a note on decomposition numbers for reductive algebraic groups in a recent paper 7 james proves two theorems on the decomposition numbers for the general linear groups and working within the context of green s monograph 5 derives properties of the decomposition numbers for the symmetric groups this note grew out of an attempt to formulate for arbitrary reductive groups these theorems of james we prove a natural result on decomposition numbers for reductive groups from which james s theorems are deduced the result the theorem of section 2 is the analogue of jantzen s satz 1 18 of 8 on decomposition numbers for representations of complex semisimple lie algebras also proved independently by deodhar very little direct use of 5 is made though some of the ideas are cribbed from that source as one may see by comparing the properties of the functor fa in section 2 with green s treatment see section 6 2 of s of ese theory s a ring e an idempotent

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

koszul duality and extensions of exponential functors we study koszul duality in the category of strict polynomial functors we compute koszul duals of various functors and apply these results to the problem of calculating ext groups between exponential functors the main application is a full description of the ext groups between twisted exterior and divided powers and between twisted symmetric and divided powers which completes the program started in v franjou e friedlander a scorichenko a suslin general linear and functor cohomology over finite fields ann of math 150 2 1999 663 728

weyl modules and the mod 2 steenrod algebra abstract this paper continues our study of the action of the mod 2 steenrod algebra a 2 on the polynomial algebra p n f 2 t 1 t n we obtain further partial results on the hit problem of f p peterson which asks for a minimal generating set for p n as an a 2 module we also study the structure of the quotient by the hit elements as a graded representation of the finite general linear group g n gl n f 2 i e as a module over the finite group algebra f 2 g n these results were obtained in previous work of the authors for the special case of the steinberg module for g n by extending the scalars to f 2 the algebraic closure of f 2 we obtain commuting actions of a 2 and g n on p n f 2 t 1 t n while this makes no essential difference to the representation theory of g n or to the hit problem it allows us to treat the action of g n on p n as the restriction of that of the algebraic group g n gl n f 2 in particular we make use of tilting modules for g n to show that for every irreducible representation l of g n a minimal set of a 2 generators of p n must contain a copy of the corresponding dual weyl module

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

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

quantum weyl reciprocity for cohomology we study the relationship between the cohomology of q schur algebras and hecke algebras in type a a key tool which has independent interest involves a specific hecke algebra resolution introduced in a special case by deodhar this resolution is somewhat akin to the koszul resolution for lie algebra cohomology in that it has an explicit description yet applies for all modules an application of the properties of this resolution establishes a quantum weyl reciprocity connecting in a range of degrees the cohomology of the q schur algebra with that of the hecke algebra this work also provides independently of the resolution cohomological inequalities which apply in all degrees and which sometimes can be related to kazhdan lusztig polynomials finally we extend to cohomology a decomposition number method of erdmann and illustrate its usefulness in computations this method enables us to exhibit some properties of symmetric group cohomology which if true would imply the sl n version of lusztig s characteristic p character conjecture

schur algebras of brauer algebras i a classical problem of invariant theory and of lie theory is to determine endomorphism rings of representations of classical groups for instance of tensor powers of the natural module schur weyl duality or of full direct sums of tensor products of exterior powers ringel duality in this article the endomorphism rings of full direct sums of tensor products of symmetric powers over symplectic and orthogonal groups are determined these are shown to be isomorphic to schur algebras of brauer algebras as defined in henke and koenig math z 272 3 4 729 759 2012 this implies structural properties of the endomorphism rings such as double centraliser properties quasi hereditary and a universal property as well as a classification of simple modules

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

a clebsch gordan decomposition in positive characteristic abstract let k be an algebraically closed field and g sl 2 k let e be the natural module and s r e the rth symmetric power we consider here for r s 0 the tensor product of s r e and the dual of s s e in characteristic zero this tensor product decomposes according to the clebsch gordan formula we consider here the situation when k is a field of positive characteristic we show that each indecomposable component occurs with multiplicity one and identify which modules occur for given r and s

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

irreducible a1 subgroups of exceptional algebraic groups abstract a closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup in this paper we classify all irreducible a 1 subgroups of exceptional algebraic groups g consequences are given concerning the representations of such subgroups on various g modules for example the conjugacy classes of irreducible a 1 subgroups are determined by their composition factors on the adjoint module of g

high order linear multistep methods with general monotonicity and boundedness properties we consider linear multistep methods that possess general monotonicity and boundedness properties strict monotonicity in terms of arbitrary starting values for the multistep schemes is only valid for a small class of methods under very stringent step size restrictions this makes them uncompetitive with the strong stability preserving ssp runge kutta methods by relaxing these strict monotonicity requirements a larger class of methods can be considered including many methods of practical interest in this paper we construct linear multistep methods of high order up to six that possess relaxed monotonicity or boundedness properties with optimal step size conditions numerical experiments show that the new schemes perform much better than the classical monotonicity preserving multistep schemes moreover there is a substantial gain in efficiency compared to recently constructed ssp runge kutta ssprk methods

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

gaussian fields and random packing consider sequential packing of unit volume balls in a large cube in any dimension and with poisson input we show after suitable rescaling that the spatial distribution of packed balls tends to that of a gaussian field in the thermodynamic limit the results cover related applied models including ballistic deposition and spatial birth growth models

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