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

a note on k theory and triangulated categories we provide an example of two closed model categories having equivalent homotopy categories but different waldhausen k theories we also show that there cannot exist a functor from small triangulated categories to spaces which recovers quillen s k theory for exact categories and which satisfies localization

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

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

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

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

on the intersection of a clarke cone with a boltyanskii cone we provide an example of two closed sets s 1 s 2 subset r 4 such that s 1 cap s 2 0 yet at the origin a boltyanskii tangent cone c 1 to s 1 and the clarke tangent cone c 2 to s 2 are strongly transversal this settles a question originally proposed by h sussmann

generic cohomology for twisted groups let g be a simple algebraic group defined and split over ko fp and let a be a surjective endomorphism of g with finite fixed point set go we give conditions under which cohomology groups of g are isomorphic to cohomology groups of go let g be a simple algebraic group defined and split over k fp and for q pm let g q be the subgroup of fq rational points for a finite dimensional rational g module v and a nonnegative integer e let v e be the g module obtained by twisting the original g action on v by the frobenius endomorphism x x1p of g cline parshall scott and van der kallen proved in 2 that for sufficiently large q and e depending on v and n the restriction map induces an isomorphism from the rational cohomology group h g v e to h g q v e this implies that as q increases the groups h g q v e have a stable or generic value h en g v in this paper we prove an analogous theorem with g q replaced by go for a surjective endomorphism a of g having finite fixed point set the first section of the paper summarizes the basic results on endomorphisms of algebraic groups required for the proof some arithmetic facts are established in the second section and the main theorem is proved in the third section the author is grateful to leonard l scott for several helpful conversations and to the referee for comments on an earlier version of this paper 1 endomorphisms of algebraic groups we briefly review here some results on endomorphisms of algebraic groups which will be needed later we refer the reader to 4 and 5 for more details let k be the algebraic closure of the prime field ko fp and let g be a simple algebraic group defined and split over ko if a is a surjective rational endomorphism of g having finite fixed point set g0 a stabilizes a borel subgroup b and a maximal torus t ua is a t equivariant isomorphism then axa u x cauq a for some received by the editors may 5 1980 and in revised form november 12 1980 1980 mathematics subject classification primary 20g10

controllability along a trajectory a variational approach this paper unifies and improves most sufficient conditions of local controllability both along a trajectory and at a point this is accomplished by defining high order variations that can be continuously summed the peculiar property of these variations is that they may be generated by thin conditions as relations in the lie algebra associated with a control system this property leads to a variational interpretation of sussmann type sufficient conditions of local controllability neutralization of obstructions and it allows the use of different weights for neutralization

the t coefficient method ii a new series expansion formula of theta function products and its implications abstract by means of jacobi s triple product identity and the t coefficient method we establish a general series expansion formula with five free parameters for the product of arbitrary two jacobi theta functions it embodies the triple quintuple sextuple and septuple theta function product identities and the generalized schroter formula as further applications we also set up a series expansion formula for the product of three theta functions it not only generalizes ewell s and chen chen huang s octuple product identities but also contains three cubic theta function identities due to farkas kra and ramanujan respectively and the macdonald identity for the root system a 2 as special cases in the meantime many other new identities including a new short expression of the triple theta series of andrews are also presented

partial theta functions i beyond the lost notebook it is shown how many of the partial theta function identities in ramanujan s lost notebook can be generalized to infinite families of such identities key in our construction is the bailey lemma and a new generalization of the jacobi triple product identity by computing residues around the poles of our identities we find a surprising connection between partial theta function identities and garrett ismail stanton type extensions of multisum rogers ramanujan identities

optimization and controllability without differentiability assumptions let mathcal x and mathcal y be real normed vector spaces k subset mathcal x convex and compact c subset mathcal y a convex body mathcal u subset k and phi phi k to mathbb r m times mathcal y a function that can be appropriately approximated by functions phi i phi i whose compositions with linear maps of small finite dimensional simplices are c 1 we derive sufficient conditions for phi to be locally controllable on mathcal u subject to the restriction phi u in c and obtain as a corollary corresponding necessary conditions for a related restricted minimum these conditions are formulated in terms of directional derivate containers which are a type of set valued derivatives of phi phi and they improve on and extend previously obtained results they are used elsewhere to obtain new conditions for controllability and restricted minimum in nonsmooth optimal control problems defined by differential or functional integral equatio

homeomorphisms and local c 1 approximations soient 7b h et 7b y des espaces de banach v 7b h ouvert et f v 7b y continue on demontre que les extensions de john du theoreme de hadamard sont applicables a des fonctions entre des espaces de banach avec des conteneurs de derivees locaux inversibles

the t coefficient method iii a general series expansion for the product of theta functions with different bases and its applications by means of jacobi s triple product identity and the t coefficient method we establish a general series expansion formula for the product of arbitrary two theta functions with bases p and q az p bzt q n p n az n 1 a tbpt t 1 2 tn pt2q from which some new results on products of arbitrary finitely many theta functions and theta identities associated with ramanujan s circular summation can be derived among them the most interesting ones include k 0mn 1 i 1n ayi k q mn 0 y1 y2 yn y1y2 yn mamnqnm22 nm2 qnm2 where is a primitive mn th root of unity and s y1 y2 yn i1 i2 in sq12 j 1nij2 j 1nyjij furthermore for y1y2y3 qy1y2y3 q and exp i 3 exp i 3 3 0 y1 y2 y3 q 3 2 3 1 y1 y2 y3 1 q q 3 i 02 j 13 yj 2i q the former contains chan and liu s circular summation cf chan and liu 2010 12 as a special case y1y2 yn 1 the latter generalizes both borweins and garvan s well known cubic theta function identity and schultz s bivariate generalization cf borwein et al 1994 9 and schultz 2013 28 in which y1 y2 y3 q1 3y1 y2 y3 q1 3 and y1 y3 z1 y2 y3 z2 y33 qz1z2 respectively

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

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 t coefficient method to partial theta function identities and ramanujan s 1ψ1 summation formula this paper is concerned with a uniform approach the t coefficient method to basic hypergeometric series by virtue of this method some new and elementary proofs for four partial theta function identities due to alladi and berkovich 2004 8 andrews and warnaar 2007 4 berkovich 5 and warnaar 2003 2 as well as for ramanujan s 1 1 summation formula are obtained

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

set separation approximating multicones and the lipschitz maximum principle abstract we present a general necessary condition for separation of the reachable set of a lipschitz control system from another given set of states expressed in terms of an approximating multicone to the set in a sense that contains as special cases the clarke and mordukhovich cones we then show how this separation result implies a strengthened form of the usual sufficient condition for local controllability along the reference curve and the necessary condition for optimality

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

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

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

an euler lagrange inclusion for optimal control problems a new first order necessary condition is proved for nonsmooth nonlinear optimal control problems with general endpoint constraints and for which the velocity set may be possibly nonconvex it is in the nature of a generalization of the euler lagrange equation of the calculus of variations to optimal control it resembles the weak form of the maximum principle but it is distinct from it because it employs a total generalized gradient instead of the customary product of partial generalized gradients the optimality condition is shown to be sufficient for optimality when it is specialized to apply to normal convex problems a counterexample illustrates that for such problems the maximum principle is not a sufficient condition

support varieties for modules over chevalley groups and classical lie algebras let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 g 1 be the first frobenius kernel and g f p be the corresponding finite chevalley group let m be a rational g module in this paper we relate the support variety of m over the first frobenius kernel with the support variety of m over the group algebra kg f p this provides an answer to a question of parshall applications of our new techniques are presented which allow us to extend results of alperin mason and janiszczak jantzen and to calculate the dimensions of support varieties for finite chevalley groups

a high order test for optimality of bang bang controls for control systems of the form dot x x x sum i 1 m y i x u i a strengthened version of the classical pontryagin maximum principle is proved the necessary condition for optimality given here is obtained using functional analytic techniques and quite general high order perturbations of the reference control as shown by an example this test is particularly effective when applied to bang bang controls a case where other high order tests do not provide additional information

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