integral equations the main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved finally numerical examples are presented to certify convergence and accuracy of the method

provably sublinear point multiplication on koblitz curves and its hardware implementation we describe algorithms for point multiplication on koblitz curves using multiple base expansions of the form k sigmaplusmntaua tau 1 b and k sigmaplusmntaua tau mu b tau2 mutau 1 c we prove that the number of terms in the second type is sublinear in the bit length of k which leads to the first provably sublinear point multiplication algorithm on koblitz curves for the first type we conjecture that the number of terms is sublinear and provide numerical evidence demonstrating that the number of terms is significantly less than that of tau adic nonadjacent form expansions we present details of an innovative fpga implementation of our algorithm and performance data demonstrating the efficiency of our method we also show that implementations with very low computation latency are possible with the proposed method because parallel processing can be exploited efficiently

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

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described