explicit description of generic representations for quivers of type an or dn we describe explicitly a generic representation for dynkin quivers of type an or dn for any dimension vector

lower bounds for the complexity of polynomials abstract we prove new lower bounds for the complexity of polynomials e g for polynomials with 0 1 coefficients divisors and multiples of given polynomials our results hold for computations over an arbitrary field and cover all complexity measures

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

log canonical singularities are du bois a recurring difficulty in the minimal model program mmp is that while log terminal singularities are quite well behaved for instance they are rational log canonical singularities are much more complicated they need not even be cohenmacaulay the aim of this paper is to prove that as conjectured in kol92 1 13 log canonical singularities are du bois the concept of du bois singularities abbreviated as db was introduced by steenbrink in ste83 as a weakening of rationality it is not clear how to define du bois singularities in positive characteristic so we work over a field of characteristic 0 throughout the paper the precise definition is rather involved see 1 10 but our main applications rely only on the following consequence

a characterization of rational singularities is given bythe reynolds operator a well known and widely used theoremstates that in characteristic 0 canonicalsingularities are cohen macaulay and therefore rational see e and kmm the original proofs are based on a very clever use of grothendieck duality simulta neously for a resolution and its restriction onto the exceptional divisor and on a doubleloop induction kollar gave a simpler proof in k2 11 without using derived cate gories but still relying on a technically hard vanishing theorem recently kollar andmori found a simple proof allowing nonempty boundaries they do not use derivedcategories either but restrict to the projective case see km 5 18 these proofs areingenious but one would like to have a simple natural proof at least in the classical case when the boundary is empty as an application of theorem1 a simple proof is given here in the classical case but without the projective assumption this proof seems even simpler than that ofkollar and mori derived categories and grothendieck duality are used but in sucha simple way that one is tempted to say that this proof is the most natural one notealso that everything used here was already available when the question was raised forthe rst time a statement similar to theorem 1 was given in k2 3 12 some ideas of the

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

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

buchi context free languages we define context free grammars with buchi acceptance condition generating languages of countable words we establish several closure properties and decidability results for the class of buchi context free languages generated by these grammars we also define context free grammars with muller acceptance condition and show that there is a language generated by a grammar with muller acceptance condition which is not a buchi context free language

general representations of quivers we investigate general representations of a quiver q there are two main questions that we shall address the first which was raised by kac is to find the dimension vectors of the summands of a general representation of dimension vector a the second is to find the dimension vectors of the subrepresentations of a general representation of dimension vector a these two questions are closely related one consequence of our investigation is to find an algorithm to compute the canonical decomposition of an arbitrary dimension vector in terms of the euler form of the quiver

on the direct summand conjecture and its derived variant andre recently gave a beautiful proof of hochster s direct summand conjecture in commutative algebra using perfectoid spaces his two main results are a generalization of the almost purity theorem the perfectoid abhyankar lemma and a construction of certain faithfully flat extensions of perfectoid algebras where discriminants acquire all p power roots in this paper we explain a quicker proof of hochster s conjecture that circumvents the perfectoid abhyankar lemma instead we prove and use a quantitative form of scholze s hebbarkeitssatz the riemann extension theorem for perfectoid spaces the same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de jong

locally finite languages we investigate properties of locally finite languages introduced by ressayre j symbolic logic 53 4 1988 1009 1026 these languages are defined by locally finite sentences and generalize languages recognized by automata or defined by monadic second order sentences we give many examples showing that numerous context free languages are locally finite then we study closure properties of the family loc of locally finite languages and show that most undecidability results that hold for context free languages may be extended to locally finite languages in a second part we consider an extension of these languages to infinite and transfinite length words we prove that each language which is recognized by a buchi automaton where is an ordinal and is defined by a locally finite sentence this result combined with a preceding one of finkel and ressayre j symbolic logic 61 2 1996 563 585 provides a generalization of buchi s result about decidability of monadic second order theory of the structure

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

generalized semi invariant distributions on p adic spaces in this paper we investigate some methods on calculating the spaces of generalized semi invariant distributions on p adic spaces using homological methods we give a criterion of automatic extension of generalized semi invariant distributions based on the meromorphic continuations of igusa zeta integrals we give another criteria with purely algebraic geometric conditions on the extension of generalized semi invariant distributions

framed sheaves on projective stacks abstract given a normal projective irreducible stack x over an algebraically closed field of characteristic zero we consider framed sheaves on x i e pairs e e where e is a coherent sheaf on x and e is a morphism from e to a fixed coherent sheaf f after introducing a suitable notion of semi stability we construct a projective scheme which is a moduli space for semistable framed sheaves with fixed hilbert polynomial and an open subset of it which is a fine moduli space for stable framed sheaves if x is a projective irreducible orbifold of dimension two and f a locally free sheaf on a smooth divisor d x satisfying certain conditions we consider d f framed sheaves i e framed sheaves e e with e a torsion free sheaf which is locally free in a neighbourhood of d and e d an isomorphism these pairs are stable for a suitable choice of a parameter entering the semi stability condition and of the polarization of x this implies the existence of a fine moduli space parameterizing isomorphism classes of d f framed sheaves on x with fixed hilbert polynomial which is a quasi projective scheme in an appendix we develop the example of stacky hirzebruch surfaces this is the first paper of a project aimed to provide an algebro geometric approach to the study of gauge theories on a wide class of 4 dimensional riemannian manifolds by means of framed sheaves on stacky compactifications of them in particular in a subsequent paper 20 these results are used to study gauge theories on ale spaces of type a k

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

from m clusters to m noncrossing partitions via exceptional sequences let w be a finite crystallographic reflection group the generalized catalan number of w coincides both with the number of clusters in the cluster algebra associated to w and with the number of noncrossing partitions for w natural bijections between these two sets are known for any positive integer m both m clusters and m noncrossing partitions have been defined and the cardinality of both these sets is the fuss catalan number cm w we give a natural bijection between these two sets by first establishing a bijection between two particular sets of exceptional sequences in the bounded derived category db h for any finite dimensional hereditary algebra h

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

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

finite automata and ordinals abstract several definitions of automata on words indexed by ordinals have been proposed previously the first one was introduced by buchi to prove the decidability of the monadic second order theory of denumerable ordinals wojciechowski studied the properties of these automata independently of the length of the input the second definition proposed by choueka works only on words of length less than n in this paper we restrict the domain of wojciechowski automata to the domain of choueka s ones that is given n we keep only sequences for n 1 in the language defined by a wojciechowski automaton in order to prove the equivalence between choueka automata and wojciechowski automata then we obtain the closure under complementation of the class of wojciechowski s definable sets and finally we give an algorithm for determinizing wojciechowski automata

on characters and formal degrees of discrete series of affine hecke algebras of classical types we address two fundamental questions in the representation theory of affine hecke algebras of classical types one is an inductive algorithm to compute characters of tempered modules and the other is the determination of the constants in the formal degrees of discrete series in the form conjectured by reeder j reine angew math 520 37 93 2000 the former is completely different from the lusztig shoji algorithm shoji in invent math 74 239 267 1983 lusztig in ann math 131 355 408 1990 and it is more effective in a number of cases the main idea in our proof is to introduce a new family of representations which behave like tempered modules but for which it is easier to analyze the effect of parameter specializations our proof also requires a comparison of the c theoretic results of opdam delorme slooten solleveld j inst math jussieu 3 531 648 2004 arxiv 0909 1227 int math res not 2008 adv math 220 1549 1601 2009 acta math 205 105 187 2010 and the geometric construction from kato duke math j 148 305 371 2009 am j math 133 518 553 2011 ciubotaru and kato adv math 226 1538 1590 2011

quiver representations with an irreducible null cone abstract let d be a prehomogeneous dimension vector for a quiver q there is an action of the product gl d of linear groups on the vector space rep q d of representations of q with dimension vector d and there is a representation t with a dense gl d orbit in rep q d we give a construction for a dense subset f q d of the variety z q d of common zeros of all semi invariants in k rep q d of positive degree and we show that this set is stable for big dimension vectors i e f q n d x t n 1 x f q d moreover we show that the existence of a dense orbit in z q d depends on a quiver q such that the category of representations of q is equivalent to the right perpendicular category t

relative invariants and subgeneric orbits of quivers of finite and tame type the notion of a quiver was introduced by gabriel in 2 we are dealing with the determination of relative invariants of representation varieties attached to quivers to be more precise let us first recall the basic concepts a quiver q consists of two finite sets qo the set of points and q1 the set of arrows and that for all e q1 there is assigned its starting point e qo and its end point e e qo let k be an algebraically closed field a representation of q is given by finite dimensional k vectorspaces v for l i e qo and k linear maps a v v for all e q1 it is well known that the representations of q form an abelian category usually denoted by q and that q is of finite type those where q admits only a finite number of indecomposable objects iff the underlying graph of q is a dynkin diagram and that q is of tame type iff the underlying graph of q is a euclidean diagram let v vi be a representation of q then the dimension type of v is this is a positive integral vector in qn defined as dim v dim k vi i s qo

perpendicular categories null cones and dense orbits abstract let q be a quiver k an algebraically closed field and d a prehomogeneous dimension vector for q with canonical decomposition d i 1 r i dim t i we denote by z q d the set of common zeros of all semi invariants in k rep q d of positive degree we study the relations between the existence of a dense gl d orbit in the null cone z q d and the position of the representations t 1 t r in the auslander reiten quiver associated to q

on the representation of rational functions of bounded complexity abstract the main result of this article is the representation theorem which characterizes families of rational functions of bounded complexity by appropriate mappings this description is not only independent of the characteristic of the underlying field but also of the specific complexity measure under consideration as an application of the representation theorem we derive good lower complexity bounds

codimension 1 orbits and semi invariants for the representations of an equioriented graph of type _ on considere le diagramme de dynkin d n equioriente et la variete hom v 1 v 3 i 2 n hom v i v i 1 v j un espace vectoriel sur k sur lequel agit le groupe g i 1 n gl v i on determine l orbite maximale et les orbites de codimension 1 de cette action

the b functions of semi invariants of quivers abstract in this paper we compute b functions or bernstein sato polynomials of various semi invariants of quivers the main tool is an explicit relation for the b functions between semi invariants that correspond to each other under reflection functors or castling transforms this enables us to compute recursively the b functions for all dynkin quivers and extended dynkin quivers with prehomogeneous dimension vectors

f singularities via alterations we give characterizations of test ideals and f rational singularities via regular alterations formally the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions lastly we establish nadel type vanishing theorems up to finite maps for test ideals and further demonstrate how these vanishing theorems may be used to extend sections

a modular compactification of m_ 1 n from a_ infty structures we show that a certain moduli space of minimal a structures coincides with the modular compactification m1 n n 1 of m1 n constructed by smyth in 22 in addition we describe these moduli spaces and the universal curves over them by explicit equations prove that they are normal and gorenstein show that their picard groups have no torsion and that they have rational singularities if and only if n 11

tree automatic scattered linear orders tree automatic linear orders on regular tree languages are studied it is shown that there is no tree automatic scattered linear order and therefore no tree automatic well order on the set of all finite labeled trees and that a regular tree language admits a tree automatic scattered linear order if and only if for some n no binary tree of height n can be embedded into the union of the domains of its trees hence the problem whether a given regular tree language can be ordered by a scattered linear order or a well order is decidable moreover sharp bounds for tree automatic well orders on some regular tree languages are computed by connecting tree automata with automata on ordinals the proofs use elementary techniques of automata theory

geometric aspects of iterated matrix multiplication abstract this paper studies geometric properties of the iterated matrix multiplication polynomial and the hypersurface that it defines we focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the polynomial the dual variety and the jacobian loci of the hypersurface that are computed with the aid of representation theory of quivers

pluricanonical maps of stable log surfaces abstract stable surfaces and their log analogues are the type of varieties naturally occurring as boundary points in moduli spaces we extend classical results of kodaira and bombieri to this more general setting if x is a stable log surface with reduced boundary possibly empty and i is its global index then 4 i k x is base point free and 8 i k x is very ample these bounds can be improved under further assumptions on the singularities or invariants for example 5 k x is very ample if x has semi canonical singularities

three kinds of mutation for a finite dimensional hereditary algebra h we consider exceptional sequences in mod h silting objects in the bounded derived category d and m cluster tilting objects in the m cluster category cm d 1 m there are mutation operations on both the set of m cluster tilting objects and the set of exceptional sequences it is also possible to define a mutation operation for silting objects we compare these three different notions of mutation