pinecone/core-2020-05-10-deduplication huggingface dataset:
Ideals in Triangulated Categories: Phantoms, Ghosts and Skeleta

AbstractWe begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik–Schnirelmann category of a space, and we calculate this stable analogue for low-dimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case ofA∞modules over anA∞ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps fromXtoYcan always be described as alim←1group. The last two sections focus on algebraic examples. In the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of Kelly on the vanishing of composites of such maps. We also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non-trivially
Linear Maps betweenC*-Algebras Whose Adjoints Preserve Extreme Points of the Dual Ball

AbstractWe give a structural characterization of linear operators from oneC*-algebra into another whose adjoints map extreme points of the dual ball onto extreme points. We show that up to a ∗-isomorphism, such a map admits of a decomposition into a degenerate and a non-degenerate part, the non-degenerate part of which appears as a Jordan ∗-morphism followed by a “rotation” and then a reduction. In the case of maps whose adjoints preserve pure states, the degenerate part does not appear, and the “rotation” is but the identity. In this context the results concerning such pure state preserving maps depend on and complement those of Størmer [1963,Acta Math.110, 233–278, 5.6 and 5.7]. In conclusion we consider the action of maps with “extreme point preserving” adjoints on some specificC*-algebras
Convexity Properties of the Moment Mapping Re-examined

AbstractConsider a Hamiltonian action of a compact Lie group on a compact symplectic manifold. A theorem of Kirwan's says that the image of the momentum mapping intersects the positive Weyl chamber in a convex polytope. I present a new proof of Kirwan's theorem, which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold. It also applies to some interesting classes of noncompact or singular Hamiltonian spaces, such as cotangent bundles and complex affine varieties
Recovering Symbolically Dated, Rooted Trees from Symbolic Ultrametrics

AbstractA well known result from cluster theory states that there is a 1-to-1 correspondence between dated, compact, rooted trees and ultrametrics. In this paper, we generalize this result yielding a canonical 1-to-1 correspondence between symbolically dated trees and symbolic ultrametrics, using an arbitrary set as the set of (possible) dates or values. It turns out that a rather unexpected new condition is needed to properly define symbolic ultrametrics so that the above correspondence holds. In the second part of the paper, we use our main result to derive, as a corollary, a theorem by H. J. Bandelt and M. A. Steel regarding a canonical 1-to-1 correspondence between additive trees and metrics satisfying the 4-point condition, both taking their values in abelian monoids
Pattern Avoidance and Rational Smoothness of Schubert Varieties

AbstractLetwbe an element of the Weyl groupSn, and letXwbe the Schubert variety associated towin the flag manifoldSLn(C)/B. Lakshmibai and Sandhya showed thatXwis smooth if and only ifwavoids the patterns 4231 and 3412. Using two tests for rational smoothness from Carrell and Peterson, we show that rational smoothness ofXwis characterized by pattern avoidance for typesBandCas well. A key step in the proof of this result is a sequence of rules for factoring the Poincaré polynomials for the cohomology ring ofXw, generalizing the work of Gasharov. Patterns can also be used to characterize actual smoothness of Schubert varietie
Representations of Symmetric Groups and Free Probability

AbstractWe consider representations of symmetric groupsSqfor largeq. We give the asymptotic behaviour of the characters when the corresponding Young diagrams, rescaled by a factorq−1/2, converge to some prescribed shape. This behaviour can be expressed in terms of the free cumulants for a probability measure associated with the limit shape of the diagram. We also show that the basic operations of representation theory, like taking tensor products, restriction, or induction, have a limiting behavior which can be described using the free probability theory of D. Voiculescu
The Combinatorics behind Number-Theoretic Sieves

AbstractEver since Viggo Brun's pioneering work, number theorists have developed increasingly sophisticated refinements of the sieve of Eratosthenes to attack problems such as the twin prime conjecture and Goldbach's conjecture. Ever since Gian-Carlo Rota's pioneering work, combinatorialists have found more and more areas of combinatorics where sieve methods (or Möbius inversion) are applicable. Unfortunately, these two developments have proceeded largely independently of each other even though they are closely related. This paper begins the process of bridging the gap between them by showing that much of the theory behind the number-theoretic refinements carries over readily to many combinatorial settings. The hope is that this will result in new approaches to and more powerful tools for sieve problems in combinatorics such as the computation of chromatic polynomials, the enumeration of permutations with restricted position, and the enumeration of regions in hyperplane arrangements
All Abelian Quotient C.I.-Singularities Admit Projective Crepant Resolutions in All Dimensions

AbstractFor Gorenstein quotient spaces Cd/G, a direct generalization of the classical McKay correspondence in dimensionsd⩾4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces that would satisfy the above property. We prove that the underlying spaces of all Gorenstein abelian quotient singularities, which are embeddable as complete intersections of hypersurfaces in an affine space, have torus-equivariant projective crepant resolutions in all dimensions. We use techniques from toric and discrete geometry
Quasifinite Representations of Classical Lie Subalgebras of W1+∞

AbstractWe show that there are precisely two, up to conjugation, anti-involutionsσ±of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension D±of the Lie subalgebra of this algebra fixed by −σ±, and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the algebra C[u]/(um+1) and its classical Lie subalgebras ofB,CandDtypes. Character formulas forpositive primitiverepresentations of D±(including all the unitary ones) are obtained. We also realize a class of primitive representations of D±in terms of free fields and establish a number of duality results between these primitive representations and finite-dimensional irreducible representations of finite-dimensional Lie groups and supergroups. We show that the vacuum moduleVcof D±carries a vertex algebra structure and establish a relationship betweenVcforc∈12Z and W-algebras
Canonical Basis and Macdonald Polynomials

AbstractIn the basic representation of[formula]realized via the algebra of symmetric functions, we compare the canonical basis with the basis of Macdonald polynomials wheret=q2. We show that the Macdonald polynomials are invariant with respect to the bar involution defined abstractly on the representations of quantum groups. We also prove that the Macdonald scalar product coincides with the abstract Kashiwara form. This implies, in particular, that the Macdonald polynomials form an intermediate basis between the canonical basis and the dual canonical basis, and the coefficients of the transition matrix are necessarily bar invariant. We also verify that the Macdonald polynomials (after a natural rescaling) form a sublattice in the canonical basis lattice which is invariant under the divided powers action. The transition matrix with respect to this rescaling is integral and we conjecture its positivity. For levelk, we expect a similar relation between the canonical basis and Macdonald polynomials withq2=tk
Representation of Symmetric Functions as Gram Determinants

AbstractWe introduce a representation of symmetric functions as determinants of Gram matrices on the space of rational functions in one variable. We use a bilinear map on the space of rational functions that allows us to identify rational functions with linear functionals. Using basic properties of Gram matrices and rational functions we obtain results about symmetric functions, like a general Jacobi–Trudi identity, and an expansion formula for general symmetric functions in terms of Schur functions. Our approach is suitable to study in a unified way the confluent “symmetric” (or pseudo-symmetric) functions proposed by Rota, and several kinds of generalized symmetric functions, like the factorial Schur functions and symmetric functions in several sets of indeterminates
Finite Dimensional Filters with Nonlinear Drift, XI Explicit Solution of the Generalized Kolmogorov Equation in Brockett–Mitter Program

AbstractEver since the technique of the Kalman–Bucy filter was popularized, there has been an intense interest in finding new classes of finite-dimensional recursive filters. In the late 1970s the concept of the estimation algebra of a filtering system was introduced. Brockett, Clark, and Mitter proposed to use the Wei–Norman approach to solve the nonlinear filtering problem. In 1990, Tam, Wong, and Yau presented a rigorous proof of the Brocket–Mitter program which allows one to construct finite-dimensional recursive filters from finite–dimensional estimation algebras. Later Yau wrote down explicitly a system of ordinary differential equations and generalized Kolmogorov equation to which the robust Duncan–Mortenser– Zakai equation can be reduced. Thus there remains three fundamental problems in Brockett–Mitter program. The first is the problem of finding explicit solution to the generalized Kolmogorov equation. The second is the problem of finding real-time solution of a system of ODEs. The third is the Brockett's problem of classification of finite–dimensional estimation algebras. In this paper, we solve the first problem
Acoustic Scattering and the Extended Korteweg– de Vries Hierarchy

AbstractThe acoustic scattering operator on the real line is mapped to a Schrödinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transfor- mation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa–Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals
The Combinatorics of a Class of Representation Functions

AbstractIn the study of the irreducible representations of the unitary groupU(n), one encounters a class of polynomials defined onn2indeterminateszij, 1⩽i,j⩽n, which may be arranged into ann×nmatrix arrayZ=(zij). These polynomials are indexed by double Gelfand patterns, or equivalently, by pairs of column strict Young tableaux of the same shape. Using the double labeling property, one may define a square matrixD(Z), whose elements are the double-indexed polynomials. These matrices possess the remarkable “group multiplication property”D(XY)=D(X)D(Y) for arbitrary matricesXandY, even though these matrices may be singular. ForZ=U∈U(n), these matrices give irreducible unitary representations ofU(n). These results are known, but not always fully proved from the extensive physics literature on representation of the unitary groups, where they are often formulated in terms of the boson calculus, and the multiplication property is unrecognized. The generality of the multiplication property is the key to understanding group representation theory from the purview of combinatorics. The combinatorial structure of the general polynomials is expected to be intricate, and in this paper, we take the first step to explore the combinatorial aspects of a special class which can be defined in terms of the set of integral matrices with given row and column sums. These special polynomials are denoted byLα,β(Z), whereαandβare integral vectors representing the row sums and column sums of a class of integral matrices. We present a combinatorial interpretation of the multiplicative properties of these polynomials. We also point out the connections with MacMahon's Master Theorem and Schwinger's inner product formula, which is essentially equivalent to MacMahon's Master Theorem. Finally, we give a formula for the double Pfaffian, which is crucial in the studies of the generating function of the 3n−jcoefficients in angular momentum theory. We also review the background of the general polynomials and give some of their properties
Bridged Links and Tangle Presentations of Cobordism Categories

AbstractWe develop a calculus of surgery data, calledbridged links, which involves besides links also pairs of balls that describe one-handle attachments. As opposed to the usual link calculi of Kirby and others this description uses only elementary,localmoves (namely modifications and isolated cancellations), and it is valid also on non-simply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3-manifold by doing surgery on any other 3-manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerf-theoretical derivation of presentations of 2+1-dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules. As an application we give a different, more natural proof of the Matveev–Polyak presentations of the mapping class groups, and, furthermore, find systematically surgery presentations of general mapping tori. We discuss a natural extension of the Reshetikhin Turaev invariant to the calculus of bridged links. Invariance follows now—similar as for knot invariants—from simple identifications of the elementary moves with elementary categorical relations for invariances or cointegrals, respectively. Hence, we avoid the lengthy computations and the unnatural Fenn–Rourke reduction of the original proofs. Moreover, we are able to start from a much weaker “modularity”-condition, which implies the one of Turaev. Generalizations of the presentation to cobordisms of surfaces with boundaries are outlined
The Lie Algebra of Homeomorphisms of the Circle

AbstractWe define and study an infinite-dimensional Lie algebrahomeo+which is shown to be naturally associated to thetopologicalLie grouphomeo+of all orientation-preserving homeomorphisms of the circle. Roughly, we rely on the universal decorated Teichmüller theory developed before as motivation to provide Fréchet coordinates on the homogeneous space given byhomeo+modulo the group of real fractional linear transformations, whose corresponding vector fields on the circle we then extend by the usual Lie algebrasl2of real traceless two-by-two matrices in order to definehomeo+. Surprisingly,homeo+turns out to be equal to the algebra of all vector fields on the circle which are “piecewisesl2” in the obvious sense. It is evidently important to consider the relationship between our new Fréchet coordinates and the usual trigonometric functions on the circle, and we undertake here both natural infinitesimal calculations. We finally apply some further previous work in order to give sufficient conditions on the Fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory
An Algebraic Framework for Group Duality

AbstractA Hopf algebra is a pair (A,Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A,Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â,Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â,Δ) is canonically isomorphic with the original multiplier Hopf algebra (A,Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting
Some Minimization Problems for the Free Analogue of the Fisher Information

AbstractWe consider the free non-commutative analogueΦ*, introduced by D. Voiculescu, of the concept of Fisher information for random variables. We determine the minimal possible value ofΦ*(a, a*), ifais a non-commutative random variable subject to the constraint that the distribution ofa*ais prescribed. More generally, we obtain the minimal possible value ofΦ*({aij, a*ij}1⩽i, j⩽d), if {aij}1⩽i, j⩽dis a family of non-commutative random variables such that the distribution ofA*Ais prescribed, whereAis the matrix (aij)di, j=1. Thed×d-generalization is obtained from the cased=1 via a result of independent interest, concerning the minimal value ofΦ*({aij, a*ij}1⩽i, j⩽d) when the matrixA=(aij)di, j=1and its adjoint have a given joint distribution. (A version of this result describes the minimal value ofΦ*({bij}1⩽i, j⩽d) when the matrixB=(bij)di, j=1is selfadjoint and has a given distribution.) We then show how the minimization results obtained forΦ* lead to maximization results concerning the free entropyχ*, also defined by Voiculescu
Basis for Power Series Solutions to Systems of Linear, Constant Coefficient Partial Differential Equations

AbstractUsing the theory of generalized functions and the theory of Fourier transforms in several complex variables, previous authors developed a nonconstructive, integral representation for power series solutions to a given system of linear, constant coefficient partial differential equations (PDEs). For a variety of reasons that theory is quite technical. In this paper we describe an algorithm which gives a constructive, countable basis for the set of power series solutions to a given system of linear, constant coefficient PDEs
Branching Functions ofA(1)n−1and Jantzen–Seitz Problem for Ariki–Koike Algebras

AbstractWe study the restrictions of simple modules of Ariki–Koike algebras Hm(v) with set of parametersv=(ζ; ζv0, …, ζvl−1), whereζis annth root of unity, to their subalgebras Hm−j(v). Using a theorem of Ariki and the crystal basis theory of Kashiwara, we relate this problem to the calculation of tensor product multiplicities of highest weight irreductible representations of the affine Lie algebraA(1)n−1. These multiplicities have a combinatorial description in terms of higher level paths or highest-lift multipartitions. This enables us to solve the Jantzen–Seitz problem for Ariki–Koike algebras, that is, to determine which irreducible representations of Hm(v) restrict to irreducible representations of Hm−1(v). From a combinatorial point of view, this problem is identical to that of computing the tensor product of anA(1)n−1-module of levelland one of level 1. We also consider natural generalisations of the Jantzen–Seitz problem corresponding to the product of a levellmodule by a levell′>1 module, and from the commutativity of tensor products, we deduce a remarkable symmetry between the generalised Jantzen–Seitz conditions and the sets of parameters of the Ariki–Koike algebras
Poisson Harmonic Forms, Kostant Harmonic Forms, and the S1-Equivariant Cohomology ofK/T

AbstractWe characterize the harmonic forms on a flag manifoldK/Tdefined by Kostant in 1963 in terms of a Poisson structure. Namely, they are “Poisson harmonic” with respect to the so-called Bruhat Poisson structure onK/T. This enables us to give Poisson geometrical proofs of many of the special properties of these harmonic forms. In particular, we construct explicit representatives for the Schubert basis of theS1-equivariant cohomology ofK/T, where theS1-action is defined byρ. Using a simple argument in equivariant cohomology, we recover the connection between the Kostant harmonic forms and the Schubert calculus onK/Tthat was found by Kostant and Kumar in 1986. By using a family of symplectic structures onK/T, we also show that the Kostant harmonic forms are limits of the more familiar Hodge harmonic forms with respect to a family of Hermitian metrics onK/T