the gl 2 c mckay correspondence in this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution as a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph also for any finite subgroup g of rm gl 2 mathbb c it means that the endomorphism ring of the special cm mathbb c x y g modules can be used to build the dual graph of the minimal resolution of mathbb c 2 g extending mckay s observation mckay proc symp pure math 37 183 186 1980 for finite subgroups of rm sl 2 mathbb c to all finite subgroups of rm gl 2 mathbb c

minimal resolutions of algebras abstract a method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself the method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations that is either homogeneous or admissible with some additional finiteness restrictions in the latter case in particular it applies to any finite dimensional algebra over an algebraically closed field the method is illustrated by a number of examples viz truncated algebras monomial algebras and koszul algebras with the aim of unifying existing treatments of these in the literature

explicit generators of the invariants of finite groups abstract let r denote a commutative and associative ring with 1 and let a denote a finitely generated commutative r algebra let g denote a finite group of r algebra automorphisms of a in the case that r is a field of characteristic 0 noether constructed a finite set of r algebra generators of the invariants of g this paper proves that the same construction produces a set of generators of the invariants of g when g is invertible in r generators of the invariants of g are also explicitly described in the case that g is solvable and g is invertible in r

computation of invariants for reductive groups abstract we will give an algorithm for computing generators of the invariant ring for a given representation of a linearly reductive group the algorithm basically consists of a single grobner basis computation we will also show a connection between some open conjectures in commutative algebra and finding good degree bounds for generating invariants

castelnuovo mumford regularity by approximation abstract the castelnuovo mumford regularity of a module gives a rough measure of its complexity we bound the regularity of a module given a system of approximating modules whose regularities are known such approximations can arise naturally for modules constructed by inductive combinatorial means we apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups

towers and injective cohomology algebras since carlsson s paper 5 and the work of lannes and zarati 9 topo1ogists have learned to recognize many spaces y so h y is injective in u for example if g is a finite group with p sylow subgroup z p kn the h bg is injective further infinite examples include eb z p k and any of the many wedge summands of these spaces a complete classification of u injectives is given in 18 and from this it can be seen that the only such injectives which are not a unbounded are the brown gitler modules these have been adequately discussed elsewhere see 6 among many an immediate consequence of theorem a will be the following result suppose y satisfies the same hypotheses as in theorem a

contractions and deformations suppose that f is a projective birational morphism with at most one dimensional fibres between d dimensional varieties x and y satisfying rf ox oy consider the locus l in y over which f is not an isomorphism taking the scheme theoretic fibre c over any closed point of l we construct algebras afib and acon which prorepresent the functors of commutative deformations of c and noncommutative deformations of the reduced fibre respectively our main theorem is that the algebras acon recover l and in general the commutative deformations of neither c nor the reduced fibre can do this as the d 3 special case this proves the following contraction theorem in a neighbourhood of the point the morphism f contracts a curve without contracting a divisor if and only if the functor of noncommutative deformations of the reduced fibre is representable

on the faber transform and efficient numerical rational approximation algorithms are presented for numerical type m n rational approximation m geqq n 1 which can be effectively applied either in general simply connected regions of the complex plane or as a special case on intervals of the real line in the former case particularly the methods are much more efficient than those previously proposed but they are also competitive even for intervals the approximations are based on the observation that the faber transform of a rational function is itself rational and are of two types a class of pade like approximants and approximations of rational caratheodory fejer type based on the singular value decomposition of a hankel matrix of faber coefficients which generalise those recently introduced for the unit disc by trefethen numer math 37 1981 pp 297 320 these latter particularly give rise to approximations which are sufficiently near to best for practical purposes

on the castelnuovo mumford regularity of rings of polynomial invariants we show that when a group acts on a polynomial ring over a eld the ring of invariants has castelnuovo mumford regularity at most zero as a consequence we prove a well known conjecture that the invariants are always generated in degrees at most n jgj 1 where n 1 is the number of polynomial generators andjgj 1 is the order of the group we also prove some other related conjectures in invariant theory

sharpening the generalized noether bound in the invariant theory of finite groups abstract we consider linear representations of a finite group g on a finite dimensional vector space over a field f in which g is invertible by a theorem due to e noether in char 0 and to fleischmann and fogarty in general the ring of invariants is generated by homogeneous elements of degree at most g schmid domokos and hegedus sharpened noether s bound when g is not cyclic and char f 0 we prove that the sharpened bound holds over general fields if g is not cyclic and g is invertible in f then the ring of invariants is generated by elements of degree at most 3 4 g if g is even and at most 5 8 g if g is odd

two dimensional vector invariant rings of abelian p groups abstract for any faithful representation v of a non trivial p group over a field of characteristic p 0 it is known that the ring of vector invariants of m copies of v is not cohen macaulay if m 3 however much less is known about the case m 2 in this paper we show that if m 2 and the group is an abelian p group then the ring of invariants of 2v is a complete intersection in some cases and is not cohen macaulay in most cases as a corollary we obtain that if the field is f p and the ring of invariants of the representation v is a polynomial ring then the ring of invariants of 2v is either a complete intersection or not cohen macaulay

rings of generalized and stable invariants of pseudoreflections and pseudoreflection groups let g gl n f be a representation of a finite groupgover the field f and f v the space of polynomial functions onv fn we associate togan idealj g f v called the ideal of stable invariants ofg or more precisely if s gl n f is a set of pseudoreflections we associate to s an ideali s f v called the ideal of generalized invariants of s whengis a pseudoreflection group we investigatei s for various choices of s gand the relation betweenj g andi s to a representation g gl n f of a finite group respectively to a set s gl n f of pseudoreflections we also associate a ring formula respectively formula we show that formula is always a polynomial algebra over f and whenever g is generated by semisimple pseudoreflections s that formula

a note on the tau method approximations for the bessel functions y0 z and y1 z abstract this paper is to complete and improve the work reported in 1 2 using the lanczos method in coleman s version to approximate the bessel functions y 0 z and y 1 z we introduce symbolic representations of the scaled faber polynomials on any fan shaped section of the complex plane these faber polynomials are used as the perturbation terms in the method numerical comparison among the power series the chebyshev series and the method are conducted to show the accuracy improvement achieved by this new version of the method some concluding remarks and suggestions on future research are given

degree bounds for syzygies of invariants abstract suppose that g is a linearly reductive group good degree bounds for generators of invariant rings were given in proc amer math soc 129 4 2001 955 here we study minimal free resolutions of invariant rings for finite linearly reductive groups g it was recently shown in adv math 156 1 2000 23 electron res announc amer math soc 7 2001 5 adv math 172 2002 151 that rings of invariants are generated in degree at most the group order g in characteristic 0 this degree bound is a classical result by emmy noether see math ann 77 1916 89 given an invariant ring of a finite linearly reductive group g we prove that the ideal of relations of a minimal set of generators is generated in degree at most 2 g

gopakumar vafa invariants do not determine flops two 3 fold flops are exhibited both of which have precisely one flopping curve one of the two flops is new and is distinct from all known algebraic d 4 flops it is shown that the two flops are neither algebraically nor analytically isomorphic yet their curve counting gopakumar vafa invariants are the same we further show that the contraction algebras associated to both are not isomorphic so the flops are distinguished at this level this shows that the contraction algebra is a finer invariant than various curve counting theories and it also provides more evidence for the proposed analytic classification of 3 fold flops via contraction algebras

homomorphisms localizations and a new algorithm to construct invariant rings of finite groups abstract let g be a finite group acting on a polynomial ring a over the field k and let a g denote the corresponding ring of invariants let b be the subalgebra of a g generated by all homogeneous elements of degree less than or equal to the group order g then in general b is not equal to a g if the characteristic of k divides g however we prove that the field of fractions quot b coincides with the field of invariants quot a g quot a g we also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for a g we prove that there is always a nonzero transfer c a g of degree g such that the localization a g c can be generated by fractions of homogeneous invariants of degrees less than 2 g 1 if a sym v f g with finite dimensional f g module v then c can be chosen in degree one and 2 g 1 can be replaced by g let n denote the image of the classical noether homomorphism see the definition in the paper we prove that n contains the transfer ideal and thus can be used to calculate generators for a g by standard elimination techniques using grobner bases this provides a new construction algorithm for a g

real vs complex rational chebyshev approximation on an interval if e c l 1 is real valued let er f and be the errors in best approximation to in the supremum norm by rational functions of type but for no n 1 is it known whether ym inff e f er f is zero or strictly positive here we show that both are possible y0i 0 but y 0 for n m 3 related results are obtained for approximation on regions in the plane

rings of invariants of modular p groups which are hypersurfaces abstract for l a finite non modular group whose invariants form a polynomial ring and h a subgroup of l containing the derived group of l nakajima found necessary and sufficient conditions on h for its invariant ring s h to be a hypersurface in a crucial step of his proof he showed that if s h is a hypersurface then between h and l there is a group g with polynomial invariant ring such that s h s g b for g a finite modular p group over f p with polynomial invariant ring and h a subgroup of g containing the derived group of g we find necessary and sufficient conditions on h to ensure that s h s g b

vector invariants for the two dimensional modular representation of a cyclic group of prime order in this paper we study the vector invariants of the 2 dimensional indecomposable representation v2 of the cyclic group cp of order p over a field f of characteristic p f mv2 cp this ring of invariants was first studied by david richman 1990 20 who showed that the ring required a generator of degree m p 1 thus demonstrating that the result of noether in characteristic 0 that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group does not extend to the modular case he also conjectured that a certain set of invariants was a generating set with a proof in the case p 2 this conjecture was proved by campbell and hughes 1997 in 3 later shank and wehlau 2002 in 24 determined which elements in richman s generating set were redundant thereby producing a minimal generating set we give a new proof of the result of campbell and hughes shank and wehlau giving a minimal algebra generating set for the ring of invariants f mv2 cp in fact our proof does much more we show that our minimal generating set is also a sagbi basis for f mv2 cp further our results provide a procedure for finding an explicit decomposition of f mv2 into a direct sum of indecomposable cp modules finally noting that our representation of cp on v2 is as the p sylow subgroup of sl2 fp we describe a generating set for the ring of invariants f mv2 sl2 fp and show that p m 2 p 1 is an upper bound for the noether number for m 2

a survey of faber methods in numerical approximation abstract although well known in function theory faber polynomials and the faber transform have only recently been systematically applied for numerical polynomial and rational approximation of complex valued functions this paper surveys some recent work of this author and others in this area

zero separating invariants for finite groups abstract we fix a field k of characteristic p for a finite group g denote by g and g respectively the minimal number d such that for every finite dimensional representation v of g over k and every v v g 0 or v v 0 respectively there exists a homogeneous invariant f k v g of positive degree at most d such that f v 0 let p be a sylow p subgroup of g which we take to be trivial if the group order is not divisible by p we show that g p if n g p p is cyclic we show g n g p if g is p nilpotent and p is not normal in g we show g g l where l is the smallest prime divisor of g these results extend known results in the non modular case to the modular case

a gluing construction for polynomial invariants abstract we give a polynomial gluing construction of two groups g x gl l f and g y gl m f which results in a group g gl l m f whose ring of invariants is isomorphic to the tensor product of the rings of invariants of g x and g y in particular this result allows us to obtain many groups with polynomial rings of invariants including all p groups whose rings of invariants are polynomial over f p and the finite subgroups of gl n f defined by sparsity patterns which generalize many known examples

efficient generation of the ring of invariants we shall use the binet minc formula in the theory of permanents to prove david richman s theorem letgbe a finite group acting ona r a1 ar whereris any commutative ring with 1 g r then the ring of invariantsagis generated overrby g a 11a 22 a rr where 1 r g applications of permanents to other problems related to invariants are given also

three dimensional flops and noncommutative rings limburgs univ ctr dept math b 3590 diepenbeek belgium van den bergh m limburgs univ ctr dept math univ campus b 3590 diepenbeek belgium vdbergh luc ac be

the noether bound in invariant theory of finite groups abstract let r be a commutative ring v a finitely generated free r module and g glr v a finite group acting naturally on the graded symmetric algebra a sym v let ag denote the minimal number m such that the ring ag of invariants can be generated by finitely many elements of degree at most m furthermore let h g be a normal subgroup such that the index g h is invertible in r in this paper we prove the inequality a g a h g h for h 1 and g invertible in r we obtain noether s bound ag g which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order g is invertible in r

real and complex chebyshev approximation on the unit disk and interval we announce the resolution of a number of outstanding questions regarding real and complex chebyshev supremum norm approximation by rational functions on a disk and on an interval the proofs consist mainly of symmetry arguments applied to explicit examples the most important results complex rational best approximations on a disk are in general not unique real functions on an interval can in general be approximated arbitrarily much better by complex rational functions than by real ones details will appear in 3 8

negligibility of projective linear automorphisms abstract letkbe any field s aij 1 i j n 1 gln 1 k k k x1 xn 1 xn xn 1 y1 yn 1 yn yn 1 a field extension of transcendence degree 2noverk the diagonal action ofsonkis defined as equation equation theorem there existz1 zn ksuch thatk k x1 xn 1 xn xn 1 z1 zn ands zi zifor 1 i n in particular ifk x1 xn 1 xn xn 1 s is rational purely transcendental overk so isk s overk

coinvariants and the regular representation of a cyclic p group we consider an indecomposable representation of a cyclic p group z p r over a field of characteristic p we show that the top degree of the corresponding ring of coinvariants is less than frac r 2 3r p r 2 this bound also applies to the degrees of the generators for the invariant ring of the regular representation

invariants of modular groups for a faithful linear representation of a finite group g over a field of characteristic p we study the ring of invariants we especially study the polynomial and cohen macaulay properties of the invariant ring we first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the hilbert ideal conjecture is true for a class of groups in particular we prove that the conjecture is true for vector invariant rings of abelian reflection p groups then we study the relationships between the invariant ring of g and that of a subgroup of g finally we study the invariant rings of affine groups and show that over a finite field if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group

flops and clusters in the homological minimal model programme suppose that f x rightarrow mathrm spec r is a minimal model of a complete local gorenstein 3 fold where the fibres of f are at most one dimensional so by van den bergh duke math j 122 3 423 455 2004 there is a noncommutative ring lambda derived equivalent to x for any collection of curves above the origin we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of lambda is finite dimensional improving a result of donovan and wemyss contractions and deformations arxiv 1511 00406 we further show that the mutation functor of iyama and wemyss invent math 197 3 521 586 2014 6 is functorially isomorphic to the inverse of the bridgeland chen flop functor in the case when the factor of lambda is finite dimensional these results then allow us to jump between all the minimal models of mathrm spec r in an algorithmic way without having to compute the geometry at each stage we call this process the homological mmp this has several applications in git approaches to derived categories and also to birational geometry first using mutation we are able to compute the full git chamber structure by passing to surfaces we say precisely which chambers give the distinct minimal models and also say which walls give flops and which do not enabling us to prove the craw ishii conjecture in this setting second we are able to precisely count the number of minimal models and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity third we prove a bijective correspondence between maximal modifying r module generators and minimal models and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry this lifts the auslander mckay correspondence to dimension three

generation of invariants abstract let g be any group acting on the weyl algebra w1 c we shall propose a method for finding explicit generators of the ring of invariants w1 c g this method can be generalized to the case of the ring of invariants of an almost normalizing extension r over a commutative ring c under the linear action of a finite group g provided that g is invertible in c and a set of explicit generators of gr r g is known

hypersurfaces in modular invariant theory let the finite group g act linearly on the n dimensional vector space v over the field k of characteristic p 0 suppose h g is a normal subgroup of index l a prime number in this paper we shall study the relationship between the two invariant rings k v h and k v g as a corollary of our main result we get that if k v g is a polynomial algebra and k v h is factorial then k v h is a graded hypersurface algebra

the length classification of threefold flops via noncommutative algebras abstract smooth threefold flops with irreducible centres are classified by the length invariant which takes values 1 2 3 4 5 or 6 this classification by katz and morrison identifies 6 possible partial resolutions of kleinian singularities that can occur as generic hyperplane sections and the simultaneous resolutions associated to such a partial resolution produce the universal flop of length l in this paper we translate these ideas into noncommutative algebra we introduce the universal flopping algebra of length l from which the universal flop of length l can be recovered by a moduli construction and we present each of these algebras as the path algebra of a quiver with relations this explicit realisation can then be used to construct examples of nccrs associated threefold flops of any length as quiver with relations defined by superpotentials to recover the matrix factorisation description of the universal flop conjectured by curto and morrison and to realise examples of contraction algebras

a new degree bound for vector invariants of symmetric groups let r be a commutative ring v a finitely generated free rmodule and g glr v a finite group acting naturally on the graded symmetric algebra a s v let g v g denote the minirnal number m such lxhat the ring ag of invariants can be generated by finitely many elements of degree at most m for g en and v n kn the k fold direct sum of the natural perrnutation module one knows that p v n kn en n provided that n is invertible in r this was used by e noether to prove p v g igi if igl g a in this paper we prove p v n 1z en masc n kn n 1 for arbitrary commutative rings r and show equality for n ps a prime power and a z or any ring with n 1r o ou r results imply g v g maz lgl rank v lgi 1 for any ring with igi g r

a sharp bound for the castelnuovo mumford regularity of subspace arrangements abstract we show that the ideal of an arrangement of d linear subspaces of projective space is d regular in the sense of castelnuovo and mumford answering a question of b sturmfels in particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d

error estimates and evaluation of matrix functions via the faber transform the need to evaluate expressions of the form f a or f a b where f is a nonlinear function a is a large sparse n times n matrix and b is an n vector arises in many applications this paper describes how the faber transform applied to the field of values of a can be used to determine improved error bounds for popular polynomial approximation methods based on the arnoldi process applications of the faber transform to rational approximation methods and in particular to the rational arnoldi process also are discussed

rings of invariants of certainp groups over the fieldfp abstract we use nakajima s j algebra85 1983 253 286 characterization ofp groups with polynomial rings of invariants over the fieldfpto prove that maximal proper subgroups of such groups have rings of invariants that are hypersurfaces we give an explicit construction for the ring of invariants of such a group as a module over the ring of invariants of the total group

global degree bounds and the transfer principle for invariants abstract the main purpose of this paper is to verify a conjecture of derksen and kemper concerned with the boundedness of rings of invariants we shall make use of some results connected with the transfer principle in invariant theory these results are dealt with in some generality because of their independent interest and potential for further applications

the noether numbers and the davenport constants of the groups of order less than 32 abstract the computation of the noether numbers of all groups of order less than thirty two is completed it turns out that for these groups in non modular characteristic the noether number is attained on a multiplicity free representation it is strictly monotone on subgroups and factor groups and it does not depend on the characteristic algorithms are developed and used to determine the small and large davenport constants of these groups for each of these groups the noether number is greater than the small davenport constant whereas the first example of a group whose noether number exceeds the large davenport constant is found answering partially a question posed by geroldinger and grynkiewicz

on generators of the hilbert ideal for cyclic groups in modular invariant theory abstract let g be a cyclic group of order p 2 and v be a faithful indecomposable representation of g over a field f of characteristic p we show that the hilbert ideal of the invariant ring is generated by polynomials of degree at most g whenever dim v 4 p or dim v p 2 2 p proving a conjecture of derksen and kemper in this particular case

the noether number for the groups with a cyclic subgroup of index two abstract for the finite groups with a cyclic subgroup of index two the exact degree bound for the generators of rings of polynomial invariants is determined

on the rate of rational approximation to functions with a finite number of poles abstract the rate of the best rational approximation and interpolation of a meromorphic function on a compact faber set is described in terms of its growth convergence along the rows as well as the columns of the walsh array is considered emphasis is on precise o estimates instead of limit relations

the noether numbers for cyclic groups of prime order abstract the noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants we compute the noether number for an arbitrary representation of a cyclic group of prime order and as a consequence prove the 2 p 3 conjecture

quivers from matrix factorizations we discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity this method also yields explicit geometrical interpretations of d branes i e quiver representations on a resolution given in terms of grassmannians as an example we analyze some non toric singularities which are resolved by a single mathbb p 1 but have length greater than one these examples have a much richer structure than conifolds a picture is proposed that relates matrix factorizations in landau ginzburg theories to the way that matrix factorizations are used in this paper to perform noncommutative resolutions

invariants of finite groups over fields of characteristicp abstract let v denote a finite dimensional k vector space and let g denote a finite group of k linear automorphisms of v let v m denote the direct sum of m copies of v and let g act on the symmetric algebra k v m of v m by the diagonal action on v m a result of noether implies that if char k 0 then k v m g can be generated as a k algebra by polynomials whose degrees are g no matter how large m is this paper proves that this result no longer holds when the characteristic of k divides g more precisely it is proved in this case that there is a positive number depending only on g and char k such that every set of k algebra generators of k v m g contains a generator whose degree is m

a note on the hilbert ideals of a cyclic group of prime order the hilbert ideal is the ideal generated by positive degree invariant polynomials of a finite group for a cyclic group of prime order p we show that the image of the transfer lie in the ideal generated by invariants of degree at most p 1 consequently we show that the hilbert ideal corresponding to an indecomposable representation is generated by polynomials of degree at most p confirming a conjecture of harm derksen and gregor kemper for this case

on vector invariants over finite fields abstract let f denote a finite field and let s m n f denote a set of generators of the invariants of sl n f acting on mn component vectors this paper proves that if m n 1 then s m n f must contain a generator whose degree is greater than or equal to m n 2 f 1 similar results are obtained for the vector invariants of other groups of matrices with entries in f

noncommutative deformations and flops we prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33 fold is representable and hence we associate to every such curve a noncommutative deformation algebra aconacon this new invariant extends and unifies known invariants for flopping curves in 33 folds such as the width of reid and the bidegree of the normal bundle it also applies in the settings of flips and singular schemes we show that the noncommutative deformation algebra aconacon is finite dimensional and give a new way of obtaining the commutative deformations of the curve allowing us to make explicit calculations of these deformations for certain 3 1 3 1 curves we then show how our new invariant aconacon also controls the homological algebra of flops for any flopping curve in a projective 33 fold with only gorenstein terminal singularities we construct an autoequivalence of the derived category of the 33 fold by twisting around a universal family over the noncommutative deformation algebra aconacon and prove that this autoequivalence is an inverse of bridgeland s flop flop functor this demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 33 fold and thus the bridgeland stability manifold

symmetric powers and modular invariants of elementary abelian p groups abstract let e be an elementary abelian p group of order q p n let w be a faithful indecomposable representation of e with dimension 2 over a field k of characteristic p and let v s m w with m q we prove that the rings of invariants k v e are generated by elements of degree q and relative transfers this extends recent work of wehlau 18 on modular invariants of cyclic groups of order p if m p we prove that k v e is generated by invariants of degree 2 q 3 extending a result of fleischmann sezer shank and woodcock 6 for cyclic groups of order p our methods are primarily representation theoretic and along the way we prove that for any d q with d m q s d v is projective relative to the set of subgroups of e with order m and that the sequence s d v d 0 is periodic with period q modulo summands which are projective relative to the same set of subgroups these results extend results of almkvist and fossum 1 on cyclic groups of prime order