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

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

cohomology for frobenius kernels of sl2 abstract let sl 2 r be the r th frobenius kernels of the group scheme sl 2 defined over an algebraically closed field of characteristic p 0 in this paper we give for r 1 a complete description of the cohomology groups for sl 2 r we also prove that the reduced cohomology ring h sl 2 r k red is cohen macaulay geometrically we show for each r 1 that the maximal ideal spectrum of the cohomology ring for sl 2 r is homeomorphic to the fiber product g b u r finally we adapt our calculations to obtain analogous results for the cohomology of higher frobenius lusztig kernels of quantized enveloping algebras of type sl 2

ringel duality as an instance of koszul duality abstract in 30 s koenig s ovsienko and the second author showed that every quasi hereditary algebra is morita equivalent to the right algebra i e the opposite algebra of the left dual of a coring let a be an associative algebra and v an a coring whose right algebra r is quasi hereditary in this paper we give a combinatorial description of an associative algebra b and a b coring w whose right algebra is the ringel dual of r we apply our results in small examples to obtain restrictions on the a structure of the ext algebra of standard modules over a class of quasi hereditary algebras related to birational morphisms of smooth surfaces

varieties of nilpotent elements for simple lie algebras i good primes let g be a simple algebraic group over k or fp where p is good set g lie g given r and a faithful restricted representation g gl v one can define a variety of nilpotent elements nr g x g x r 0 in this paper we determine this variety when is an irreducible representation of minimal dimension or the adjoint representation

cohomology of infinitesimal and discrete groups an infinitesimal algebraic group over a field k is a connected algebraic group scheme whose coordinate ring is a finite dimension al k algebra if one begins with a reductive algebraic group g defined over fp then one has a close relationship between the representation theory of g that of its infinitesimal subgroups g defined as the kernels of the frobenius endomorphisms rr g g and that of its finite chevalley subgroups g fq of fq rational points q pd a basic theorem of cline et al i 9 theorem 6 6 relates the cohomology of g in a rational representation to that of the finite groups g fq related work of cline et al 6 and of the authors 1 16 relates the cohomology of g to that of its infinitesimal subgroups a determination of the cohomology of the first infinitesimal subgroup g 1 of g with trivial coefficients has enabled the authors in 14 to make certain explicit cohomology calculations for the finite groups g fq with certain nontrivial coefficients in this paper we continue our study of infinitesimal subgroups of simple algebraic groups by proving various qualitative properties of their cohomology and by extending our computational knowledge in particular we provide a computation which applies in a range of degrees to chevalley groups over the prime field we also investigate the cohomology of infinitesimal subgroups of nilpotent groups enabling us for example to determine the cohomology of their lie algebras in certain cases we apply this lie algebra cohomology computation to determine the integral cohomology of various torsion free nilpotent groups modulo small prime torsion thereby extending a theorem of dwyer 12 in more detail the paper begins in sect 1 with a spectral sequence converging to the cohomology of the first infinitesimal subgroup g which was previously studied in 14 from this spectral sequence it easily follows that h g1 m is a noetherian h g1 k module for each finite dimensional rational gl module m we then apply the work of andersen and jantzen 1 on the gl cohomology of induced modules to obtain the analogous result for the second infinitesimal subgroup gz our analysis also permits a determination of the generic

complexity for extended dynamical systems we consider dynamical systems for which the spatial extension plays an important role for these systems the notions of attractor entropy and topological entropy per unit time and volume have been introduced previously in this paper we use the notion of kolmogorov complexity to introduce for extended dynamical systems a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems we introduce this notion as an almost sure limit on orbits of the system moreover we prove a kind of variational principle for this complexity

on the character of some modular indecomposable tilting modules for sl3 abstract we calculate explicitly the character of some indecomposable tilting modules for sl 3 k where k is a field of characteristic p 2

extensions of simple modules for finite chevalley groups let g be an almost simple algebraic group over an algebraically closed field k of positive characteristic p we assume that g is defined and split over the prime field 5 for it 3 1 we denote by g n the finite group consisting of the points of g over the field with p elements our first result states that for a certain class of simple modules for g n the only extensions that can occur are those which are restrictions of g extensions see theorem 2 8 the full story about g extensions is not known yet except for a few rather small groups however much more is known about extensions for the algebraic group than about those for the finite group for instance the linkage principle l gives an upper bound for which g extensions may occur moreover a proof of the lusztig conjecture on the irreducible characters for g also implies strong statements about g extensions see 3 proposition 2 81 it is an open question whether all extensions between simple g n modules can be found via the extension theory for g we prove that in generic cases this is in fact so see theorem 3 2 moreover for the groups x 3 p and sp 4 p we are able to determine completely all extensions between simple modules from the corresponding known results for the algebraic groups x 3 k and sp 4 k our results extend previous results by cline parshall and scott s 91 who consider extensions of the trivial module by a minimal module and smith ls who considers extensions between certain minimal modules the method we use is a generalization of the one used by humphreys 131 who proves the existence of certain selfextensions for sp 4 p it should be pointed out that our methods do not work for p small on the other hand everything we do generalizes easily to the twisted chevalley groups see theorem 3 4 we demonstrate this by working out all extensions for su 3 p see section 4 3

the attractor for a nonlinear reaction diffusion system in an unbounded domain in this paper the quasi linear second order parabolic systems of reaction diffusion type in an unbounded domain are considered our aim is to study the long time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions to this end we give a systematic study of given parabolic systems and their attractors in weighted sobolev spaces dependence of the hausdorff dimension of attractors on the weight of the sobolev spaces is considered 2001 john wiley sons inc

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

p filtrations of dual weyl modules abstract let g be a semisimple algebraic group over a field of characteristic p 0 we prove that the dual weyl modules for g all have p filtrations when p is not too small moreover we give applications of this theorem to p n filtrations for n 1 to modules containing the steinberg module as a tensor factor and to the donkin conjecture on modules having p filtrations

scheduling with controllable release dates and processing times makespan minimization abstract the paper deals with the single machine scheduling problem in which job processing times as well as release dates are controllable parameters and they may vary within given intervals while all release dates have the same boundary values the processing time intervals are arbitrary it is assumed that the cost of compressing processing times and release dates from their initial values is a linear function of the compression amount the objective is to minimize the makespan together with the total compression cost we construct a reduction to the assignment problem for the case of equal release date compression costs and develop an o n 2 algorithm for the case of equal release date compression costs and equal processing time compression costs for the bicriteria version of the latter problem with agreeable processing times we suggest an o n 2 algorithm that constructs the breakpoints of the efficient frontier

second cohomology groups for frobenius kernels and related structures abstract let g be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p let b be a borel subgroup of g and u be its unipotent radical in this paper we determine the second cohomology groups of b and its frobenius kernels for all simple b modules we also consider the standard induced modules obtained by inducing a simple b module to g and compute all second cohomology groups of the frobenius kernels of g for these induced modules also included is a calculation of the second ordinary lie algebra cohomology group of lie u with coefficients in k

unipotent elements tilting modules and saturation the results in this paper were motivated by work of serre 19 where semisimplicity results for representations of arbitrary groups in positive characteristic were established a key ingredient in the arguments was the notion of saturation where one embeds a unipotent element of prime order of a simple algebraic group in a 1 dimensional unipotent subgroup one of our goals here is to show that saturation can be achieved rather generally and to establish a uniqueness result for the resulting unipotent subgroups as a by product we show for a simple algebraic group in good characteristic a unipotent element of prime order is contained in a particularly nice subgroup of type a1 and that the lie algebra of the ambient algebraic group usually affords a tilting module for this subgroup the tilting decompositions are applied to show that the centralizers of the unipotent element a corresponding 1 dimensional unipotent group and its lie algebra all coincide and are closely related to the centralizer of the a1 subgroup we also establish a convenient factorization of the centralizer of the unipotent element results for finite groups of lie type are also obtained

minimization of the makespan in a two machine problem under given resource constraints in the paper the classical two machine flow shop problem was generalized to the case when job processing times may be reduced linearly by the application of a limited continuously divisible resource e g financial outlay energy fuel catalyzer etc it is proved that the decision form of this problem is np complete even for the fixed job processing times on one of the machines and identical job reduction rates on another some polynomially solvable cases of the problem are identified four simple and modified approximate algorithms are presented together with their worst case and experimental analysis also a fast exact algorithm of the branch and bound type based on the shown elimination properties of the problem considered is presented some computational results and generalizations e g bicriterial approach are included as well

some remarks on the uniqueness of liftings of principal indecomposable modules for infinitesimal subgroups of semisimple groups abstract this note is in the nature of an appendix to the recent paper by paul sobaje 5 in that paper section 2 2 the question of the uniqueness of g structures on or liftings of principal indecomposable g 1 module in the case g sl 3 is raised we here answer this question positively as well as making some more general observations

scheduling two agents with controllable processing times abstract we consider several two agent scheduling problems with controllable job processing times where agents a and b have to share either a single machine or two identical machines in parallel while processing their jobs the processing times of the jobs of agent a are compressible at additional cost the objective function for agent b is always the same namely a regular function f max several different objective functions are considered for agent a including the total completion time plus compression cost the maximum tardiness plus compression cost the maximum lateness plus compression cost and the total compression cost subject to deadline constraints the imprecise computation model all problems are to minimize the objective function of agent a subject to a given upper bound on the objective function of agent b these problems have various applications in computer systems as well as in operations management we provide np hardness proofs for the more general problems and polynomial time algorithms for several special cases of the problems

job shop scheduling with resource time models of operations abstract the job shop problems with allocation of continuously divisible nonrenewable resource is considered the mathematical models of operations are linear decreasing functions with respect to an amount of resource the objective is sequencing operations and allocation of constrained resource such that the project duration is minimized thus the problem considered is a generalization of the classical job shop problem some properties of the optimal solution are presented the algorithm of solving this problem is based on the disjunctive graphs theory and branch and bound technique the theory of the algorithm is based on the critical path concept using the segment system approach the special feature of the algorithm is that it gives a complete solution which is associated with each node of the enumeration tree possible generalizations of the results presented are indicated

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

extensions for frobenius kernels 1 1 let g be a simple algebraic group over an algebraically closed field k of characteristic p 0 for r 1 let gr be the rth frobenius kernel of g it is well known that the representations for g1 are equivalent to the restricted representations for lie g historically the cohomology for frobenius kernels has been best understood for large primes friedlander and parshall fp first computed the cohomology ring h g1 k for p 3 h 1 where h is the coxeter number of the underlying root system they proved that the cohomology ring can be identified with the coordinate algebra of the nullcone andersen and jantzen aj later verified this fact for p h furthermore they generalized this calculation by looking at h g1 h0 where h0 indb for p h their results had some restrictions on the type of root system involved kumar lauritzen and thomsen klt removed the restrictions on the root systems through the use of frobenius splittings the cohomology ring h g1 k modulo nilpotents can be identified in general with the coordinate algebra of the restricted nullcone n1 x lie g x p 0 for good primes nakano parshall and vella npv proved that this variety is irreducible and can be identified with the closure of some richardson orbit recently carlson lin nakano and parshall clnp have given an explicit description of n1 these recent results provide some indication that one can systematically study extensions of frobenius kernels for small primes by using general formulas which exhibit generic behavior for large primes

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

second cohomology groups for algebraic groups and their frobenius kernels let g be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic p 0 let t be a maximal split torus in g b t be a borel subgroup of g and u its unipotent radical let f g g be the frobenius morphism for r 1 define the frobenius kernel gr to be the kernel of f iterated with itself r times define ur respectively br to be the kernel of the frobenius map restricted to u respectively b let x t be the integral weight lattice and x t be the dominant integral weights the computations of particular importance are h2 u1 k h2 br for x t h2 gr h0 for x t and h2 b for x t the above cohomology groups for the case when the field has characteristic 2 are computed in this paper these computations complete the picture started by bendel nakano and pillen 2007 5 for p 3 furthermore the computations show h2 gr h0 has a good filtration

attractors of reaction diffusion systems in unbounded domains and their spatial complexity the nonlinear reaction diffusion system in an unbounded domain is studied it is proven that under some natural assumptions on the nonlinear term and on the diffusion matrix this system possesses a global attractor in the corresponding phase space since the dimension of the attractor happens to be infinite we study its kolmogorov s entropy upper and lower bounds of this entropy are obtained moreover we give a more detailed study of the attractor for the spatially homogeneous rde in n in this case a group of spatial shifts acts on the attractor in order to study the spatial complexity of the attractor we interpret this group as a dynamical system with multidimensional time if n 1 acting on a phase space it is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy in order to clarify the nature of this chaos we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic and even lipschitz continuous embedding of this system into the spatial shifts on the attractor finally we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution 2003 wiley periodicals inc

on good p r filtrations for rational g modules abstract in this paper we investigate donkin s p r filtration conjecture and present two proofs of the if direction of the statement when p 2 h 2 one proof involves the investigation of when the tensor product between the steinberg module and a simple module has a good filtration one of our main results shows that this holds under suitable requirements on the highest weight of the simple module the second proof involves recasting donkin s conjecture in terms of the identifications of projective indecomposable g r modules with certain tilting g modules and establishing necessary cohomological criteria for the p r filtration conjecture to hold

algebraic group actions in the cohomology theory of lie algebras of cartan type abstract given a lie algebra g of cartan type we construct an infinite dimensional cocommutative hopf algebrad g u g which is the analog of the distribution algebra of a connected reductive algebraic group we show that the simpleu g modules lift to ad g u g structure this additional structure is used to formulate and prove relative projectivity theorems for lie algebras of cartan type the support varieties of certaind g u g modules are computed by using these algebraic group techniques

random exponential attractor for stochastic reaction diffusion equation with multiplicative noise in r3 abstract in this paper we first improve the existing conditions for the existence of a random exponential attractor for a continuous cocycle on a separable banach space then we consider the existence of a random exponential attractor for stochastic non autonomous reaction diffusion equation with multiplicative noise defined in r 3 which implies the existence of a random attractor with finite fractal dimension the essential difficulty here is the continuity of the spectrum of the linear part of the equation which can be overcome by the tail estimation of solutions of equation and carefully decomposing the solution into a sum of three parts of whose one part is finite dimensional and other two parts are quickly decay in mean sense

on the existence of the compact global attractor for semilinear reaction diffusion systems on rn we show that a class of reaction diffusion systems on rngenerates an asymptotically compact semiflow on the banach space of bounded uniformly continuous functions if such a semiflow is dissipative then a unique non empty compact minimal attractor is known to exist we apply this abstract result to obtain the existence of the compact minimal attractor for reaction diffusion systems on rnthat contain appropriate weight functions we also state conditions which guarantee that the attractor has finite hausdorff dimension

on the cohomology of algebraic and related finite groups in the above theorem h gl m denotes the rational cohomology of the algebraic group gl with coefficients in m the rational gl module obtained from the adjoint module m m through twisting by the r th power of the frobenius endomorphism on gl the algebra structure on the cohomology is induced from the associative algebra structure on m on the other hand we provide the following systematic determination of certain rational cohomology groups h g v r where v r is the rational gmodule obtained from the rational g module v by applying the r th frobenius twist see 1 2 this is a somewhat sharpened version of theorem 4 of 36

finitistic dimension of properly stratified algebras abstract we prove that the finitistic dimension of a properly stratified algebra having a simple preserving duality and for which every tilting module is cotilting equals twice the projective dimension of the characteristic tilting module as a corollary we get that the global dimension of a quasi hereditary algebra with duality equals twice the projective dimension of the characteristic tilting module as another corollary we obtain an affirmative answer to the conjecture of erdmann and parker finally we calculate the finitistic dimension of the blocks of certain parabolic generalizations of the category o

sheaf cohomology on gb and tensor products of weyl modulus let k be an algebraically closed field it is assumed that char k p 0 unless otherwise indicated g is a simply connected semisimple algebraic group over k t is a fixed maximal torus and b a bore1 subgroup of g containing t let qi g t be the root system of g d the set of all simple roots and and the sets of all positive and negative roots relative to d respectively we assume that the set a is chosen such that b corresponds to 0 let x be the weight lattice of sp which may be considered as the character group of t or b therefore if 1 e x by abuse of notation we also denote by l the one dimensional tor b module determined by a let x be the set of all dominant weights and w the fundamental weight corresponding to a simple root a e a we denote by 6 the half sum of all positive roots and we know that 6 is the sum of all fundamental weights let w be the weyl group of 0 the reflection with respect to a root a is denoted by s the longest element of w by w there are two ways of w acting on x i the ordinary action if 1 e x m e w the image of i under w is denoted by 2 ii dot action w a 5 2 6 s in particular let a w a the dual root system of is denoted by and the dual root of a e by a i e u 2a a u where is the inner product invariant under the ordinary action of w of the euclidean space spanned by for 1 e x a e let a a 1 a all modules of an algebraic group we consider in this paper are assumed to be rational in particular for an infinite dimensional module the term rational module means this module is a direct limit of finite dimensional rational modules let h be a closed subgroup of g containing t such as g b or t m an h module we define for l e x

invariant forms on irreducible modules of simple algebraic groups let g be a simple linear algebraic group over an algebraically closed field k of characteristic p 0p 0 and let v be an irreducible rational g module with highest weight when v is self dual a basic question to ask is whether v has a non degenerate g invariant alternating bilinear form or a non degenerate g invariant quadratic form if p 2p 2 the answer is well known and easily described in terms of in the case where p 2p 2 we know that if v is self dual it always has a non degenerate g invariant alternating bilinear form however determining when v has a non degenerate g invariant quadratic form is a classical problem that still remains open we solve the problem in the case where g is of classical type and is a fundamental highest weight i i and in the case where g is of type alal and r s r s for 1 r s l1 r s l we also give a solution in some specific cases when g is of exceptional type as an application of our results we refine seitz s 1987 description of maximal subgroups of simple algebraic groups of classical type one consequence of this is the following result if x y sl v x y sl v are simple algebraic groups and v xv x is irreducible then one of the following holds 1 v yv y is not self dual 2 both or neither of the modules v yv y and v xv x have a non degenerate invariant quadratic form 3 p 2p 2 x so v x so v and y sp v y sp v

the gln q module structure of the symmetric algebra around the steinberg module abstract we determine the graded composition multiplicity in the symmetric algebra s v of the natural gl n q module v or equivalently in the coinvariant algebra of v for a large class of irreducible modules around the steinberg module this was built on a computation via connections to algebraic groups of the steinberg module multiplicity in a tensor product of s v with other tensor spaces of fundamental weight modules

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

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

bases for coordinate rings of conjugacy classes of nilpotent matrices abstract a general conjecture is given for an explicit basis of the coordinate ring of the closure of the conjugacy class of a nilpotent matrix this conjecture is proven when the partition given by the transpose jordan type of the nilpotent matrix is a hook or has two parts

complete reducibility in good characteristic let g be a simple algebraic group of exceptional type over an algebraically closed field of characteristic p 0 a closed subgroup h of g is called g completely reducible g cr if whenever h is contained in a parabolic subgroup p of g it is contained in a levi subgroup of p in this paper we determine the g conjugacy classes of non g cr simple connected subgroups of g when p is good for g for each such subgroup x we determine the action of x on the adjoint module l g and the connected centraliser of x in g as a consequence we classify all non g cr reductive subgroups of g and we additionally show the existence of and classify the following restricted non g cr semisimple subgroups non g cr reductive subgroups with trivial connected centraliser and subgroups which are maximal among reductive subgroups of g

the frobenius morphism in invariant theory abstract let r be the homogeneous coordinate ring of the grassmannian g gr 2 n defined over an algebraically closed field of characteristic p 0 in this paper we give a completely characteristic free description of the decomposition of r considered as a graded r p module into indecomposables frobenius summands as a corollary we obtain a similar decomposition for the frobenius pushforward of the structure sheaf of g and we obtain in particular that this pushforward is almost never a tilting bundle on the other hand we show that r provides a noncommutative resolution for r p when p n 2 generalizing a result known to be true for toric varieties in both the invariant theory and the geometric setting we observe that if the characteristic is not too small the frobenius summands do not depend on the characteristic in a suitable sense in the geometric setting this is an explicit version of a general result by bezrukavnikov and mirkovic on frobenius decompositions for partial flag varieties we are hopeful that it is an instance of a more general p uniformity principle

graded analogues of one parameter subgroups and applications to the cohomology of glm n r abstract we introduce a family m r f of infinitesimal supergroup schemes which we call multiparameter supergroups that generalize the infinitesimal frobenius kernels g a r of the additive group scheme g a then following the approach of suslin friedlander and bendel we use functor cohomology to define characteristic extension classes for the general linear supergroup g l m n and we calculate how these classes restrict along homomorphisms m r f g l m n finally we apply our calculations to describe up to a finite surjective morphism the spectrum of the cohomology ring of the r th frobenius kernel g l m n r of the general linear supergroup g l m n

on the representation theory of lie triple systems in this paper we take a new look at the representation theory of lie triple systems we consider both ordinary lie triple systems and restricted lie triple systems in the sense of hodge 2001 in a final section we begin a study of the cohomology of lie triple systems

attractors of partial differential evolution equations in an unbounded domain there is a large number of papers in which attractors of parabolic reaction diffusion equations in bounded domains are investigated in this paper these equations are considered in the whole unbounded space and a theory of attractors of such equations is built while investigating these equations specific difficulties arise connected with the noncompactness of operators with the continuity of their spectra etc therefore some new conditions on nonlinear terms arise in this paper weighted spaces are widely applied an important feature of this problem is worth mentioning namely properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ

differentiating the weyl generic dimension formula with applications to support varieties abstract the authors compute the support varieties of all irreducible modules for the small quantum group u g where g is a finite dimensional simple complex lie algebra and is a primitive l th root of unity with l larger than the coxeter number of g the calculation employs the prior calculations and techniques of ostrik and of nakano parshall and vella as well as deep results involving the validity of the lusztig character formula for quantum groups and the positivity of parabolic kazhdan lusztig polynomials for the affine weyl group analogous support variety calculations are provided for the first frobenius kernel g 1 of a reductive algebraic group scheme g defined over the prime field f p

reductive groups the loop grassmannian and the springer resolution in this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic p bigger than the coxeter number and a derived category of equivariant coherent sheaves on the springer resolution or a parabolic counterpart in the case of the principal block combined with previous results this provides a modular version of celebrated constructions due to arkhipov bezrukavnikov ginzburg for lusztig s quantum groups at a root of unity as an application we prove a graded version of a conjecture of finkelberg mirkovic describing the principal block in terms of mixed perverse sheaves on the dual affine grassmannian and deduce a new proof of lusztig s conjecture in large characteristic

dynamics of strongly damped wave equations in locally uniform spaces attractors and asymptotic regularity this paper is dedicated to analyzing the dynamical behavior of strongly damped wave equations with critical nonlinearity in locally uniform spaces after proving the global well posedness we first establish the asymptotic regularity of the solutions which appears to be optimal and the existence of a bounded in h 2 lu r n x h 1 lu r n subset which attracts exponentially every initial h 1 lu r n x l 2 lu r n bounded set with respect to the h 1 lu r n x l 2 lu r n norm then we show there is a h 1 lu r n x l 2 lu r n h 1 r n x h 1 r n global attractor which reflects the strongly damped property of u t to some extent

a note on the tensor product of restricted simple modules for algebraic groups let g be a semisimple simply connected algebraic group over an algebraically closed field of positive characteristic p denote by g1 its first frobenius kernel in this note we determine for which group g the restriction to g1 of any indecomposable g summand of the tensor product of any two restricted simple g modules remains indecomposable

tensor products of quantized tilting modules letuk denote the quantized enveloping algebra corresponding to a finite dimensional simple complex lie algebral assume that the quantum parameter is a root of unity ink of order at least the coxeter number forl also assume that this order is odd and not divisible by 3 if typeg2 occurs we demonstrate how one can define a reduced tensor product on the familyf consisting of those finite dimensional simpleuk modules which are deformations of simplel and which have non zero quantum dimension this together with the work of reshetikhin turaev and turaev wenzl prove that uk f is a modular hopf algebra and hence produces invariants of 3 manifolds also by recent work of duurhus jakobsen and nest it leads to a general topological quantum field theory the method of proof explores quantized analogues of tilting modules for algebraic groups

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

infinite dimensional exponential attractors for nonlinear reaction diffusion systems in unbounded domains and their approximation in this paper we study the long time behaviour of solutions of reaction diffusion equations rdes in unbounded domains of r n in particular we prove that under appropriate assumptions on the nonlinear interaction function and on the external forces these equations possess infinite dimensional exponential attractors whose kolmogorov entropy satisfies an estimate of the same type as that obtained previously for the entropy of the global attractor moreover we also study the problem of the approximation of these infinite dimensional exponential attractors by finite dimensional ones associated with the same rdes in bounded domains

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

extensive properties of the complex ginzburg landau equation we study the set of solutions of the complex ginzburg landau equation in r d d 3 we consider the global attracting set i e the forward map of the set of bounded initial data and restrict it to a cube q l of side l we cover this set by a minimal number n q l of balls of radius in l infin q l we show that the kolmogorov entropy per unit length exists in particular we bound by which shows that the attracting set is smaller than the set of bounded analytic functions in a strip we finally give a positive lower bound

existence and finite dimensionality of the global attractor for evolution equations on unbounded domains abstract this paper is devoted to the asymptotic behavior for some evolution equations when the underlying euclidean domain is unbounded we present a method that is able to overcome the lack of compactness of the trajectories in such situations and that allows us to obtain some results on the global attractor similar to those on bounded domains this method is based on the utilization of an appropriate weight integration function that depends on x and t and becomes effective in x for large t s

some remarks on branching rules and tensor products for algebraic groups let f be an algebraically closed field of characteristic p 0 in bk1 bk2 we have revealed and exploited various relations between the branching rules from gln f to its levi subgroups on one hand and decompositions of tensor products over gln f itself on the other for example if l is some irreducible rational gln f module and v is the natural gln f module there is a close relationship between the highest weight vectors relative to gln 1 f in the restriction l gln 1 f and the highest weight vectors relative to gln f in the tensor product l v in this paper we obtain more results in this direction some of which are valid for an arbitrary type to describe our main results we adopt standard lie theoretic notation let g be a connected reductive algebraic group over f as in j r denotes the root system of g with respect to a fixed maximal torus t r r denotes the set of positive roots determined by a choice of borel subgroup b containing t and 1 r is the corresponding base for r denote the highest long root of r by 0 and the longest element of the weyl group w ng t t by w0 we write x t for the character group hom t f y t for the cocharacter group hom f t and let be the natural pairing x t y t z for r denotes the corresponding coroot in y t and x t denotes the set x t i 0 i 1 of dominant weights all g modules are assumed to be rational for x t we have the g modules l and which are the irreducible the standard or weyl and the costandard gmodules with highest weight let dist g be the algebra of distributions of g as in j i 7 which is generated by dist t and the divided power root generators x y n for r n 1 write x i x n i yi y n i for i 1 if g is semisimple and simply connected which we may assume for the proofs dist g coincides with the hyperalgebra of g arising from the chevalley construction we note that any g module is a dist g module in a natural way see j i 7 11 ii 1 20 given a weight x t and a g module m m will denote the weight space of m if in addition x t is a dominant weight we define