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

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

prinjective modules reflection functors quadratic forms and auslander reiten sequences let a b be artinian rings and let a m b be an a b bimodule which is a finitely generated left a module and a finitely generated right b module a right a m b prinjective module is a finitely genrated module x r x a x b x a a m b x b over the triangular matrix ring r a o a m b b such that x a is a projective a module x b is an injective b module and is a b homomorphism we study the category prin r b a of right a m b prinjective modules it is an additive krull schmidt subcategory of mod r closed under extensions

stanley symmetric functions and quiver varieties abstract we prove that the coefficients obtained when stanley symmetric functions are expanded in the basis of schur functions are special cases of the coefficients in the formula for quiver varieties given by the author and w fulton 1999 invent math 135 665 687 and we discuss the relations of this with a conjectured littlewood richardson rule for these quiver coefficients in addition we generalize a factorization formula for schubert polynomials of a product of two permutations

spectral correspondences for affine hecke algebras we introduce the notion of spectral transfer morphisms between normalized affine hecke algebras and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine hecke algebras involved we define a partial ordering on the set of isomorphism classes of normalized affine hecke algebras which plays an important role for the langlands parameters of lusztig s unipotent representations

the derivation of cubic splines with obstacles by methods of optimization and optimal control interpolating splines which are restricted in their movement by the presence of obstacles are investigated for simplicity we mainly treat cubic splines which are required to be non negative the extension to splines of higher order and to certain other forms of obstacles is straightforward methods of optimization and of optimal control are used to obtain necessary optimality criteria these criteria are applied to derive an algorithm to compute splines which are restricted to constant lower or upper bounds there is a numerical example which illustrates the method presented

green functions and character sheaves purely combinatorial way using earlier ideas of p hall in the general case they were introduced 3 in terms of l adic cohomology as values of certain virtual representations of r o t is a maximal torus defined over fq at unipotent elements the characters of rg 0 at arbitrary elements were then expressible in a simple form in terms of green functions of g and of smaller groups one drawback of the definition of 3 was that it did not allow one to compute the green functions except in some simple cases another definition of green functions was proposed by springer first in terms of trigonometric sums on the lie algebra and later in geometrical terms 17 this definition was applicable only in large characteristic and in that case it was identified with the definition in 3 by springer 17 and kazhdan 10 in 12 i observed that green functions for gln can be redefined in terms of intersection cohomology this led me to a new definition of green functions

chern class formulas for quiver varieties our goal in this paper is to prove a formula for the general degeneracy locus xr associated to an oriented quiver of type an if we are given a sequence of vector bundles and vector bundle maps e0 1 e1 2 e2 y y eny1 n en on an algebraic variety x and a collection r rij 0 i j n of non negative integers there is a degeneracy locus xr xr e xr e de ned by xr fx 2 x j rank ei x ej x rij 8 i jg 1 1

homological algebra for affine hecke algebras abstract in this paper we study homological properties of modules over an affine hecke algebra h in particular we prove a comparison result for higher extensions of tempered modules when passing to the schwartz algebra s a certain topological completion of the affine hecke algebra the proof is self contained and based on a direct construction of a bounded contraction of certain standard resolutions of h modules this construction applies for all positive parameters of the affine hecke algebra this is an important feature since it is an ingredient to analyse how the irreducible discrete series representations of h arise in generic families over the parameter space of h for irreducible non simply laced affine hecke algebras this will enable us to give a complete classification of the discrete series characters for all positive parameters we will report on this application in a separate article

nonnegative splines in particular of degree five we investigate splines from a variational point of view which have the following properties a they interpolate given data b they stay nonnegative when the data are positive c for a given integer k 0 they minimize the functional f x int a b x k t 2 dt for all nonnegative interpolating x in w 2 k a b we extend known results for k 2 to larger k in particular to k 3 and we find general necessary conditions for solutions of this restricted minimization problem these conditions imply that solutions are splines in an augmented grid in addition we find that the solutions are in c 2k 2 a b and consist of piecewise polynomials in pi 2k 1 with respect to the augmented grid we find that for general odd k ge3 there will be no boundary arcs which means nontrivial subintervals in which the spline is identically zero we show also that the occurrence of a boundary arc in an interval between two neighboring knots prohibits the existence of any further knot in that interval for k 3 we show that between given neighboring interpolation knots the augmented grid has at most two additional grid points in the case of two interpolation knots the local problem we develop polynomial equations for the additional grid points which can be used directly for numerical computation for the general global problem we propose an algorithm which is based on a newton iteration for the additional grid points and which uses the local spline data as an initial guess there are extensions to other types of constraints such as two sided restrictions also ones which vary from interval to interval as an illustration several numerical examples including graphs of splines manufactured by matlab and fortran programs are given

corrigendum to orbit closures in the enhanced nilpotent cone adv math 219 1 2008 27 62 abstract in this note we point out an error in the proof of theorem 4 7 of p n achar a henderson orbit closures in the enhanced nilpotent cone adv math 219 1 2008 27 62 a statement about the existence of affine pavings for fibres of a certain resolution of singularities of an enhanced nilpotent orbit closure we also give independent proofs of later results that depend on that statement so all other results of that paper remain valid

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

a c 1 rational spline in range restricted interpolation of scattered data a range restricted c1 interpolation local scheme to scattered data is developed c rational spline function classes over triangles and quadrilaterals have been investigated by the first author in this paper we concentrate on the c1 rational spline over triangles and represent it as piecewise a convex combination of three cubic bezier triangular patches further with such an equivalent form of c1 rational spline the construction of a range restricted c1 interpolant subject to polynomial constraint surface of degree up to three is considered in the end of this paper some numerical examples are illustrated

divergence free bases and multiresolution methods for reduced order flow modeling although reduced order modeling methods have e ourished in the study of structural system dynamics similar methodologies for the representation of e ow are few in comparison a novel methodology based on biorthogonal wavelet constructions and multiresolution analysis for order reduction of velocity e elds obtained via digital particle image velocimetry or computational e uid dynamics is presented the derivation algorithmic implementation and experimental verie cation of fast projective e ltering methods based on divergence free wavelets are discussed experimental evidence presented demonstrates the feasibility of achieving several hundredfold compression for typical velocity e elds obtained via experiments using particle image velocimetry the results also suggest the feasibility and amenability of wavelet methods for diverse problems of interest to the aerospace community including 1 experimental data denoising 2 local cross correlation calculations and 3 feature extraction

analytic r groups of affine hecke algebras we define analytic r groups for affine hecke algebras and prove the analog of the knapp stein dimension theorem as a corollary we prove that the commutant algebra of a unitary principal series representation is isomorphic to the complex group algebra of the r group twisted by a certain 2 cocycle gamma for classical hecke algebras we prove that gamma is always trivial

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

a reduction functor tameness and tits form for a class of orders abstract let d be a complete discrete valuation domain which is an algebra over an algebraically closed field k we study in the paper a class of suborders of tiled d orders by means of a rational quadratic form q associated to and a finite poset i having exactly two maximal elements and criteria for finite lattice type and for tame lattice type of are given in terms of the form q and of the category i spr of socle projective k linear representations of the poset i the shape of auslander reiten quiver latt is described in corollary 3 2 a reduction functor h latt i spr preserving representation types is constructed it is shown in corollary 2 10 that for any poset i having exactly two maximal elements there exists a d order in our class an additive functor latt i spr preserving representation types and a poset isomorphism i i

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

a survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

formal degrees of unipotent discrete series representations and the exotic fourier transform we introduce a notion of elliptic fake degrees for unipotent elliptic representations of a semisimple p adic group we conjecture and verify in some cases that the relation between the formal degrees of unipotent discrete series representations of a semisimple p adic group and the elliptic fake degrees is given by the exotic fourier transform matrix introduced by lusztig in the study of representations of finite groups of lie type

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

on the variety of complexes publisher summary this chapter focuses on the characteristic free study of the buchsbaum eisenbud varieties of complexes in detail with the use of young diagrams and young tableaux to give an explicit basis for buchsbaum and eisenbud have studied the varieties of complexes from the point of view of resolutions and have proven the results for the variety w k 1 a young tableau is a filling of the boxes of a young diagram with integers out of 1 2 and is called the shape of the young tableau the chapter presents a proposition that states that ach element in can be written as a linear combination of standard multitableaux

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

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 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

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

characters of springer representations on elliptic conjugacy classes for a weyl group w we investigate simple closed formulas valid on elliptic conjugacy classes for the character of the representation of w in the homology of a springer fiber we also give a formula valid again on elliptic conjugacy classes of the w character of an irreducible discrete series representation with real central character of a graded affine hecke algebra with arbitrary parameters in both cases the pin double cover of w and the dirac operator for graded affine hecke algebras play key roles

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

using the refinement equation for evaluating integrals of wavelets the wavelet galerkin method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets this paper studies the problem from the point of view of stationary subdivision schemes one of the main results is to identify these integrals as components of the unique solution of a certain eigenvector moment problem associated with the coefficients of the refinement equation asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach

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

orbit closures in the enhanced nilpotent cone we study the orbits of g gl v in the enhanced nilpotent cone where is the variety of nilpotent endomorphisms of v these orbits are parametrized by bipartitions of n dimv and we prove that the closure ordering corresponds to a natural partial order on bipartitions moreover we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of kostka polynomials defined by shoji finally we make a connection with kato s exotic nilpotent cone in type c proving that the closure ordering is the same and conjecturing that the intersection cohomology is the same but with degrees doubled

positivity of quiver coefficients through thom polynomials abstract we use the thom polynomial theory developed by feher and rimanyi to prove the component formula for quiver varieties conjectured by knutson miller and shimozono this formula expresses the cohomology class of a quiver variety as a sum of products of schubert polynomials indexed by minimal lace diagrams and implies that the quiver coefficients of buch and fulton are non negative we also apply our methods to give a new proof of the component formula from the grobner degeneration of quiver varieties and to give generating moves for the kms factorizations that form the index set in k theoretic versions of the component formula

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

particle image velocimetry via wavelet analysis novel analytical and image processing methods derived as part of the development of a digital particle image velocimetry system based on multiresolution analysis are presented the derivation of wavelet based multiresolution methods for velocity e eld reconstruction is addressed and the experimental setup used for validation of results is described new techniques are proposed with improved spatial resolution and reliability over some existing methods the techniques are based on wavelet based representations of digital particle image data that are used to calculate spatially localized and frequency localized e ltered correlations of successive images an essential feature of the method is the development of windowed cross correlation expressions for wavelet based expansions that are not orthogonal or biorthogonal over the cross correlation window the methodology makes use of recently introduced ree nable functions and generalized connection coefe cients derived in wavelet based e nite element methods a conventional charge coupled device camera is used with a frame rate of 30 frames s and pixel resolution of 512 480 per frame the images are acquired in pairs at 30 frames s with a user dee ned time delay between image pairs to capture the e owe eld structure evolution the e ow illumination was achieved using a 5 w argon ion laser finally hardware and algorithm performance is demonstrated via sample water tunnel experiments

matrix product states and the quantum max flow min cut conjectures in this note we discuss the geometry of matrix product states with periodic boundary conditions and provide three infinite sequences of examples where the quantum max flow is strictly less than the quantum min cut in the first we fix the underlying graph to be a 4 cycle and verify a prediction of hastings that inequality occurs for infinitely many bond dimensions in the second we generalize this result to a 2d cycle in the third we show that the 2d cycle with periodic boundary conditions gives inequality for all d when all bond dimensions equal two namely a gap of at least 2d 2 between the quantum max flow and the quantum min cut

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

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

linear preservers and representations with a 1 dimensional ring of invariants we determine the group of linear transformations on a vector space v that preserve a polynomial function f on v for several interesting pairs v f using the theory of semisimple algebraic groups in an 1897 paper fro97 frobenius proved that every linear transformation of the n by n real matrices that preserves the determinant is of the form x 7 axb or x 7 ax t b for some a b gln r such that det ab 1 this is the basic example of a linear preserver problem lpp one is given a finite dimensional vector space v over a field k and a polynomial function f v k and one wants to determine the linear transformations of v that preserve f since frobenius many such prob lems have been solved see for example the surveys lp01 pll 92 lt92 and mar62 we develop here a general method that solves several new problems see examples 3 14 and 3 15 and corollaries 7 10 8 3 8 7 8 5 8 8 and 8 11 our method is to introduce an auxiliary group g gl v that is semisimple and such that v is an irreducible representation or weyl module of g in section 2 we determine the normalizer ngl v g of g in gl v we prove in theorem 3 2 that this subgroup equals the stabilizer of a closed g orbit o in the projective space p v i e stabgl v o ngl v g using this result in sections 7 and 8 we solve two families of lpps by reducing the problems in each family to determining this stabilizer these two families consist of representations v with a 1 dimensional ring of g invariant functions generated by f and are examples of prehomogenous vector spaces of parabolic type the two families correspond to the cases where the unipotent radical u of the parabolic subgroup is abelian i e u u 0 or u u is 1 dimensional respectively and we use the general results on representations in these families from rrs92 roh93 and hel12 besides obtaining new results we also recover many known solutions to linear preserver problems the generality of our method is in contrast to many of the proofs in the literature which typically are highly dependent on the particular choice of v and f the arguments in pd95 and gur97 are notable exceptions further we require only very weak assumptions about the field k at most we require that the characteristic is 6 2 3 and determine the preserver precisely and not just its identity component or lie algebra

the existence of hall polynomials for posets of finite prinjective type abstract we prove the existence of hall polynomials for prinjective representations of finite partially ordered sets of finite prinjective type in section 4 we shortly discuss consequences of the existence of hall polynomials in particular we are able to define a generic ringel hall algebra for prinjective representations of posets of finite prinjective type

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

discrete series characters for affine hecke algebras and their formal degrees we introduce the generic central character of an irreducible discrete series representation of an affine hecke algebra using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine hecke algebras except for the types e n 1 n 6 7 8 with arbitrary positive parameters and we prove an explicit product formula for their formal degrees in all cases

an overview of mathematical issues arising in the geometric complexity theory approach to mathbf vp neq mathbf vnp we discuss the geometry of orbit closures and the asymptotic behavior of kronecker coefficients in the context of the geometric complexity theory program to prove a variant of valiant s algebraic analogue of the mathbf p neq mathbf np conjecture we also describe the precise separation of complexity classes that their program proposes to demonstrate

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

topology of tensor ranks abstract we study path connectedness and homotopy groups of sets of tensors defined by tensor rank border rank multilinear rank as well as their symmetric counterparts for symmetric tensors we show that over c the set of rank r tensors and the set of symmetric rank r symmetric tensors are both path connected if r is not more than the complex generic rank these results also extend to border rank and symmetric border rank over c over r the set of rank r tensors is path connected if it has the expected dimension but the corresponding result for symmetric rank r symmetric d tensors depends on the order d connected when d is odd but not when d is even border rank and symmetric border rank over r have essentially the same path connectedness properties as rank and symmetric rank over r when r is greater than the complex generic rank we are unable to discern any general pattern for example we show that border rank three tensors in r 2 r 2 r 2 fall into four connected components for multilinear rank the manifold of d tensors of multilinear rank r 1 r d in c n 1 c n d is always path connected and the same is true in r n 1 r n d unless n i r i j i r j for some i 1 d beyond path connectedness we determine over both r and c the fundamental and higher homotopy groups of the set of tensors of a fixed small rank and taking advantage of bott periodicity those of the manifold of tensors of a fixed multilinear rank we also obtain analogues of these results for symmetric tensors of a fixed symmetric rank or a fixed symmetric multilinear rank

deformations of nilpotent cones and springer correspondences let g mathop sp 2n be the symplectic group over bbb z we present a certain kind of deformation of the nilpotent cone of g with g action this enables us to make direct links between the springer correspondence of frak sp 2n over bbb c that over characteristic two and our exotic springer correspondence as a by product we obtain a complete description of our exotic springer correspondence

geometric complexity theory ii towards explicit obstructions for embeddings among class varieties in k d mulmuley and m sohoni siam j comput 31 2001 pp 496 526 henceforth referred to as part i we suggested an approach to the p vs np and related lower bound problems in complexity theory through geometric invariant theory in particular it reduces the arithmetic characteristic zero version of the np not subseteq p conjecture to the problem of showing that a variety associated with the complexity class np cannot be embedded in a variety associated with the complexity class p we shall call these class varieties associated with the complexity classes p and np this paper develops this approach further reducing these lower bound problems which are all nonexistence problems to some existence problems specifically to proving the existence of obstructions to such embeddings among class varieties it gives two results towards explicit construction of such obstructions the first result is a generalization of the borel weil theorem to a class of orbit closures which include class varieties the second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class nc these results indicate that the fundamental lower bound problems in complexity theory are in turn intimately linked with explicit construction problems in algebraic geometry and representation theory the results here were announced in k d mulmuley and m sohoni in advances in algebra and geometry hyderabad 2001 hindustan book agency new delhi india 2003 pp 239 261

range restricted interpolation using gregory s rational cubic splines abstract the construction of range restricted univariate and bivariate interpolants to gridded data is considered we apply gregory s rational cubic c1 splines as well as related rational quintic c2 splines assume that the lower and upper obstacles are compatible with the data set then the tension parameters occurring in the mentioned spline classes can be always determined in such a way that range restricted interpolation is successful

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

convergence of newton s method for convex best interpolation in this paper we consider the problem of finding a convex function which interpolates given points and has a minimal l2 norm of the second derivative this problem reduces to a system of equations involving semismooth functions we study a newton type method utilizing clarke s generalized jacobian and prove that its local convergence is superlinear for a special choice of a matrix in the generalized jacobian we obtain the newton method proposed by irvine et al and settle the question of its convergence by using a line search strategy we present a global extension of the newton method considered the efficiency of the proposed global strategy is confirmed with numerical experiments

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

a vanishing theorem for dirac cohomology of standard modules abstract this paper studies the dirac cohomology of standard modules in the setting of graded hecke algebras with geometric parameters we prove that the dirac cohomology of a standard module vanishes if and only if the module is not twisted elliptic tempered the proof makes use of two deep results one is some structural information from the generalized springer correspondence obtained by s kato and lusztig another one is a computation of the dirac cohomology of tempered modules by barbasch ciubotaru trapa and ciubotaru we apply our result to compute the dirac cohomology of ladder representations for type a n for each of such representations with non zero dirac cohomology we associate to a canonical weyl group representation we use the dirac cohomology to conclude that such representations appear with multiplicity one

computing algebraic formulas using a constant number of registers it is shown that over an arbitrary ring the functions computed by polynomial size algebraic formulas are also computed by polynomial length algebraic straight line programs that use only three registers this was previously known for boolean formulas d a barrington j comput system sci 38 1989 pp 150 164 which are equivalent to algebraic formulas over the ring gf 2 for formulas over arbitrary rings the result is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas in order to obtain polynomial length straight line programs moreover the straight line programs that arise in these constructions have the property that they consist of statements whose actions on the registers are linear and bijective a consequence of this is that the problem of determining the iterated product of n3 times 3 matrices is complete under p projections for algebraic nc 1 also when the ring is gf 2 the programs that arise in the c

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed