the boundary of the irreducible components for invariant subspace varieties given partitions alpha beta gamma the short exact sequences begin aligned 0 longrightarrow n alpha longrightarrow n beta longrightarrow n gamma longrightarrow 0 end aligned of nilpotent linear operators of jordan types alpha beta gamma respectively define a constructible subset mathbb v alpha gamma beta of an affine variety geometrically the varieties mathbb v alpha gamma beta are of particular interest as they occur naturally and since they typically consist of several irreducible components in fact each littlewood richardson tableau vargamma of shape alpha beta gamma contributes one irreducible component overline mathbb v vargamma we consider the partial order vargamma le mathsf boundary widetilde vargamma on lr tableaux which is the transitive closure of the relation given by mathbb v widetilde vargamma cap overline mathbb v vargamma ne emptyset in this paper we compare the boundary relation with partial orders given by algebraic combinatorial and geometric conditions it is known that in the case where the parts of alpha are at most two all those partial orders are equivalent we prove that those partial orders are also equivalent in the case where beta setminus gamma is a horizontal and vertical strip moreover we discuss how the orders differ in general

the closure of the set of periodic modules over a concealed canonical algebra is regular in codimension one abstract let be a concealed canonical algebra and d the dimension vector of a module which is periodic respect to the action of the auslander reiten translation in the paper we investigate the union of the closures of the orbits of the periodic modules of dimension vector d we show that this set is closed and regular in codimension one

voronoi means moving averages and power series we introduce a non regular generalisation of the norlund mean and show its equivalence with a certain moving average the abelian and tauberian theorems establish relations with convergent sequences and certain power series a strong law of large numbers is also proved

degenerations in a class of matrix varieties and prinjective modules abstract let k be an algebraically closed field and let i be a finite partially ordered set to any vector v n i we associate the irreducible variety mat v i with an action of the algebraic group g v i see section 6 the main aim of this paper is to give a description of degenerations of orbits of this action in terms of homomorphisms and extensions of prinjective ki modules

classifying representations by way of grassmannians let a be a finite dimensional algebra over an algebraically closed field criteria are given which characterize existence of a fine or coarse moduli space classifying up to isomorphism the representations of a with fixed dimension d and fixed squarefree top t next to providing a complete theoretical picture some of these equivalent conditions are readily checkable from quiver and relations of a in the case of existence of a moduli space unexpectedly frequent in light of the stringency of fine classification this space is always projective and in fact arises as a closed subvariety brassy of a classical grassmannian even when the full moduli problem fails to be solvable the variety grass t d is seen to have distinctive properties recommending it as a substitute for a moduli space as an application a characterization of the algebras having only finitely many representations with fixed simple top is obtained in this case of finite local representation type at a given simple t the radical layering j l m j l 1 m l 0 is shown to be a classifying invariant for the modules with top t this relies on the following general fact obtained as a byproduct proper degenerations of a local module m never have the same radical layering as m

on open orbits and their complements 1 1 we fix an algebraically closed field k and a quiver q without oriented cycles whose vertex set q we identify with 1 n for a in the set q of arrows of q we denote by ta and ha the tail and the head of a respectively a representation x of q consists of a family x i in qo of finite dimensional k vector spaces and a family x a x ta x ha a e q1 of k linear maps a morphism f x y between two representations of q is a family f i x i y i in qo of k linear maps satisfying f ha ox a y a of ta for all arrows a we denote the category of representations of q by mod q it is an abelian category of global dimension at most 1 we will denote the only possibly non trivial extension group ext by ext the dimension vector of a representation x of q is the vector dim x dim x l dim x n in n and the dimension of x is the natural number dim x dim x 1 dim x n the support of x is the full subquiver of q whose vertices are i x i zo for a vector d d d e n we define r q d to be the set of representations x of q such that x i k for ie qo thus r q d is the product r q d n hom kdru kdhe

top stable degenerations of finite dimensional representations i abstractgiven a finite dimensional representation m of a finite dimensional algebra two hierarchies of degenerations of m are analyzed in the context of their natural orders the poset of those degenerations of m which share the top m jm with m here j denotes the radical of the algebra and the sub poset of those which share the full radical layering gl j l m j l 1 mgr l 0 with m in particular the article addresses the existence of proper top stable or layer stable degenerations more generally it addresses the sizes of the corresponding posets including bounds on the lengths of saturated chains as well as structure and classification

the complexity of minimal degenerations for representation finite selfinjective algebras abstract minimal degenerations of modules over all but one exceptional class of representation finite selfinjective algebras are given by extensions g zwara colloq math 75 1998 91 this means in terms of the complexity of degenerations that these minimal degenerations have complexity one the minimal degenerations of modules for the exceptional class of representation finite selfinjective algebras are studied by r aehle j london math soc 66 2002 73 in this paper it is shown that these degenerations have complexity less or equal to two

twisted frobenius schur indicators of finite symplectic groups denote by e the classical frobenius schur indicator of the irreducible representation v with character that is e 1 g g g g 2 and frobenius and schur proved that e 1 if v is a real representation e 1 if is real valued but is not a real representation and e 0 otherwise then the conclusion of theorem 1 1 is equivalent to the statement that every irreducible representation of g satisfies e 1 gow obtained the following intriguing result from theorem 1 1

real representations of the finite orthogonal and symplectic groups of odd characteristic let q be a power of an odd prime p let sp 2n q denote the symplectic group of degree 2n over gf q let o n q denote the split orthogonal group of degree n over gf q corresponding to a symmetric form of maximal witt index and 0 n q denote the non split orthogonal group unless we need to be more specific we will simply write o n q for either of these two orthogonal groups in addition we will often abbreviate these symbols further to o n and sp 2n whenever q is understood to be fixed it is known from conjugacy class considerations that all complex irreducible characters of the finite orthogonal groups o n are real valued the same is true of the symplectic groups sp 2n q provided that q z 1 mod 4 when q 3 mod 4 not all characters of sp 2n q are realvalued however as the total number of irreducible characters is a manic polynomial in q of degree n and this is true of the number of real valued irreducible characters we can assert that the majority of characters is realvalued in this case it is of interest to know whether or not the real valued characters of these families of groups are the characters of representations that are defined over the real numbers this paper gives a solution to this problem by proving the following theorem

on degenerations and extensions of finite dimensional modules abstract we derive a cancellation theorem for degenerations of modules that says in particular that projective or injective common direct summands can always be neglected combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease for representation directed algebras whence in particular for path algebras of dynkin quivers we provide an alternative proof which shows in addition that any minimal degenerationnofmcomes from an exact sequence with middle termmwhose end terms add up ton by a careful examination the same is true for degenerations of matrix pencils having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences in particular we show that any non simple indecomposable over a tame quiver is an extension of an indecomposable and a simple

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

on degenerations between preprojective modules over wild quivers abstract we study minimal degenerations between preprojective modules over wild quivers asymptotic properties of such degenerations are studied with respect to codimension and numbers of indecomposable direct summands we provide families of minimal disjoint degenerations of arbitrary codimension for almost all wild quivers and show that no such examples exist in the remaining cases

linear degenerations of flag varieties partial flags defining equations and group actions we continue generalize and expand our study of linear degenerations of flag varieties from cerulli irelli et al math z 287 1 2 615 654 2017 we realize partial flag varieties as quiver grassmannians for equi oriented type a quivers and construct linear degenerations by varying the corresponding quiver representation we prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration the former is the partial analogue of the mf degenerate flag variety and the latter coincides with the partial pbw degenerate flag variety we compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus we also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus finally we prove an analogue of the borel weil theorem for the flat irreducible locus

on minimal disjoint degenerations for preprojective representations of quivers we derive a root test for degenerations as described in the title in the case of dynkin quivers this leads to a conceptual proof of the fact that the codimension of a minimal disjoint degeneration is always one for euclidean quivers it enables us to show a periodic behaviour this reduces the classification of all these degenerations to a finite problem that we have solved with the aid of a computer it turns out that the codimensions are bounded by two somewhat surprisingly the regular and preinjective modules play an essential role in our proofs

tauberian theorems for power series methods applied to double sequences abstract we discuss the relations between power series methods weighted mean methods and ordinary convergence for double sequences in particular we study tauberian theorems for methods being products of the related one dimensional summability methods

on relations between weighted mean and power series methods of summability soit p n une suite de nombres non negatifs avec p 0 0 et p n k 0 n p k p x n 0 p u x n pour 0 x 1 soit s n une suite de nombres reels on etudie le lien entre la methode de sommabilite en moyenne carree et la methode de serie de puissances

products of two involutions in classical groups of characteristic 2 in a corresponding result for fields not of characteristic 2 wonenburger showed in 6 that every orthogonal transformation is the product of two orthogonal involutions whereas every symplectic transformation is the product of two skew symplectic involutions a skew symplectic transformation t satisfies b tu tu u u for u v in v it follows from our theorem that if k is a finite field of even characteristic each element of the symplectic group or of the two nonisomorphic orthogonal groups is conjugate to its inverse and thus the complex characters of these three families of finite groups are all real valued 583 0021 8693 81 080583 09 02 00 0

perturbation theory of matrix pencils through miniversal deformations abstract we give new proofs of several known results about perturbations of matrix pencils in particular we give a direct and constructive proof of andrzej pokrzywa s theorem 1983 in which the closure of the orbit of each kronecker canonical matrix pencil is described in terms of inequalities for invariants of matrix pencils a more abstract description is given by klaus bongartz 1996 by methods of representation theory we formulate and prove pokrzywa s theorem in terms of successive replacements of direct summands in a kronecker canonical pencil first we show that it is sufficient to prove pokrzywa s theorem in two cases for matrices under similarity and for each matrix pencil p q that is a direct sum of two indecomposable pencils then we calculate the kronecker canonical form of pencils that are close to p q in fact the kronecker canonical form is calculated for only those pencils that belong to a miniversal deformation of p q this is sufficient since all pencils in a neighborhood of p q are reduced to them by a smooth strict equivalence transformation

multiple flag varieties of finite type abstract we classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group we also classify the orbits in each case and construct explicit representatives

tight monomials and the monomial basis property abstract we generalize a criterion for tight monomials of quantum enveloping algebras associated with symmetric generalized cartan matrices and a monomial basis property of those associated with symmetric classical cartan matrices to their respective symmetrizable case we then link the two by establishing that a tight monomial is necessarily a monomial defined by a weakly distinguished word as an application we develop an algorithm to compute all tight monomials in the rank 2 dynkin case

degenerations over a singularities and construction of degenerations over commutative rings abstract we give a necessary condition of degeneration via matrix representations and consider degenerations of indecomposable cohen macaulay modules over hypersurface singularities of type a we also provide a method to construct degenerations of finitely generated modules over commutative rings

dominated variation and related concepts and tauberian theorems for laplace transforms abstract some abelian and tauberian theorems are proved under conditions of dominated variation and related concepts for example u is dominatedly varying if and only if its laplace stieltjes transform is dominatedly varying

the bruhat chevalley order of parabolic group actions in general linear groups and degeneration for δ filtered modules abstract let k be an algebraically closed field and v a finite dimensional k space let gl v be the general linear group of v and p a parabolic subgroup of gl v now p acts on its unipotent radical p u and on p u lie p u the lie algebra of p u via the adjoint action more generally we consider the action of p on the l th member of the descending central series of p u denoted by p l u all instances when p acts on p l u for l 0 with a finite number of orbits are known in this note we give a complete combinatorial description of the closure relation on the set of p orbits on p l u i e the bruhat chevalley order for every finite case there is a canonical bijection between the set of p orbits on p l u and the set of isomorphism classes of filtered modules of a particular dimension vector e of a certain quasi hereditary algebra a t l these isomorphism classes in turn are given by the orbits of a reductive group g e on the variety r e of all a t l modules with filtration and dimension vector e the subcategory of a t l mod of all filtered a t l modules of dimension vector e is denoted by f e in our chief result theorem 1 1 we show that provided there is only a finite number of isomorphism classes of indecomposable modules in f the following three posets coincide 1 the bruhat chevalley order on the set of p orbits on p l u 2 the bruhat chevalley order on the set of g e orbits on r e 3 the poset opposite to the so called hom order on the set of isomorphism classes of f e the advantage of this hom order is that it is given purely by discrete invariants and that it can be computed explicitly for any given finite case theorem 1 1 then in turn allows us to explicitly determine the closure relations for the p orbits on p l u with the aid of this hom order we present some examples of hasse diagrams in an appendix

tauberian conditions under which the original convergence of double sequences follows from the statistical convergence of their weighted means abstract in this paper we introduce a new type of slow oscillation and slow decrease conditions we prove that these or their variants are tauberian conditions from s m n s t s to s m n s we also prove that they are tauberian conditions from t m n 11 s t s to s m n s where t m n 11 are the weighted means of the double sequence s m n m n 0 our results not only generalize well known results but also solve the conjecture of moricz posed in f moricz tauberian theorems for double sequences that are statistically summable c 1 1 j math anal appl 286 2003 340 350

irreducible components of module varieties an example abstract we determine the irreducible components of the variety of d dimensional modules over the algebra k 2 2 q with q k for any d and we describe any intersection of irreducible components as the zero set of some ideal we show that any such intersection is irreducible to our knowledge this is the first time irreducible components are described by equations for modules over an algebra which is neither representation finite nor hereditary

complementary spaces and multipliers of double fourier series for functions of bounded variation complementary spaces for fourier series were introduced by g goes and generalized by m tynnov in this paper we investigate a notion of complementary space for double fourier series of functions of bounded variation various applications are given

an explicit description of the irreducible components of the set of matrix pencils with bounded normal rank abstract the set of m n singular matrix pencils with normal rank at most r is an algebraic set with r 1 irreducible components these components are the closure of the orbits under strict equivalence of r 1 matrix pencils which are in kronecker canonical form in this paper we provide a new explicit description of each of these irreducible components which is a parametrization of each component therefore one can explicitly construct any pencil in each of these components the new description of each of these irreducible components consists of the sum of r rank 1 matrix pencils namely a column polynomial vector of degree at most 1 times a row polynomial vector of degree at most 1 where we impose one of these two vectors to have degree zero the number of row vectors with zero degree determines each irreducible component

on degenerations of cohen macaulay modules abstract in relation to degenerations of modules we introduce several partial orders on the set of isomorphism classes of finitely generated modules over a noetherian commutative local ring our main theorem says that under several special conditions any degenerations of maximal cohen macaulay modules are essentially obtained by the degenerations of auslander reiten sequences

on minimal disjoint degenerations of modules over tame path algebras abstract for representations of tame quivers the degenerations are controlled by the dimensions of various homomorphism spaces furthermore there is no proper degeneration to an indecomposable therefore up to common direct summands any minimal degeneration from m to n is induced by a short exact sequence 0 u m v 0 with indecomposable ends that add up to n we study these building blocs of degenerations and we prove that the codimensions are bounded by two therefore a quiver is dynkin resp euclidean resp wild iff the codimension of the building blocs is one resp bounded by two resp unbounded we explain also that for tame quivers the complete classification of all the building blocs is a finite problem that can be solved with the help of a computer

geometry of directing modules over tame algebras throughout the paper k denotes a fixed algebraically closed field by an algebra we mean a finite dimensional k algebra associative with an identity and by a module a finite dimensional left a module the class of algebras may be divided into two disjoint subclasses one class consists of tame algebras for which the indecomposable modules occur in each dimension in a finite number of discrete and a finite number of one parameter families the second class is formed by the wild algebras whose representation theory is as complicated as the study of finite dimensional vector spaces together with two noncommuting endomorphisms for which the classification up to isomorphism is a well known unsolved problem hence we can realistically hope to describe modules only for tame algebras given an algebra a and a nonnegative vector d in the grothendieck group k a of a it is an interesting task to study the 0 affine variety mod d of a modules of dimension vector d and the action a of the corresponding product g d of general linear groups for example we may ask when the variety mod d is irreducible smooth complete a intersection gorenstein cohen macaulay normal the main aim of this paper is to describe the geometry of module varieties mod d for the dimension vectors d of arbitrary directing moda w x ules over tame algebras recall that following 20 an indecomposable a module m is said to be directing if it does not belong to a cycle m a m a a m a m of nonzero nonisomorphisms between inde1 r composable a modules directing modules have played an important role

some tauberian theorems for the weighted mean methods of summability in this paper we introduce the concept of the weighted generator sequence and prove that certain conditions in terms of weighted generator sequences are tauberian conditions for the weighted mean summability method

tauberian theorems for jp summability abstract we prove tauberian theorems for jp summability methods of power series type with respect to ordinary convergence under a o tauberian condition on the underlying sequence our tauberian conclusion is valid for all jp methods relaxing the tauberian assumption to a o type condition a related result is shown for methods with moderate weight sequences

minimal singularities in orbit closures of matrix pencils abstract we show that all singularities occurring in minimal degenerations of matrix pencils are cohen macaulay and regular in codimension 1

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