characteristic free decomposition of skew schur functors this note is a generalization of our thesis brandeis university 1984 whose material can be found in 2 our aim is to show that by the methods presented there one can produce a characteristic free filtration of the skew schur functor l f whose associated graded module is the direct sum of schur modules l f one would expect from the characteristic zero theory as in 2 notations and facts are freely borrowed from l

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

on relations on minors of generic symmetric matrices let r be a commutative ring with 1 and fix an integer n b 1 suppose x are variables with 1 d i j n we denote by s r x the polynomial ring over r with n n 1 2 variables x assume x x when 1 d i j n then x is the generic n x n symmetric matrix with entries in s for a positive integer p such that 1 dp n let j be the ideal of s defined by all p minors of x the rings s j have been the classical objects of intensive study for instance by the first and the second fundamental theorems 2 111 it is well known that when an orthogonal group acts on a certain polynomial ring in a certain way the ring of invariants is described in the form of s j furthermore it has been proved that when r is cohen macaulay s j is also cohen macaulay and the equality depth j proj dim s j n p 1 n p 2 2 holds 7 in s jbzetiak pragacz and weyman constructed the minimal free resolution of s j for any n and p when r contains the rational number field the minimal free resolution means the resolution such that its entries of boundary maps do not have constant terms but over an arbitrary commutative ring r minimal free resolutions have not been obtained in general in the cases where p 1 and p n we can construct the minimal free resolutions of s j from the koszul complex further when p n 1 minimal free resolutions of s j were explicitly constructed in 3 41 in this article we prove that the relation module of minors of x is generated by degree 0 and 1 relations on minors and explicitly describe those relations moreover by using this result we prove the existence of minimal free resolutions of s j when p n 2 the basic idea is the same as that of 6 but we use plethysm formulas in place of cauchy s formula used in 6 388 0021 8693 89 3 00

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

geometric characterizations of the representation type of hereditary algebras and of canonical algebras abstract we show that a finite connected quiver q with no oriented cycles is tame if and only if for each dimension vector d and each integral weight of q the moduli space m q d s s of semi stable d dimensional representations of q is just a projective space in order to prove this we show that the tame quivers are precisely those whose weight spaces of semi invariants satisfy a certain log concavity property furthermore we characterize the tame quivers as being those quivers q with the property that for each schur root d of q the field of rational invariants k rep q d gl d is isomorphic to k or k t next we extend this latter description to canonical algebras more precisely we show that a canonical algebra is tame if and only if for each generic root d of and each indecomposable irreducible component c of rep d the field of rational invariants k c gl d is isomorphic to k or k t along the way we establish a general reduction technique for studying fields of rational invariants on schur irreducible components of representation varieties

geometry and the zero sets of semi invariants for homogeneous modules over canonical algebras abstract we characterize the canonical algebras such that for all dimension vectors of homogeneous modules the corresponding module varieties are complete intersections respectively normal we also investigate the sets of common zeros of semi invariants of non zero degree in important cases in particular we show that for sufficiently big vectors they are complete intersections and calculate the number of their irreducible components

flat ideals and stability in integral domains abstract we introduce the concept of quasi stable ideal in an integral domain d a nonzero fractional ideal i of d is quasi stable if it is flat in its endomorphism ring i i and study properties of domains in which each nonzero fractional ideal is quasi stable we investigate some questions about flatness that were raised by s glaz and w v vasconcelos in their 1977 paper 17

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

normality of orthogonal and symplectic nilpotent orbit closures in positive characteristic abstract in this note we investigate the normality of closures of orthogonal and symplectic nilpotent orbits in positive characteristic we prove that the closure of such a nilpotent orbit is normal provided that neither type d nor type e minimal irreducible degeneration occurs in the closure and conversely if the closure is normal then any type e minimal irreducible degeneration does not occur in it here the minimal irreducible degenerations of a nilpotent orbit are introduced by w hesselink in 7 or see 11 from which we take table 1 for the complete list of all minimal irreducible degenerations our result is a weak version in positive characteristic of 11 theorem 16 2 ii one of the main results of 11 over complex numbers

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

the zero set of semi invariants for extended dynkin quivers we show that the set of common zeros z q d of all semi invariants vanishing at 0 on the variety rep q d of all representations with dimension vector d of an extended dynkin quiver q under the group gl d is a complete intersection if d is big enough in case rep q d does not contain an open gl d orbit which is the case not considered so far the number of irreducible components of z q d grows with d except if q is an oriented cycle

resolutions of determinantal ideals the submaximal minors for many years there has been considerable interest in finding a resolution of the ideal generated by the minors of order p of a generic m x n matrix to put the problem more precisely suppose r is a commutative ring and x are variables with 1 i m and 1 j q n if we let r r x be the polynomial ring over r then we have the generic matrix x and we may form the ideal zp in r generated by the p x p minors of this matrix the problem then is to find an explicit free resolution of the ideal zp over the ring r it was proved by eagon and hochster lo that r i has a resolution of length m p l n p l but their proof consisted in showing that the ideal zp is perfect it did not provide a construction of the resolution in fact it is not known whether the betti numbers of the ideal zp depend on the characteristic of the ground ring r in 121 lascoux succeeded in giving an explicit resolution provided that the ground ring r contained the field q of rational numbers his construction rests heavily on the theory of schur functors and the fact that in characteristic zero the schur functors are the irreducible representations of the general linear group over the integers however the construction breaks down despite the fact that one can define the schur functors over an arbitrary commutative ring see 1 2 14 151 for various constructions of schur functors in analyzing the work of lascoux and its subsequent reworking by nielsen 131 some basic facts seemed to clamor for attention one was that within a resolution of r z there appeared to be two types of boundary maps one of degree 1 and one of degree p the maps of degree 1 were maps between sums of schur functors of fixed durfee square k see section 2 for definitions while the maps of degree p were from sums of schur functors of

the strong global dimension of piecewise hereditary algebras let a be a finite dimensional piecewise hereditary algebra over an algebraically closed field this text investigates the strong global dimension of a this invariant is characterised in terms of the lengths of sequences of tilting mutations relating a to a hereditary abelian category in terms of the generating hereditary abelian subcategories of the derived category of a and in terms of the auslander reiten structure of that derived category

a short proof of a theorem of m hashimoto abstract recently m hashimoto announced that the resolutions of determinantal ideals depend on the characteristic of the base field the purpose of this note is to give a short proof of this result one should stress that even though the method we use is different from hashimoto s his result provided us with the clue where the additional relation exists

on odd symplectic schur functions abstract proctor defined combinatorially a family of laurent polynomials called odd symplectic schur functions indexed by pairs c where is partition andcis a column length of a conjecture of proctor invent math 92 1988 307 332 includes the statement that the odd symplectic schur functions are actually characters ofsp 2n 1 c the purpose of the present note is to prove this

a smooth space of tetrahedra abstract we construct a smooth symmetric compactification of the space of all labeled tetrahedra in p 3

acyclicity of schur complexes and torsion freeness of schur modules abstract let r be a noetherian commutative ring and m a r module with pd r m 1 that has rank necessary and sufficient conditions were provided in 1 for an exterior power k m to be torsion free when m is an ideal of r similar necessary and sufficient conditions were provided in 2 for a symmetric power s k m to be torsion free we extend these results to a broad class of schur modules l m en route for any map of finite free r modules f g we also study the general structure of the schur complexes l and provide necessary and sufficient conditions for the acyclicity of any given l by computing explicitly the radicals of the ideals of maximal minors of all its differentials

module varieties over canonical algebras abstract the main purpose of this paper is the study of module varieties over the class of canonical algebras providing a rich source of examples of varieties with interesting properties our main tool is a stratification of module varieties which was recently introduced by richmond this stratification does not require a precise knowledge of the module category if it is finite then it provides a method to classify irreducible components we determine the canonical algebras for which this stratification is finite in this case we describe the algorithm for calculating the dimension of the variety and the number of irreducible components of maximal dimension for an infinite family of examples we give easy combinatorial criteria for irreducibility cohen macaulay and normality

on the resolutions of the powers of the pfaffian ideal x is a generic skew symmetric matrix and we let s stand for the polynomial ring r x r z the pfaffians of the 2k order principal submatrices of x generate an ideal of s the pfalfian ideal denoted by i it is well known that i is a generically perfect gorenstein prime ideal of grade 3 in fact a prototype of all the ideals of this kind cf b e 31 hence i has a finite free s resolution of length 3 a resolution which looks the same regardless of the ring r

integral domains of finite t character abstract an integral domain d is said to be of finite t character if each nonzero nonunit of d is contained in only finitely many maximal t ideals of d for example noetherian domains and krull domains are of finite t character in this paper we study several properties of integral domains of finite t character we also show when the ring d s d x d s x is of finite t character where x is an indeterminate over d and s is a multiplicative subset of d

koszul modules and green s conjecture we prove a strong vanishing result for finite length koszul modules and use it to derive green s conjecture for every g cuspidal rational curve over an algebraically closed field mathbf k with text char mathbf k 0 or text char mathbf k ge frac g 2 2 as a consequence we deduce that the general canonical curve of genus g satisfies green s conjecture in this range our results are new in positive characteristic whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by voisin our strategy involves establishing two key results of independent interest 1 we describe an explicit characteristic independent version of hermite reciprocity for mathfrak s mathfrak l 2 representations 2 we completely characterize in arbitrary characteristics the non vanishing behavior of the syzygies of the tangential variety to a rational normal curve

semi invariants for quivers with relations for a given quiver q and a dimension vector let si q be the ring of semi invariants for dimensional representations schofield defined in s for any dimensional representation v with 0 a semi invariant c si q we recall that we proved in dw that the ring of semi invariants si q is generated by such schofield semi invariants in this note we generalize the results of dw to quivers with relations in the case when the base field has characteristic zero we make this assumption throughout the paper let q be a quiver and let i be an ideal generated by admissible relations the variety rep q i of representations of q satisfying the relations from i does not have to be irreducible we are interested in the generators of rings of semiinvariants on the irreducible components of rep q i it is clear that the ring of semi invariants on such components will be generated by the restrictions of the semi invariants c where v is a representation of q however such description is not very useful we might have a case when the quiver q with relations i is of a finite or tame type but q is wild in such case one would still like to describe the semi invariants of q i explicitly but the description of indecomposable representations of q is impossible therefore it

invariant theory of canonical algebras abstract based on the first fundamental theorem of classical invariant theory we present a reduction technique for computing relative invariants for quivers with relations this is applied to the invariant theory of canonical algebras and yields an explicit construction of the moduli spaces together with the quotient morphisms from the corresponding representation spaces for families of modules with a fixed dimension vector belonging to the central sincere separating subcategory by means of a tilting process we extend these results to the invariant theory of concealed canonical algebras thus covering the cases of tame hereditary tame concealed and tubular algebras respectively our approach yields in particular a uniform treatment to an essential part of the invariant theory of extended dynkin quivers a topic popular over the years but stretches far beyond since also concealed canonical algebras of tubular or wild representation type are covered

bazzoni s conjecture a conjecture posed 11 years ago by s bazzoni is solved by showing that a prufer domain with the property that every locally finitely generated ideal is finitely generated is in fact a prufer domain of finite character

a straightening algorithm for row convex tableaux abstract we produce a new basis for the schur and weyl modules associated to a row convex shape d the basis is indexed by a new class of straight tableaux which we introduce by weakening the usual requirements for standard tableaux spanning is proved via a new straightening algorithm for expanding elements of the representation into this basis for skew shapes this algorithm specializes to the classical straightening law the new straight basis is used to produce bases for flagged schur and weyl modules to provide groebner and sagbi bases for the homogeneous coordinate rings of some configuration varieties and to produce a flagged branching rule for row convex representations systematic use of supersymmetric letterplace techniques enables the representation theoretic results to be applied to representations of the general linear lie superalgebra as well as to the general linear group

addendum specht modules for column convex diagrams characteristic free results for weyl modules abstract it is shown that the results in reiner and shimozono 12 have characteristic free analogues for weyl modules

concealed canonical algebras and separating tubular families we characterise those concealed canonical algebras which arise as endomorphism rings of tilting modules all of whose indecomposable summands have strictly positive rank as those artin algebras whose module categories have a separating exact subcategory that is a separating tubular family of standard tubes this paper develops further the technique of shift automorphisms which arises from the tubular structure it is related to the characterisation of hereditary noetherian categories with a tilting object as the categories of coherent sheaves on a weighted projective line

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

borel weil theorem for configuration varieties and schur modules we present a generalization of the classical schur modules ofgl n exhibiting the same interplay among algebra geometry and combinatorics a generalized young diagramdis an arbitrary finite subset ofn n for eachd we define the schur modulesdofgl n we introduce a projective variety fdand a line bundle ld and describe the schur module in terms of sections of ld for diagrams with the northeast property i1 j1 i2 j2 d min i1 i2 max j1 j2 d which includes the skew diagrams we resolve the singularities of fdand show analogues of bott s and kempf s vanishing theorems finally we apply the atiyah bott fixed point theorem to establish a weyl type character formula of the form charsd x txwt t i j 1 xix 1j dij t wheretruns over certain standard tableaux ofd our results are valid over fields of arbitrary characteristic

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 orthogonal invariants in characteristic 2 abstract working over an algebraically closed base field k of characteristic 2 the ring of invariants r g is studied where g is the orthogonal group o n or the special orthogonal group so n acting naturally on the coordinate ring r of the m fold direct sum k n k n of the standard vector representation it is proved for o 2 o 3 so 3 so 4 and o 4 that there exists an m linear invariant with m arbitrarily large which is not expressible as a polynomial of invariants of lower degree this is in sharp contrast with the uniform description of the ring of invariants valid in all other characteristics and supports the conjecture that the same phenomena occur for all n for general even n new o n invariants are constructed which are not expressible as polynomials of the quadratic invariants in contrast with these results it is shown that rational invariants have a uniform description valid in all characteristics similarly if m n then r o n is generated by the obvious invariants for all n the algebra r g is a finitely generated module over the subalgebra generated by the quadratic invariants and for odd n the square of any so n invariant is a polynomial of the quadratic invariants finally we mention that for even n an n linear so n invariant is given which distinguishes between so n and o n just like the determinant in all characteristics different from 2

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

primary ideals associated to the linear strands of lascoux s resolution and syzygies of the corresponding irreducible representations of the lie superalgebra gl m n abstract we investigate equivariant koszul duality between primary ideals i a b of s s m m n associated to rectangular young diagrams a b and the corresponding atypical irreducible mixed supertensor representations x a b of gl m n in characteristic zero we show i a b to be h 0 s x a b of the koszul dual s x a b and we compute all the higher cohomology of s x a b to be direct sums of i a r b r with multiplicities being given by coefficients of gauss polynomials utilizing this we are able to describe the equivariant syzygies of x a b over s m m n and determine the homological dimension of i a b over s

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

rings of invariants of 2 2 matrices in positive characteristic abstract we prove that the rings of invariants of 2 2 matrices over an infinite field are cohen macaulay this result generalizes the similar theorem of mehta and ramadas in odd characteristics our approach is more elementary and it uses only some standard facts from the theory of modules with good filtrations and the theory of determinantal rings

good filtrations of symmetric algebras and strong f regularity of invariant subrings in this paper we prove that a linear action of a reductive group on a polynomial ring with good filtrations over a field of characteristic p 0 yields a strongly f regular in particular cohen macaulay invariant subring the strongly f regular property of some known examples of invariant subrings such as the coordinate rings of schubert varieties in grassmannians are recovered a similar result over a field of characteristic zero is also proved

clifford regular domains abstract the class semigroup of a commutative integral domain r is the semigroup s r of the isomorphism classes of the nonzero ideals of r with operation induced by multiplication a domain r is said to be clifford regular if s r is a clifford semigroup i e s r is the disjoint union of the subgroups associated to the idempotent elements in this paper we characterize the noetherian and the integrally closed clifford regular domains and find some properties of an arbitrary clifford regular domain

on conjugating representations and adjoint representations of semisimple groups where ea is a direct sum of irreducible g modules of highest weight 2 here we consider the conjugating representation in arbitrary characteristic and prove the appropriate version of richardson s theorem under small and almost certainly unnecessary characteristic root system restrictions one can no longer decompose a g into homogeneous components but instead we prove the existence of an ascending

invariants of quivers under the action of classical groups abstract we consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical linear groups instead of a product of the general linear groups and by considering the dual action of groups on vertex vector spaces together with the usual action a generating system for the corresponding algebra of invariants is found in particular semi invariants of supermixed representations of quivers are established as a consequence a generating system for the algebra of so n invariants of several matrices is constructed over a field of characteristic different from 2 the proof uses the reduction to semi invariants of mixed representations and the decomposition formula that generalizes amitsur s formula for the determinant

homotopy equivalence of two families of complexes an explicit homotopy equivalence is established between two families of complexes both of which generalize the classical koszul complex

moduli of vector bundles frobenius splitting and invariant theory let x be an irreducible projective curve of genus g over an algebraically closed field of positive characteristic 2 3 in part i we prove adapting the degeneration arguments of n tr that moduli spaces of rank two parabolic bundles on x are frobenius split m r for generic smooth x a similar result holds for x nodal a consequence is the verlinde formula in positive characteristic in part ii we give a direct proof of the fact that the local rings of the moduli spaces are f split and further that they are cohen macaulay this involves showing that the ring of invariants of g copies of 2 x 2 matrices under the adjoint action of sl 2 is f split and cohen macaulay

class semigroups of prufer domains abstract the class semigroup of a commutative integral domainris the semigroup s r of the isomorphism classes of the nonzero ideals ofrwith the operation induced by multiplication the aim of this paper is to characterize the prufer domainsrsuch that the semigroup s r is a clifford semigroup namely a disjoint union of groups each one associated to an idempotent of the semigroup we find a connection between this problem and the following local invertibility property an idealiofris invertible if and only if every localization ofiat a maximal ideal ofris invertible we consider the property introduced in 1967 for prufer domainsr stating that if 1and 2are two distinct sets of maximal ideals ofr then rm m 1 rm m 2 let c be the class of prufer domains satisfying the separation property or with the property that each localization at a maximal ideal if finite dimensional we prove that ifrbelongs to c then the local invertibility property holds onrif and only if every nonzero element ofris contained only in a finite number of maximal ideals ofr moreover ifrbelongs to c then s r is a clifford semigroup if and only if every nonzero element ofris contained only in a finite number of maximal ideals ofr

the monoidal structure on strict polynomial functors abstract the category of strict polynomial functors inherits an internal tensor product from the category of divided powers to investigate this monoidal structure we consider the category of representations of the symmetric group s d which admits a tensor product coming from its hopf algebra structure it is classical that there exists a functor f from the category of strict polynomial functors to the category of representations of the symmetric group our main result is that this functor f is monoidal in addition we study the relations under f between projective strict polynomial functors and permutation modules and the link to symmetric functions

affine strict polynomial functors and formality abstract we introduce the notion of an affine strict polynomial functor we show how this concept helps to understand homological behavior of the operation of frobenius twist in the category of strict polynomial functors over a field of positive characteristic we also point out for an analogy between our category and the category of representations of the group of algebraic loops on g l n

moduli spaces for representations of concealed canonical algebras abstract continuing m domokos and h lenzing j algebra 228 2000 738 762 we investigate the relative invariants and moduli spaces introduced in a d king quart j math oxford ser 2 45 1994 515 530 which are related to a separating family of stable tubes in the module category of a concealed canonical algebra the class of concealed canonical algebras consists of representation infinite algebras of tame or wild representation type and contains in particular the representation theory of extended dynkin quivers we present a uniform characteristic free approach avoiding case by case analysis we introduce the notion of admissible weight for such algebras these are the weights i e elements of the dual of the grothendieck group of the module category attached to the infinite moduli spaces for families of modules from the separating subcategory by using an explicit description of the semigroup of admissible weights the corresponding notion of semistability is analyzed and is related to the formation of perpendicular categories we construct relative invariants belonging to admissible weights and prove that the constructed system is complete in the case of canonical algebras it turns out that for any concealed canonical algebra and for any admissible weight all the corresponding moduli spaces are isomorphic to some projective space moreover any such moduli space can be naturally identified with the moduli space belonging to a special weight called the rank as a consequence we show that in the case of a tame concealed algebra any infinite moduli space for families of modules is a projective space and all fields of rational invariants on irreducible components of representation spaces are purely transcendental

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

the space of triangles vanishing theorems and combinatorics abstract we consider compactifications of pn 3 ij the space of triples of distinct points in projective space one such space is a singular variety of configurations of points and lines another is the smooth compactification of fulton and macpherson and a third is the triangle space of schubert and semple we compute the sections of line bundles on these spaces and show that they are equal as gl n representations to the generalized schur modules associated to bad generalized young diagrams with three rows borel weil theorem on the one hand this yields weyl type character and dimension formulas for the schur modules on the other a combinatorial picture of the space of sections cohomology vanishing theorems play a key role in our analysis

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

the minimal free resolution of the migliore peterson rings in the case that the reflexive sheaf has even rank abstract letrbe a commutative noetherian ring and f gbe a homomorphism of freer modules where rank f fand rank g g fix an elementbg 1 g 1fand a generator g for gg the module action of f on fproduces the elementb1 g g bg 1 inf letjdenote the image ofb1 f r assume that gradej f g which is the largest grade possible and is attained in the generic case the idealjmay be interpreted as the defining ideal of the degeneracy locus of a regular section of a rankf greflexive sheaf it may also be interpreted as the order ideal of an element in a second syzygy module of rankf g also jmay be interpreted as the defining ideal for the symmetric algebra of a module of projective dimension two migliore and peterson have studied the idealjunm which is the unmixed part ofj under geometric hypotheses they have shown thatr junmis a cohen macaulay ring and they have resolved this ring furthermore iff gis odd thenjunmis a gorenstein ideal and is not equal toj on the other hand iff gis even thenjunm j in the present paper we produce the resolution ofr jby freer modules in the case thatf gis even and f g 2 is a unit inr our resolution is minimal whenever the data are local or homogeneous our resolution is built from the differential graded algebra f x1 xg d where the restriction ofdto f is the koszul complex associated tob1 f rand the degree two divided power variablesx1 xghave been adjointed in order to kill the cycles g 1f the acyclicity lemma is used to prove exactness ifg 1 then the idealjis equal to the huneke ulrich almost complete intersection ideali1 yx whereyis a 1 fmatrix andxis anf falternating matrix the resolution of this ideal is already known

a note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup abstract let g be a connected semisimple simply connected algebraic group over an algebraically closed field k of positive characteristic the action of g on itself by conjugation induces on the coordinate algebra k g and hence on the coordinate algebra k g r of the infinitesimal subgroup g r the structure of a rational g module in the recent paper 3 by bendel nakano pillen and sobaje the authors ask whether the tensor product of k g r with the rth steinberg module st r has a good filtration here we show that this is the case provided that the characteristic of k is good for g we do so by relating k g r to a certain quotient of the coordinate algebra k g of the lie algebra g of g and using a koszul resolution we note that the argument may also be applied to certain polynomial modules occurring in the context of doty s conjecture 17 4 2 11

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

explicit generalized pieri maps abstract the purpose of this paper is to describe explicitly in a characteristic free setting certain maps between skew weyl modules for the general linear group as an application we indicate briefly how our generalized pieri maps are utilized to obtain new results concerning nonzero homomorphisms between certain skew weyl modules over fields of positive characteristics in particular we obtain a new proof of the special case of the carter payne theorem considered in this paper

generating invariant rings of quivers in arbitrary characteristic abstract it is well known that the ring of polynomial invariants of a reductive group is finitely generated however it is difficult to give strong upper bounds on the degrees of the generators especially over fields of positive characteristic in this paper we make use of the theory of good filtrations along with recent results on the null cone to provide polynomial bounds for matrix semi invariants in arbitrary characteristic and consequently for matrix invariants our results generalize to invariants and semi invariants of quivers

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal

remark on the branching theorem and supersymmetric algebras supersymmetric algebras have already proved useful in giving trans parent proofs of a number of basic results of representation theory 3 7 9 1 l l 51 specifically the technique of introducing virtual variables which may have a different signature than the signature of variables to be delt with often cuts down the amount of computation furthermore the extension of results of representation theory to the superalgebraic setting sheds new light and permits us to establish natural correspondences that were formally missing in this note we carry out this program by deriving a superalgebraic version of the branching rules for the representations of the general linear group while the statement of supersymmetric branching rules is in all respects similar to the ordinary one and differing from it in our allowing variables of two signature the proof yields a useful dividend namely a simple combinatorial construction of a canonical basis for the decomposi tion of a restriction of a representation as an application we give a supersymmetric generalization of pieri s formula as well as a proof of this formula which is perhaps as short as it can be whittled down to this application has been inspired by some recent work of bofli s 61 we have benefited from the pioneering work of berele and regev 3 who were first to state such a supersymmetric extension of branching rules as well as from the insights of balentekin and bars 2 255

geometry of regular modules over canonical algebras we classify canonical algebras such that for every dimension vector of a regular module the corresponding module variety is normal respectively a complete intersection we also prove that for the dimension vectors of regular modules normality is equivalent to irreducibility