unipotent elements and generalized exponential maps abstract let g be a simple and simply connected algebraic group over an algebraically closed field k of characteristic p 0 assume that p is good for the root system of g and that the covering map g s c g is separable in previous work we proved the existence of a not necessarily unique springer isomorphism for g that behaved like the exponential map on the restricted nullcone of g in the present paper we give a formal definition of these maps which we call generalized exponential maps we provide an explicit and uniform construction of such maps for all root systems demonstrate their existence over z p and give a complete parameterization of all such maps one application is that this gives a uniform approach to dealing with the saturation problem for a unipotent element u in g providing a new proof of the known result that u lies inside a subgroup of c g u that is isomorphic to a truncated witt group we also develop a number of other explicit and new computations for g and for g this paper grew out of an attempt to answer a series of questions posed to us by p deligne who also contributed several of the new ideas that appear here

stratification for module categories of finite group schemes the second author was partly supported by nsf grant dms 1503044 the fourth author was partly supported by nsf grant dms 0953011

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

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

the varieties and the cohomology ring of a module let g be a finite group and let k be a field of characteristic p 0 recently several studies have focused on the homological invariants of finitely generated kg modules in l alperin proposed the study of the complexity of a module the complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module see definition 2 4 in 2 alperin and evens proved that the complexity of a kg module m is equal to the maximum of the complexities of the restrictions of m to elementary abelian p subgroups of g one of the roots of the alperin evens theorem is quillen s dimension theorem see 21 or 22 which with some difficulty can be interpreted as saying the same thing for the special case of the trivial kg module alperin and evens 3 and avrunin 4 have further studied the annihilator in ext k k of the cohomology of m and have produced theorems of a similar nature these are related to the work in 9 on the structure of the cohomology ring ext m m the purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module we concentrate on the case in which g is an elementary abelian p group however using such results as those mentioned above we can apply the present work to more general groups even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p subgroups we show here that in the case of an elementary abelian group much of this information can be found by looking at the restrictions to cyclic p subgroups of the group of units of kg some of the results in this paper were announced in s let g xl xn be an elementary abelian group of order p suppose that k is algebraically closed to any finitely generated kg module m we

a simplex based algorithm for the lexicographically extended linear maxmin problem abstract let f x cx c where c is a real m n matrix c a constant m vector and x a variable n vector g x denotes the vector with components f i x in order of ascending magnitude the problem is to maximize g x over x in the lexicographic sense where x is the set of all points such that ax b x 0 a a real r n matrix it is known that this problem can be reduced to a sequence of m maximum problems where the maxmin problems can be solved by the well known simplex method of linear optimization a prerequisite such a procedure is the knowledge of at least one common index per stage it is shown that these common indices can readily be obtained from the respective optimal simplex tableaux some simple tests help to obtain more than just one common index per stage

on solving linear programs with the ordered weighted averaging objective abstract the problem of aggregating multiple criteria to form overall objective functions is of considerable importance in many disciplines the most commonly used aggregation is based on the weighted sum the ordered weighted averaging owa aggregation introduced by yager uses the weights assigned to the ordered values i e to the worst value the second worst and so on rather than to the specific criteria this allows to model various aggregation preferences preserving simultaneously the impartiality neutrality with respect to the individual criteria in this paper we analyze solution procedures for linear programs with the owa objective functions two alternative linear programming formulations are introduced and their computational efficiency is analyzed

cohomological finite generation for finite supergroup schemes abstract in this paper we compute extension groups in the category of strict polynomial superfunctors and thereby exhibit certain universal extension classes for the general linear supergroup some of these classes restrict to the universal extension classes for the general linear group exhibited by friedlander and suslin while others arise from purely super phenomena we then use these extension classes to show that the cohomology ring of a finite supergroup scheme equivalently of a finite dimensional cocommutative hopf superalgebra over a field is a finitely generated algebra implications for the rational cohomology of the general linear supergroup are also discussed

equitable aggregations and multiple criteria analysis abstract in the past decade increasing interest in equity issues resulted in new methodologies in the area of operations research this paper deals with the concept of equitably efficient solutions to multiple criteria optimization problems multiple criteria optimization usually starts with an assumption that the criteria are incomparable however many applications arise from situations which present equitable criteria moreover some aggregations of criteria are often applied to select efficient solutions in multiple criteria analysis the latter enforces comparability of criteria possibly rescaled this paper presents aggregations which can be used to derive equitably efficient solutions to both linear and nonlinear multiple optimization problems an example with equitable solutions to a capital budgeting problem is analyzed in detail an equitable form of the reference point method is introduced and analyzed

on exponentiation and infinitesimal one parameter subgroups of reductive groups abstract let g be a reductive algebraic group over an algebraically closed field k of characteristic p 0 and assume p is good for g let p be a parabolic subgroup with unipotent radical u for r 1 denote by g a r the r th frobenius kernel of g a we prove that if the nilpotence class of u is less than p then any embedding of g a r in u lies inside a one parameter subgroup of u and there is a canonical way in which to choose such a subgroup applying this result we prove that if p is at least as big as the coxeter number of g then the cohomological variety of g r is homeomorphic to the variety of r tuples of commuting elements in n 1 g the p nilpotent cone of lie g

complexity periodicity and one parameter subgroups using the variety of infinitesimal one parameter subgroups introduced in 55 56 by suslin friedlander bendel we define a numerical invariant for representations of an infinitesimal group scheme g for an indecomposable g module m of complexity 1 this number which may also interpreted as the height of a vertex um g is related to the period of m in the context of the frobenius category of grt modules associated to a smooth reductive group g and a maximal torus t g our methods give control over the behavior of the heller operator of such modules as well as precise values for the periodicity of their restrictions to gr applications include the structure of stable auslander reiten components of grt modules as well as the distribution of baby verma modules

asymptotic prime power divisibility of binomial generalized binomial and multinomial coefficients this paper presents asymptotic formulas for the abundance of binomial generalized binomial multinomial and generalized multinomial coefficients having any given degree of prime power divisibility in the case of binomial coefficients for a fixed prime p we consider the number of x y with 0 xv y pn for which x y is divisible by pzf but not pzf l when zn is an integer and oa z 3 say by means of a classical theorem of kummer and the probabilistic theory of large deviations we show that this number is approximately p d 43 where d cx 3 sup d z oa z 3 and d is given by an explicit formula we also develop a p adic multifractal theory and show how d may be interpreted as a multifractal spectrum of divisibility dimensions we then prove that essentially the same results hold for a large class of the generalized binomial coefficients of knuth and wilf including the q binomial coefficients of gauss and the fibonomial coefficients of lucas and finally we extend our results to multinomial coefficients and generalized multinomial coefficients introduction the purpose of this paper is to present asymptotic formulas regarding the primepower divisibility of binomial coefficients and their generalizations that follow from a classical theorem of kummer and its generalizations specifically for a fixed prime p and large n we consider initially how many binomial coefficients x with 0 x y pfn are divisible by pzn but not pzn l when zn is an integer and ae z 3 say we will find that this number is approximately pnd a 8 where d a 3 sup d z a z and d is a function we will determine explicitly first we will derive the exact answer by combinatorial means for ordinary binomial coefficients and then starting afresh we will find the asymptotic result by means of the probabilistic theory of large deviations in yet a third approach we will show that the function d represents the p adic multifractal spectrum of divisibility dimensions then we will show that essentially the same results hold for a large class of generalized binomial coefficients as defined by knuth and wilf 32 by means of their received by the editors december 8 1995 and in revised form march 26 1996 1991 mathematics subject classification primary 05a16 secondary 11k16

selecting preferred solutions in the minimax approach to dynamic programming problems under flexible constraints abstract dynamic programming is a powerful approach to the optimization of sequential or multistage decision processes e g in planning or in system control in this paper we consider both theoretical and algorithmic issues in sequential decision processes under flexible constraints such processes must attain a given goal within some tolerance tolerances or preferences also apply to the values the decision variables may take or on the action chosen at each step such problems boil down to maximin optimization unfortunately this approach suffers from the so called drowning effect lack of discrimination and the optimality principle of dynamic programming is not always verified in this context we introduce a general framework for refined minimax optimization procedures in order to compare and select preferred alternatives this framework encompasses already introduced methods such as leximin and discrimin but it allows their extension to the comparison of vectors of unequal lengths we show that these refined comparisons restore compatibility with the optimality principle and that classical algorithms can be adapted to compute such preferred solutions by exploiting existing results on idempotent semirings

refining aggregation operator based orderings in multifactorial evaluation part i continuous scales aggregation operators are often needed when building preference relations in multicriteria decision making problems most existing approaches have limitations due to incomparability between decisions or ties due to the use of some aggregation operations that produce a ranking the natural way of overcoming the lack of discrimination power is to refine the obtained ranking we bring an overview of methods that enable aggregation based rankings to be refined generalizing concepts like discrimin max leximin max and lorentz orderings that refine such aggregation operations like the minimum the maximum and the sum

representation theoretic support spaces for finite group schemes we introduce the space p g of abelian p points of a finite group scheme over an algebraically closed field of characteristic p 0 we construct a homeomorphism g p g proj g from p g to the projectivization of the cohomology variety for any finite group g for an elementary abelian p group respectively an infinitesimal group scheme p g can be identified with the projectivization of the variety of cyclic shifted subgroups resp variety of 1 parameter subgroups for a finite dimensional g module m g restricts to a homeomorphism p g m proj g m thereby giving a representation theoretic interpretation of the cohomological support variety

balanced symmetric functions over hbox gf p under mild conditions on n p we give a lower bound on the number of n variable balanced symmetric polynomials over finite fields gf p where p is a prime number the existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound furthermore we prove that x 2t 2t 1lscr 1 are balanced and conjecture that these are the only balanced symmetric polynomials over gf 2 where x d n sigma1lesi 1 i 2 hellip i d lesnxi 1xi 2hellipxi d

on support varieties for lie superalgebras and finite supergroup schemes abstract we study the spectrum of the cohomology rings of cocommutative hopf superalgebras restricted and non restricted lie superalgebras and finite supergroup schemes we also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory we completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero while for non restricted lie superalgebras in positive characteristic we obtain descriptions that are suggestive of varieties previously defined by duflo and serganova in characteristic zero but which have no known cohomological interpretation our computations for restricted lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of friedlander and parshall and of bendel friedlander and suslin in the classical setting

representations of elementary abelian p groups and bundles on grassmannians abstract we initiate the study of representations of elementary abelian p groups via restrictions to truncated polynomial subalgebras of the group algebra generated by r nilpotent elements k t 1 t r t 1 p t r p we introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras we also introduce modules of constant radical and socle type generalizing modules of constant jordan type and provide several general constructions of modules with these properties we show that modules of constant radical and socle type lead to families of algebraic vector bundles on grassmannians and illustrate our theory with numerous examples

modules of constant jordan type we introduce the class of modules of constant jordan type for a finite group scheme g over a field k of characteristic p 0 this class is closed under taking direct sums tensor products duals heller shifts and direct summands and includes endotrivial modules it contains all modules in an auslander reiten component which has at least one module in the class highly non trivial examples are constructed using cohomological techniques we offer conjectures suggesting that there are strong conditions on a partition to be the jordan type associated to a module of constant jordan type

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

nilpotent commuting varieties of reductive lie algebras let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 and lie g in positive characteristic suppose in addition that p is good for g and the derived subgroup of g is simply connected let denote the nilpotent variety of and nil x y x y 0 the nilpotent commuting variety of our main goal in this paper is to show that the variety nil is equidimensional in characteristic 0 this confirms a conjecture of vladimir baranovsky see 2 when applied to gl n our result in conjunction with an observation in 2 shows that the punctual local hilbert scheme n hilb n 2 is irreducible over any algebraically closed field

general linear and functor cohomology over finite fields in recent years there has been considerable success in computing extgroups of modular representations associated to the general linear group by relating this problem to one of computing ext groups in functor categories f l s f s in this paper we extend our ability to make such ext group calculations by establishing several fundamental results throughout this paper we work over fields of positive characteristic p the reader familiar with the representation theory of algebraic objects will recognize the importance of an understanding of ext groups for example the existence of nonzero ext groups of positive degree is equivalent to the existence of objects which are not direct sums of simple objects indeed a knowledge of ext groups provides considerable knowledge of compound objects in the study of modular representation theory of finite chevalley groups such as gln fq ext groups play an even more central role it has been shown in cps that a knowledge of certain ext groups is sufficient to prove lusztig s conjecture concerning the dimension and characters of irreducible representations we consider two different categories of functors the category f fq of all functors from finite dimensional fq vector spaces to fq vector spaces where fq is the finite field of cardinality q and the category p fq of strict polynomial functors of finite degree as defined in f s the category p fq presents several advantages over the category f fq from the point of view of computing extgroups these are the accessibility of injectives and projectives the existence of a base change and an even easier access to ext groups of tensor products this explains the usefulness of our comparison in theorem 3 10 of ext groups in the category p fq with ext groups in the category f fq weaker forms of this theorem have been known to us since 1995 and to s betley independently

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

the variety of an indecomposable module is connected let g be a finite group and let k be an algebraically closed field of characteristic p 0 the connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest in particular cohomology theory has been used to attach to each finitely generated kgmodule m a homogeneous affine variety v m if we let 17 m be the corresponding projective variety then our main result is the following

quillen stratification for modules let g be a finite group and k a fixed algebraically closed field of characteristic p o if p is odd let h be the subring of h g k consisting of elements of even degree following 20 221 we take h h g k if p 2 though one could just as well use the subring of elements of even degree for all p h is a finitely generated commutative k algebra 13 and we let v denote its associated affine variety max h if m is any finitely generated kg module then the cohomology variety v m of m may be defined as the support in v of the h module h g m if g is a p group and in general as the largest support of h g l m where l is any kg module 4 91 a module l with each irreducible kc module as a direct summand will serve d quillen 20 221 proved a number of beautiful results relating k to the varieties v associated with the various elementary abelian p subgroups e of g culminating in his stratification theorem 20 221 this theorem gives a piecewise description of v almost explicitly in terms of the subgroups e and their normalizers in g a well known corollary is that dim v max dim v where e

on cocommutative hopf algebras of finite representation type abstract let g be a finite algebraic group defined over an algebraically closed field k of characteristic p 0 such a group decomposes into a semidirect product g g 0 g red with a constant group g red and a normal infinitesimal subgroup g 0 if the principal block b 0 g of the group algebra h g has finite representation type then both constituents have the same property with at least one of them being semisimple we determine the structure of the infinitesimal constituent g 0 up to the classification of v uniserial groups

on nilpotent commuting varieties and cohomology of frobenius kernels abstract the paper studies the dimensions of irreducible components of commuting varieties of restricted nilpotent r tuples in a classical lie algebra g defined over an algebraically closed field k as applications we obtain some new results on the structure of the even cohomology ring of frobenius kernels g r for each r 1 where g is the simply connected simple algebraic group such that lie g g explicit calculations for rank two groups are also presented

geometric invariants of representations of finite groups abstract for a finite group and a field k of characteristic p dividing the order of we construct a map p u p y of varieties over k with unions of affine spaces as fibers and with p y p isogenous to proj h k on u we construct a universal p nilpotent operator p which leads to the construction of equivariant coherent sheaves associated to finite dimensional k modules m these coherent sheaves are algebraic vector bundles on u if m has constant jordan type for any finite dimensional k module m these coherent sheaves are vector bundles when restricted to the complements of the generalized support varieties of m for a linear algebraic group g we compare these constructions for the finite group g f p to previous constructions for the infinitesimal group scheme g r our comparison of bundle invariants for a rational g module m of exponential type p r upon restriction to g f p and to g r is a strengthened form of earlier comparisons of support varieties for m our constructions extend to semi direct products g of an infinitesimal group scheme g and a finite group thus to all finite group schemes if k is algebraically closed

stratification and π cosupport finite groups we introduce the notion of pi cosupport as a new tool for the stable module category of a finite group scheme in the case of a finite group we use this to give a new proof of the classification of tensor ideal localising subcategories in a sequel to this paper we carry out the corresponding classification for finite group schemes

constructions for infinitesimal group schemes let g be an infinitesimal group scheme over a field k of char acteristic p 0 we introduce the universal p nilpotent operatorg 2 homk k g k v g where v g is the scheme which represents1 parameter subgroups of g this operatorg applied to m encodes the local jordan type of m and leads to computational insights into the representation theory of g for certain kg modules m including those of constant jordan type we employg to associate various algebraic vector bundles on p g the pro jectivization of v g these vector bundles not only distinguish certain rep resentations with the same local jordan type but also provide a method of constructing algebraic vector bundles on p g

nilpotent matrices having a given jordan type as maximum commuting nilpotent orbit abstract the jordan type of a nilpotent matrix is the partition giving the sizes of its jordan blocks we study pairs of partitions p q where q q p is the jordan type of a generic nilpotent matrix a commuting with a nilpotent matrix b of jordan type p t kosir and p oblak have shown that q has parts that differ pairwise by at least two such partitions which are also known as super distinct or rogers ramanujan are exactly those that are stable or self large in the sense that q q q in 2012 p oblak formulated a conjecture concerning the cardinality of q 1 q when q has two parts and proved some special cases r zhao refined this to posit that the partitions in q 1 q for q u u r with u r 1 could be arranged in an r 1 u r table t q where the entry in the k th row and l th column has k l parts we prove this table theorem and then generalize the statement to propose a box conjecture for the set of partitions q 1 q for an arbitrary partition q whose parts differ pairwise by at least two

erratum to representation theoretic support spaces for finite group schemes as pointed out to us by rolf farnsteiner the results presented in our paper ajm 127 pp 379 420 require a modified definition of abelian p point with this modified definition functionally equivalent to one which we implicitly use all of the results of our paper become valid we make explicit this modified definition as well as those arguments which require this new definition and the modification of one proof of theorem 4 8 which is required

filtrations 1 parameter subgroups and rational injectivity abstract we investigate rational g modules m for a linear algebraic group g over an algebraically closed field k of characteristic p 0 using filtrations by sub coalgebras of the coordinate algebra k g of g even in the special case of the additive group g a interesting structures and examples are revealed the degree filtration we consider for unipotent algebraic groups leads to a filtration by exponential degree applicable to rational g modules for any linear algebraic group g of exponential type this filtration is defined in terms of 1 parameter subgroups and is related to support varieties introduced recently by the author for such rational g modules we formulate in terms of this filtration a necessary and sufficient condition for rational injectivity for rational g modules our investigation leads to the consideration of two new classes of rational g modules those that are mock injective and those that are mock trivial

support varieties ar components and good filtrations let g be a reductive group defined over the galois field mathbb f p with p being good for g using support varieties and covering techniques based on grt modules we determine the position of simple modules and baby verma modules within the stable auslander reiten quiver s gr of the rth frobenius kernel of g in particular we show that the almost split sequences terminating in these modules usually have an indecomposable middle term concerning support varieties we introduce a reduction technique leading to isomorphisms mathcal v g r z r lambda cong mathcal v g r d z r d mu for baby verma modules of certain highest weights lambda mu in x t which are related by the notion of depth

derived categories of grassmannians over integers and modular representation theory abstract in this paper we study the derived categories of coherent sheaves on grassmannians gr k n defined over the ring of integers we prove that the category d b gr k n has a semi orthogonal decomposition with components being full subcategories of the derived category of representations of g l k this in particular implies existence of a full exceptional collection which is a refinement of kapranov s collection 13 which was constructed over a field of characteristic zero we also describe the right dual semi orthogonal decomposition which has a similar form and its components are full subcategories of the derived category of representations of g l n k the resulting equivalences between the components of the two decompositions are given by a version of koszul duality for strict polynomial functors we also construct a tilting vector bundle on gr k n we show that its endomorphism algebra has two natural structures of a split quasi hereditary algebra over z and we identify the objects of d b gr k n which correspond to the standard and costandard modules in both structures all the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the grassmannian by extension of scalars base change similar results over fields of arbitrary characteristic were obtained independently in 7 by different methods

weil restriction and support varieties this paper extends the comparison results for cohomological support varieties for chevalley groups of the form g fp established by j carlson z lin and d nakano to chevalley groups of the form g fpd other topics considered are complexity and non maximal support varieties for modules techniques developed by j pevtsova and the author together with the use of weil restriction functors are employed

the avrunin and scott theorem and a truncated polynomial algebra we prove an analogue of a theorem of avrunin and scott for truncated polynomial algebras m k x 1 x m x 2 i over an algebraically closed field of arbitrary characteristic the avrunin and scott theorem relates the support variety for a finite dimensional ke module to its rank variety where char k p and e is an elementary abelian p group the analogue of the avrunin and scott theorem relates the support variety for a finite dimensional m module using hochschild cohomology to its rank variety developed in k erdmann m holloway rank varieties and projectivity for a class of local algebras math z 247 2004 441 460 using clifford algebras along the way to proving our main result we provide a new proof of the avrunin and scott theorem for elementary abelian p group algebras which we are then able to generalise to the setting of m algebras

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

computing improved optimal solutions to max min flexible constraint satisfaction problems abstract the formal framework for decision making in a fuzzy environment is based on a general max min bottleneck like optimization problem proposed by zadeh it is also the basis for extending the constraint satisfaction paradigm of artificial intelligence to accommodating flexible or prioritized constraints this paper surveys refinements of the ordering of solutions supplied by the max min formulation namely the discrimin partial ordering and the leximin complete preordering a general algorithm is given which computes all maximal solutions in the sense of these relations it also sheds light on the structure of the set of best solutions moreover classes of problems for which there is a unique best discrimin and leximin solution are exhibited namely continuous problems with convex domains and so called isotonic problems noticeable examples of such problems are fuzzy linear programming problems and fuzzy pert like scheduling problems

cohomology of classical algebraic groups from the functorial viewpoint abstract we prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups this result was previously known only for general linear groups we give several applications to the study of classical algebraic groups such as a cohomological stabilization property the injectivity of external cup products and the existence of hopf algebra structures on the stable cohomology of a classical algebraic group with coefficients in a hopf algebra our result also opens the way to new explicit cohomology computations we give an example inspired by recent computations of djament and vespa

generic and maximal jordan types for a finite group scheme g over a field k of characteristic p 0 we associate new invariants to a finite dimensional kg module m namely for each generic point of the projectivized cohomological variety text proj mathop h bullet g k we exhibit a generic jordan type of m in the very special case in which g e is an elementary abelian p group our construction specializes to the non trivial observation that the jordan type obtained by restricting m via a generic cyclic shifted subgroup does not depend upon a choice of generators for e furthermore we construct the non maximal support variety g m a closed subset of text proj mathop h bullet g k which is proper even when the dimension of m is not divisible by p

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

refinements of the maximin approach to decision making in a fuzzy environment abstract the most popular approach to decision making in the setting of fuzzy sets is the maximin ranking of solutions this method is natural when interpreting the fuzzy sets as flexible constraints that cannot compensate with one another however the obtained ranking of solutions is very coarse two kinds of refinements to this ordering are introduced a partial ordering according to the least satisfied discriminating constraint and a lexicographical ranking the latter refines the former and combines utilitarist and egalitarist points of view on the aggregation of feasibility degrees these orderings are characterized in several ways and their representation by means of two place numerical functions is studied dual refinements of the maximax ranking are provided

generalizing leximin to t norms and t conorms the lexit and lexis orderings the leximin procedure introduced by dubois and prade in 1996 provides a methodology for adjudicating ties in aggregated values when using the min t norm here with the introduction of the lexit procedure we generalize these ideas to provide a tool for adjudicating ties when aggregating data using any t norm we then provide the lexis procedure for adjudicating ties when aggregating arguments using t conorms

lagrangian submanifolds in para kahler manifolds abstract in this paper we initiate the study of lagrangian submanifolds in para kahler manifolds in particular we prove two general optimal inequalities for lagrangian submanifolds of the flat para kahler manifold e 2 n g p moreover we completely classify lagrangian submanifolds which satisfy the equality case of one of the two inequalities

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

complexes and vanishing of cohomology for group schemes abstract completely reducible homology complexes were originally introduced in the context of finite groups and used to study the question of vanishing of cohomology in this paper we study these complexes and the vanishing of cohomology for arbitrary finite dimensional cocommutative hopf algebras applications are later provided for infinitesimal group schemes of reductive and solvable algebraic groups

modules with small loewy length a module of complexity c for e z p r in characteristic p has loewy length at least p 1 r c 1 we study the case of equality if p is odd the only rank varieties possible are finite unions of linear subspaces of dimension c and every such rank variety occurs if p 2 the variety has to be equidimensional if such a variety is a finite union of set theoretic complete intersections then it occurs for such a module but otherwise the situation is unclear exterior algebras in any characteristic are also treated and follow the same behaviour as the case p 2 above

support varieties of line bundle cohomology groups for sl3 k abstract let g sl 3 k where k is a field of characteristic p 0 and let x t be any weight with corresponding line bundle l on g b in this paper we compute the support varieties for all modules of the form h i h i g b l over the first frobenius kernel g 1 the calculation involves certain recursive character formulas given by donkin cf 5 which can be used to compute the characters of the line bundle cohomology groups in the case where is a p regular weight and m h i 0 for some i these formulas are used to show that any pth root of unity is not a root of the generic dimension of m to handle the case where is not p regular we employ techniques similar to those used by drupieski nakano and parshall cf 7 to show that the module h i is not projective over g 1 whenever it is nonzero and lies outside of the steinberg block

varieties of modules for z 2z z 2z let k be an algebraically closed field of characteristic 2 we prove that the restricted nilpotent commuting variety c that is the set of pairs of n n matrices a b such that a2 b2 a b 0 is equidimensional c can be identified with the variety of n dimensional modules for z 2z z 2z or equivalently for k x y x2 y2 on the other hand we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic 2 we also prove that if e2 0 then the set of elements of the centralizer of e whose square is zero is equidimensional finally we express each irreducible component of c as a direct sum of indecomposable components of varieties of z 2z z 2z modules

bifunctor cohomology and cohomological finite generation for reductive groups let g be a reductive linear algebraic group over a field k let a be a finitely generated commutative k algebra on which g acts rationally by k algebra automorphisms invariant theory states that the ring of invariants ag h0 g a is finitely generated we show that in fact the full cohomology ring h g a is finitely generated the proof is based on the strict polynomial bifunctor cohomology classes constructed in 22 we also continue the study of bifunctor cohomology of gl 1

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

cohomology and support varieties for lie superalgebras unlike lie algebras the finite dimensional complex representations of a simple lie superalgebra are usually not semisimple as a consequence despite over thirty years of study these remain mysterious objects in this paper we introduce a new tool the notion of cohomological support varieties for the finite dimensional supermodules for a classical lie superalgebra g g 0 g 1 which are completely reducible over g 0 they allow us to provide a new functorial description of the previously combinatorial notions of defect and atypicality we also introduce the detecting subalgebra of g its role is analogous to the defect subgroup in the theory of finite groups in positive characteristic using invariant theory we prove that there are close connections between the cohomology and support varieties of g and the detecting subalgebra

computations of sheaves associated to the representation theory of sl2 abstract we explicitly compute examples of sheaves over the projectivization of the spectrum of the cohomology of sl 2 in particular we compute ker m for every indecomposable m and we compute f i m when m is an indecomposable weyl module and i p we also give a brief review of the classification of sl 2 modules and of the general theory of such sheaves in the case of a restricted lie algebra

classification of hamiltonian stationary lagrangian submanifolds of constant curvature in cp3 with positive relative nullity abstract a lagrangian submanifold in a kaehler manifold is said to be hamiltonian stationary if it is a critical point of the area functional restricted to compactly supported hamiltonian variations in this paper we classify hamiltonian stationary lagrangian submanifolds of constant curvature in cp 3 with positive relative nullity as an immediate by product several explicit new families of hamiltonian stationary lagrangian submanifolds in cp 3 are obtained

leximin optimality and fuzzy set theoretic operations abstract the leximin ranking of vectors of values taken from a totally ordered set is sometimes encountered in fields like operational research social choice or numerical analysis but has seldom been studied in connexion with fuzzy optimization in this paper we prove that a leximin optimal solution to a vector ranking problem on the unit hypercube can be obtained as the limit of optimal solutions to a problem of fuzzy multiple criteria optimization where fuzzy sets are aggregated either using a triangular norm or a generalized mean or an ordered weighted average owa operation

the galois group of a stable homotopy theory abstract to a stable homotopy theory a presentable symmetric monoidal stable category we naturally associate a category of finite etale algebra objects and using grothendieck s categorical machine a profinite group that we call the galois group we then calculate the galois groups in several examples for instance we show that the galois group of the periodic e algebra of topological modular forms is trivial and that the galois group of k n local stable homotopy theory is an extended version of the morava stabilizer group we also describe the galois group of the stable module category of a finite group a fundamental idea throughout is the purely categorical notion of a descendable algebra object and an associated analog of faithfully flat descent in this context

pi supports for modules for finite group schemes we introduce the space g of equivalence classes of points of a finite group scheme g and associate a subspace g m to any g module m our results extend to arbitrary finite group schemes g over arbitrary fields k of positive characteristic and to arbitrarily large g modules the basic results about cohomological support varieties and their interpretation in terms of representation theory in particular we prove that the projectivity of any possibly infinite dimensional g module can be detected by its restriction along points of g unlike the cohomological support variety of a g module m the invariant m 7 g m satisfies good properties for all modules thereby enabling us to determine the thick tensor ideal subcategories of the stable module category of finite dimensional g modules finally using the stable module category of g we provide g with the structure of a ringed space which we show to be isomorphic to the scheme projh g k

fuzzy set approach to the utility preference relations and aggregation operators abstract score x x 1 x n describing an alternative is modelled by means of a continuous quasi convex fuzzy quantity x thus allowing to compare alternatives scores by means of fuzzy ordering comparison methods applying some defuzzification method leads to the introduction of operators acting on scores a special stress is put on the mean of maxima defuzzification method allowing to introduce several averaging aggregation operators moreover our approach allows to introduce weights into above mentioned aggregation even in the non anonymous non symmetric case finally ordered weighted aggregation operators owao are introduced generalizing the standard owa operators

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

elementary subalgebras of lie algebras abstract we initiate the investigation of the projective varieties e r g of elementary subalgebras of dimension r of a p restricted lie algebra g for various r 1 these varieties e r g are the natural ambient varieties for generalized support varieties for restricted representations of g we identify these varieties in special cases revealing their interesting and varied geometric structures we also introduce invariants for a finite dimensional u g module m the local r j radical rank and local r j socle rank functions which are lower upper semicontinuous on e r g examples are given of u g modules for which some of these rank functions are constant