a mackey imprimitivity theory for algebraic groups let g be an affine algebraic group over an algebraically closed field k and let h be a closed subgroup of g if v is a rational h module a comodule for the coordinate ring of h there is a now well known notion of an induced module vi g for g defined as the space morph g v of all h equivariant morphisms from g to a finite dimensional subspace of v with obvious g action the question arises given a rational g module m how can one recognize m as an induced module vig for a finite group g the answer is the mackey imprimitivity theorem the module m is induced if and only if it is a direct sum of subspaces permuted transitively by g with h the stabilizer of one of these subspaces called v one uses this result for example in proving the famous mackey decomposition theorem which describes the restriction of any induced module to a second subgroup l as a direct sum of suitable induced modules given the imprimitivity theorem the proof is just a matter of grouping the summands permuted by g into their orbits under l in the case of algebraic groups the situation is quite different for some subgroups h all g modules are induced this occurs for example if g is connected and k g h k e g if h is parabolic also if g is a connected unipotent group then a rational g module m is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule e for the conjugation action of g with e k 1 and such that k l is precisely the subspace annihilated by the action of the lie algebra of g on e 27 cf also 5 5 below the latter is an application of a general criterion in case g h is affine a rational g module m is induced if and only if there is an action of the coordinate ring a of g h compatible with the action of g on both a and m note that this generalizes the imprimitivity theorem in the case of finite groups since then a has a k basis of g h orthogonal idempotents permuted transitively by g

coring stabilizers for a hopf algebra coaction abstract the coring stabilizer stab p is introduced for any prime ideal p of a right h comodule algebra a such that the factor ring a p is either right or left goldie this notion is used to obtain hopf algebraic analogs of two category equivalences associated with a homogeneous space the category of linearized quasicoherent sheaves on a noncommutative homogeneous space is interpreted as a suitable quotient category of the category of hopf modules birational invariance of such quotient categories is proved it is shown that for a birational h coequivariant extension b of a properly defined subsets of prime ideals of a and b correspond to each other bijectively

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

etching a two dimensional mathematical approach a mathematical model for the etching of a semi infinite active surface is presented the model assumes that the transport of the active species occurs solely by diffusion it is shown that when the diffusion field propagates much faster than the etched surface the problem can be solved by a singular perturbation technique which distinguishes a near field in the area where the moving surface and the non etchable mask meet and a far field where edge effects may be disregarded to first order the leading terms of a composite expansion are given from which the shape of the moving boundary can be determined at all times

numerical solution of an etching problem abstract after a short description of the chemical background of etching processes a simplified mathematical model is formulated this contains two parameters that determine the etching profile two approaches for obtaining a numerical solution are discussed first the problem in terms of a variational inequality is reformulated and some new results on its numerical solution are presented then how the problem can be solved by means of a moving grid method as well is explored results obtained by both methods are presented and compared the latter method is applicable to a much wider class of boundary conditions but the variational inequality approach seems attractive for other reasons finally this work is compared with some experimental results

on injective modules for infinitesimal algebraic groups ii let g be a semisimple algebraic group scheme defined and split over the prime subfield of an algebraically closed field k of characteristic p 0 if f g g is the frobenius morphism and t is a maximal split torus in g then tg is the inverse image of t under f thus tg is a non reduced subgroup scheme of g let b be a bore1 subgroup of gz containing 7 whose opposite b determines the positive system of roots in the root system of g relative to t let tb b xc tg b n tg the scheme theoretic intersection of b with 7 g it is well known j2 sect 3 1 that injective tg mods as a factor of q p coincides with the multiplicity of l p as a composition factor in lnd i at the same time if i is generic relative to an alcove in the sense defined in cps 1 appendix a the multiplicities of l p in tndyi i coincide with the multiplicities inde i l p where l f denotes the rational irreducible g module of high weight i relative to b thus in the generic case one can translate the problem of computing multiplicities of irreducibles in indg l to 2 question about the structure of rational injectivc tg modules we begin here a study of the structure of rational injectivc 7 modules from the point of view of the parabolic subgroup schemes of 7 g in this we proceed by analogy with the early work h of harish chandra sp of springer and with the more recent work rs of ronan and smith if p is a parabolic subgroup scheme of g containing t let tp p xc tg p n tg then tp has a levi factorization tp tl x lip where c denotes the unipotcnt radical of tp and tl its levi factor

a comparison of numerical models for one dimensional stefan problems in this paper we present a critical comparison of the suitability of several numerical methods level set moving grid and phase field model to address two well known stefan problems in phase transformation studies melting of a pure phase and diffusional solid state phase transformations in a binary system similarity solutions are applied to verify the numerical results the comparison shows that the type of phase transformation considered determines the convenience of the numerical techniques finally it is shown both numerically and analytically that the solid solid phase transformation is a limiting case of the solid liquid transformation

asymptotic and numerical solutions for the two dimensional solidification of a liquid half space this paper concerns the twodimensional inward solidification of a halfspace filled with molten material at its fusion temperature the temperature along half of the boundary surface is instantaneou

characteristic free representation theory of the general linear group on etudie systematiquement les z formes des representations rationnelles du groupe lineaire generale

the ariki terasoma yamada tensor space and the blob algebra we show that the ariki terasoma yamada tensor module and its permutation submodules m are modules for the blob algebra when the ariki koike algebra is a hecke algebra of type b we show that m and the standard modules have the same dimensions the same localization and similar restriction properties and are equal in the grothendieck group still we find that the universal property for fails for m making m and different modules in general finally we prove that m is isomorphic to the dual specht module for the ariki koike algebra

on quantum spin chain spectra and the structure of hecke algebras and q groups at roots of unity we combine statistical mechanical and algebraic lie theory techniques to investigate the ordinary representation theory of the quotients of the hecke algebras when q is a root of unity we show how to determine the main invariants of these algebras the standard module contents of the indecomposable projective modules in general n we give complete explicit results in the cases n 2 3 these results are used to determine the q variation energy level convergences in invariant quantum spin chains

etching through a slit the problem of etching through a slit shaped orifice is formulated in mathematical terms although a full solution to this problem can be obtained only by the application of numerical methods it is shown that certain aspects of it can be dealt with by analytical means asymptotic solutions that are valid in certain parts of the time domain are derived early intermediate and late asymptotes are considered these analytical results are compared with numerical ones and a satisfactory agreement is found emphasizing the validity of the two approaches

another characterization of parabolic subgroups let g be a reductive algebraic group defined over an algebraically closed field k let h be a closed connected subgroup of g containing a maximal torus t of g in 13 it was shown at least in characteristic zero that the parabolic subgroups of g can be characterized among all such subgroups h by a certain finiteness property of the induction functor iz and its derived functors lk g this theme is continued in the present paper where it is shown that the parabolic subgroups can be characterized by yet another familiar property of the induction functor at least in characteristic zero we also obtain several results which are independent of the characteristic by added hypotheses on h or by using the restriction functor

representations of hopf algebras arising fromlie algebras of cartan type c w x 1 1 this is a continuation of 13 in an effort to use algebraic group actions to study the representations of restricted lie algebras of cartan type in this paper we study the representation theory of certain finite diw x mensional hopf algebras that arise from the construction in 13 the representation theory of these hopf algebras closely resembles the representation theory of frobenius kernels of reductive algebraic groups in positive characteristics let g be a connected reductive algebraic group scheme over an algebraically closed field k of characteristic p 0 it is well known that if f g a g is the r th iteration of the frobenius map and g s ker f r r there is a chain of normal subgroup schemes g eg e eg in earlier 1 2 terminology this is equivalent to the existence of a chain of finite dimen sional cocommutative hopf algebras dist g dist g 1 2 dist g where dist g is known as the r th hyperalgebra of the distribur tion algebra dist g of g let g be the lie algebra of g and u g be its restricted universal enveloping algebra there exists an isomorphism of

asymptotic solutions to the stefan problem with a constant heat source at the moving boundary in 1989 huppert formulated mathematical models of the melting and or freezing that occurs when a hot fluid begins to flow turbulently over a cold substrate he solved the resulting stefan problems numerically and presented certain asymptotic results this note revisits one of his problems and adds to the mathematical analysis of its asymptotic structure generalizations and relationships with other results are also noted

a new semi analytical method for phase transformations in binary alloys in the present paper a new semi analytical method is developed to cover a wide range of phase transformation problems and their practical applications the solution procedure consists of two parts first determination of the position of the moving boundary named the homogenous part and second determination of the concentration named the non homogenous part the homogenous part leads to a system of homogenous linear equations based on the mathematical fact that a homogenous system has a non trivial solution if the determinant of the coefficient matrix equals zero this determinant leads to an ordinary differential equation for the moving boundary and its solution leads to a closed form formula for the position of the moving boundary the non homogenous part transforms the governing equations to a non homogenous linear system of equations having three unknowns that appear in the concentration profile assumed in the beginning of the proposed method solution of the non homogenous system leads to a value of these unknowns once these unknowns are computed the concentration at any time and at any point can be found easily

from submodule categories to the stable auslander algebra abstract we construct two functors from the submodule category of a representation finite self injective algebra to the module category of the stable auslander algebra of these functors factor through the module category of the auslander algebra of moreover they induce equivalences from the quotient categories of the submodule category modulo their respective kernels and said kernels have finitely many indecomposable objects up to isomorphism their construction uses a recollement of the module category of the auslander algebra induced by an idempotent and this recollement determines a characteristic tilting and cotilting module if is taken to be a nakayama algebra then said tilting and cotilting module is a characteristic tilting module of a quasi hereditary structure on the auslander algebra we prove that the self injective nakayama algebras are the only algebras with this property

a comment on on prime factor mapping for the discrete hartley transform lun and sin see ieee trans signal processing vol 40 no 6 p 1399 1411 1982 describe a method of computing prime factor decomposed discrete hartley transform dht over n points with 2n less additions compared to the existing algorithms of bracewell and sorensen 1976 the author shows how the method of lun and siu ran be extended to eliminate the 2n extra additions in the algorithms of bracewell and sorenson

analytical and numerical solutions to two dimensional moving interface problems with applications to the solidification of killed steel ingots this paper is concerned with the development of methods for investigating moving interface problems that involve more than one space variable the work was motivated by problems arising from the british steel corporation scunthorpe concerning the solidification of killed steel ingots first we investigate certain two dimensional analytical solutions which simulate particular aspects of ingot solidification these relate to the inward solidification of a liquid half space either when the wall temperature varies with position along the wall or when newton cooling takes place at the wall with the local heat transfer coefficient a function of wall position other approximate analytical approaches with use of the heat balance integral method and formulation in terms of an integro differential equation are discussed briefly and their limited application to two dimensional problems noted a computer procedure is then developed for solving these half space problems and this yields results in exact agreement with the analytical solutions in the work special attention is given to obtaining reliable information for small stefan number beta l c p t f t 0 since this relates to values found in the steel making process in the major part of the paper a two dimensional mathematical model of ingot solidification is developed numerical solutions are given for an ingot when the mould surface varies in geometry and in composition good agreement is obtained with radio isotope tracer tests performed at the british steel corporation scunthorpe

realisation functors in tilting theory derived equivalences and t structures are closely related we use realisation functors associated to t structures in triangulated categories to establish a derived morita theory for abelian categories with a projective generator or an injective cogenerator for this purpose we develop a theory of non compact or large tilting and cotilting objects that generalises the preceding notions in the literature within the scope of derived morita theory for rings we show that under some assumptions the realisation functor is a derived tensor product this fact allows us to approach a problem by rickard on the shape of derived equivalences finally we apply the techniques of this new derived morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t structures glueing to a tilting or a cotilting t structure as a further application we answer a question by xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras

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

on computing the discrete hartley transform the discrete hartley transform dht is a real valued transform closely related to the dft of a real valued sequence bracewell has recently demonstrated a radix 2 decimation in time fast hartley transform fht algorithm in this paper a complete set of fast algorithms for computing the dht is developed including decimation in frequency radix 4 split radix prime factor and winograd transform algorithms the philosophies of all common fft algorithms are shown to be equally applicable to the computation of the dht and the fht algorithms closely resemble their fft counterparts the operation counts for the fht algorithms are determined and compared to the counts for corresponding real valued fft algorithms the fht algorithms are shown to always require the same number of multiplications the same storage and a few more additions than the real valued fft algorithms even though computation of the fht takes more operations in some situations the inherently real valued nature of the discrete hartley transform may justify this extra cost

a conserving discretization for the free boundary in a two dimensional stefan problem the dissolution of a disk likeal2cuparticle is considered a characteristic property is that initially the particle has a nonsmooth boundary the mathematical model of this dissolution process contains a description of the particle interface of which the position varies in time such a model is called a stefan problem it is impossible to obtain an analytical solution for a general two dimensional stefan problem so we use the finite element method to solve this problem numerically first we apply a classical moving mesh method computations show that after some time steps the predicted particle interface becomes very unrealistic therefore we derive a new method for the displacement of the free boundary based on the balance of atoms this method leads to good results also for nonsmooth boundaries some numerical experiments are given for the dissolution of anal2cuparticle in anal cualloy

a cohomological characterization of parabolic subgroups of reductive algebraic groups let g be an algebraic group defined over an algebraically closed field k and rat g the category of rational g modules for any closed subgroup h of g and any rational h module v let v ig denote the rational g module induced from i the induction functor ig rat h rat g is left exact and we denote its derived functors by l j for n o 1 2 of course these derived functors are 0 for n 2 1 whenever 1 g is exact and by a result of cline parshall and scott 3 this happens precisely when g h is an afftne variety so the case of an afftne quotient is characterized by the vanishing of the higher derived functors in a like vein in this paper we will attempt to find a property of these functors which characterizes the case of a projective quotient since by definition g h is projective iff h p is a parabolic subgroup of g we are in effect looking for a cohomological characterization of parabolic subgroups of g the property of the functors l j we consider is a certain finiteness condition explained below the author was originally led to consider this question in 14 where induction from certain nonparabolic subgroups is studied and is shown to have this finiteness property it is well known that for every rational h module v there is an induced bundle l on g h with rank equal to dim i such that l j v identifies naturally with the sheaf cohomology group h g h l see 4 for details thus we may apply standard results from algebraic geometry to obtain some basic facts for example lg g 0 for n dim g h and serre s theorem gives the easy half of the result of cline parshall and scott men

models of quasiprojective homogeneous spaces for hopf algebras given an affine group scheme g of finite type over a field k a homogeneous space for g is a scheme x over k containing a rational point x such that g operates transitively on x assuming that g operates on the right we may identify x with the quotient k g of g by the left action of the stabilizer k of x in g the representation theoretic significance ofk g is that the induction functor indk from k modules to g modules factors as a category equivalence

on prime factor mapping for the discrete hartley transform the authors propose a new prime factor mapping scheme which requires no extra arithmetic operations for the realization of prime factor mapping for the computation of the discrete hartley transform dht it is achieved by embedding all the extra arithmetic operations into the subsequent short length computations with the computational complexities of these embedded short lengths remaining unchanged consequently the present approach significantly eliminates the burden which is introduced by the extra arithmetic operations with this mapping scheme it is further demonstrated that a prime factor mapped dht would have superb performance compared with other fast dht algorithms

stratifying endomorphism algebras associated to hecke algebras abstract let g be a finite group of lie type and let k be a field of characteristic distinct from the defining characteristic of g in studying the non describing representation theory of g the endomorphism algebra s g k end kg j ind g p j k plays an increasingly important role in type a by work of dipper and james s g k identifies with a q schur algebra and so serves as a link between the representation theories of the finite general linear groups and certain quantum groups this paper presents the first systematic study of the structure and homological algebra of these algebras for g of arbitrary type because s g k has a reinterpretation as a hecke endomorphism algebra it may be analyzed using the theory of hecke algebras its structure turns out to involve new applications of kazhdan lusztig cell theory in the course of this work we prove two stratification conjectures about coxeter group representations made by e cline b parshall and l scott mem amer math soc 591 1996 and we formulate a new conjecture about the structure of s g k we verify this conjecture here in all rank 2 examples

a direct method for solving two dimensional one phase stefan problems abstract a new algorithm is proposed for solving the stefan problem for the case when the initial region occupied by the medium in a given phase is a starshaped two dimensional domain the evolution of this region as time increases is determined by plotting the free boundary directly at each time step this is approximated by a polygon in the interior of which the heat equation is solved with piecewise linear finite elements numerical experiments indicate that the method is stable and that convergence is to be expected under suitable assumptions on the regularity of both the boundary and initial data of the problem

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

projectivity and freeness over comodule algebras let h be a hopf algebra and a an h simple right h comodule algebra it is shown that under certain hypotheses every h a hopf module is either projective or free as an a module and a is either a quasi frobenius or a semisimple ring as an application it is proved that every weakly finite in particular every finite dimensional hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras and the latter are frobenius algebras similar results are obtained for h simple h module algebras

deformations of lie algebra representations let g be a reductive algebraic group defined over an algebraically closed field k of positive characteristic p in this paper we continue our previous investigation 7 of the representation theory of the lie algebra f of g a key ingredient in our earlier work involved the consideration of a family a x s of finite dimensional algebras parameterized by the set 0 qs 1homka1g y b k of characters on a certain central subalgebra 0 6 of the universal enveloping algebra u 6 of g each irreducible module is naturally an irreducible module for exactly one such ax with the special case x 0 corresponding to restricted modules we introduce a so called deformation algebra s4 intermediate between the universal enveloping algebra u and certain of the algebras ax here sl is a quotient of u q such that each ax containing the restricted enveloping algebra v a of a fixed borel subalgebra 2a is in turn a quotient of s4 given a module mx for such an a we introduce the important notion of a deformation family of mx in definition 1 3 the hypothesis that the behavior of mx is related to that of the corresponding modules for near by algebras at motivates the present study a basic result 7 theorem 3 2 reduces the representation theory of ax for x 0 to that of an associated a for a subalgebra of to which the restriction t of x is nilpotent in the sense of 121 thus a study of the representation theory of the algebras ax for x a nilpotent character subsumes the classical problem of determining the irreducible representations of the lie algebra each g conjugacy class in the nullcone x of nilpotent elements of 0 y f defines a family of isomorphic algebras ax presumably the restricted enveloping algebra v ao corresponding to the conjugacy class o is the most complicated algebra of this kind its representation theory has been extensively investigated in recent years because of its intimate connections

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

gorenstein defect categories of triangular matrix algebras abstract we apply the technique of recollement to study the gorenstein defect categories of triangular matrix algebras first we construct a left recollement of gorenstein defect categories for a triangular matrix algebra under some conditions using it we give a categorical interpretation of the gorenstein properties of the triangular matrix algebras obtained by x w chen b l xiong and p zhang respectively second under some additional conditions a recollement of gorenstein defect categories for a triangular matrix algebra is constructed as an application for a special kind of triangular matrix algebras which are called simple gluing algebras we describe their singularity categories and gorenstein defect categories

modular representation theory of lie algebras on etudie les representations non necessairement restreintes de l algebre universelle enveloppante u 7b g pour une algebre de lie restreinte 7b g

on the generic structure of cohomology modules for semisimple algebraic groups let c be a connected semisimple algebraic group over a field of positive characteristic denote by b a borel subgroup our main result says that generically the cohomology modules for line bundles on g b have simple socles and simple heads and wc identify the corresponding highest weights as one of the conse quences we discover a certain symmetry among extensions of simple modules for g in this paper we continue the study of the 1nodule structure of the cohomology groups h x h g b t x for g a sesimple algebraic group over a field of positive characteristic p b a borel subgroup of g and 2 x the line bundle on g b induced by the character x of b for earlier work on various aspects of this question see e g 1 4 7 9 11 13 here we will consider the case where x is generic i e lies far away from the walls of the weyl chamber containing it see 2 1 for the precise condition in this case h x is nonzero in exactly one degree our main result theorem 2 2 says that this nonzero cohomology module has simple socle and simple head for x dominant it is a classical result that h x has simple socle the first nondominant case was treated in 2 where we proved that h1 x also always has a simple socle it should likewise be mentioned that j c jantzen has proved that weyl modules which may be identified with hn k n dimg b in the generic case have simple socles 12 6 3 and that j e humphreys in 10 has formulated conjectures which suggest that for x generic h x has the same module structure as a weyl module recently doty and sullivan have announced results which include special cases of our result see their forthcoming paper 8 to formulate some of the consequences of our main theorem we need a little more notation we set w equal to the weyl group of g relative to a maximal torus t in b for w e w we denote by l w the length of w and if x is a generic dominant weight then we let l xw be the simple g module which by our theorem occurs as

irreducible modules and parabolic subgroups summary it seems desirable from the point of view of finite group theorists to develop further the representation theory of finite chevalley groups in the characteristic of their definition say p by focusing attention on the full array of p local subgroups that is the parabolics rather than the usual single borel or cartan subgroup the following observation may be regarded as a generalization to arbitrary parabolic subgroups of the standard result 3 theorem 39 d for a borel subgroup namely that in an irreducible module the subspace fixed by a maximal unipotent subgroup is 1 dimensional and so affords an irreducible module for a levi complement which is an abelian cartan subgroup it seems natural to state the result first for algebraic groups

strong exact borel subalgebras of quasi hereditary algebras and abstract kazhdan lusztig theory abstract strong exact borel subalgebras and strong subalgebras are shown to exist for quasi hereditary algebras which possess exact borel subalgebras and subalgebras this implies that the algebras associated with blocks of category o have strong exact borel subalgebras and strong subalgebras the structure of these subalgebras is shown to be closely related to abstract kazhdan lusztig theory the main technical tool in this paper is a construction which has an exact borel subalgebras of a given quasi hereditary algebra as input and a strong exact borel subalgebra as output from this result morita invariance of the existence of exact borel subalgebras is derived

on root categories of finite dimensional algebras abstract for any finite dimensional algebra a over a field k of finite global dimension we investigate the root category r a as the triangulated hull of the 2 periodic orbit category of a via the construction of b keller in on triangulated orbit categories this is motivated by ringel hall lie algebras associated to 2 periodic triangulated categories as an application we study the ringel hall lie algebras for a class of finite dimensional k algebras of global dimension 2 which turns out to give an alternative answer to a question of gim lie algebras by slodowy in beyond kac moody algebra and inside

permutation complexes and modular representation theory hc g o iti has two basic functors a rg 2 di rh and a rh 2 rg given by restriction and induction which play an essential role in representation theory an important and elementary class of rg representations is permutation modules which are direct sums of modules indz r obtained by induction from the trivial rh module r for various hs g in another extreme one has rg modules which arise by induction from rh projective modules leading to the concept of relative projectivity and green s theory of vertices and sources cr gr the value of these subcategories of modules in representation theory and related areas is well known in a different direction influenced by algebraic geometry and topology one considers not only module categories but various categories of chain complexes of modules and their cohomologies this culminates in the more recent approaches to representation theory through the theory of derived categories see sc cps and their many references