pinecone/core-2020-05-10-deduplication huggingface dataset:
Factorizations in Schubert Cells

AbstractFor any reduced decomposition i=(i1, i2, …, iN) of a permutation w and any ring R we construct a bijection Pi:(x1, x2, …, xN)↦Pi1(x1)Pi2(x2)…PiN(xN) from RN to the Schubert cell of w, where Pi1(x1), Pi2(x2), …, PiN(xN) stand for certain elementary matrices satisfying Coxeter-type relations. We show how to factor explicitly any element of a Schubert cell into a product of such matrices. We apply this to give a one-to-one correspondence between the reduced decompositions of w and the injective balanced labellings of the diagram of w, and to characterize commutation classes of reduced decompositions. Copyright 2000 Academic Press.Etant donné un anneau R et une décomposition réduite i=(i1, i2, …, iN) d'une permutation w, nous construisons une bijection Pi:(x1, x2, …, xN)↦Pi1(x1)Pi2(x2)…PiN(xN) de RN vers la cellule de Schubert de w, où Pi1(x1),Pi2(x2), …, PiN(xN) sont des matrices élémentaires vérifiant des relations de type Coxeter. Nous montrons comment factoriser explicitement tout élément d'une cellule de Schubert comme un produit de matrices Pi(x). Nous utilisons ces factorisations pour établir une bijection entre les décompositions réduites de w et les remplissages injectifs équilibrés du diagramme de w et pour caractériser les classes de commutation de décompositions réduites
Galois Connections for Incidence Hopf Algebras of Partially Ordered Sets

AbstractAn important well-known result of Rota describes the relationship between the Möbius functions of two posets related by a Galois connection. We present an analogous result relating the antipodes of the corresponding incidence Hopf algebras, from which the classical formula can be deduced. To motivate the derivation of this more general result, we first observe that a simple conceptual proof of Rota's classical formula can be obtained by interpreting it in terms of bimodules over the incidence algebras. Bimodules correct the apparent lack of functoriality of incidence algebras with respect to monotone maps. The theory of incidence Hopf algebras is reviewed from scratch and centered around the notion of cartesian posets. Also, the universal multiplicative function on a poset is constructed and an analog for antipodes of the classical Möbius inversion formula is presented
Arguesian Identities in the Congruence Variety of Abelian Groups

AbstractA class of identities in the Grassmann–Cayley algebra was found by M. J. Hawrylycz (1994, “Geometric Identities in Invariant Theory,” Ph.D. thesis, Massachusetts Institute of Technology) which yields a large number of geometric theorems on the incidence of subspaces of projective spaces. In a previous paper we established a link between such identities in the Grassmann–Cayley algebra and a family of inequalities in the class of linear lattices, i.e., the lattices of commuting equivalence relations. We proved that a subclass of identities found by Hawrylycz, namely, the Arguesian identities of order 2, can be systematically translated into inequalities holding in linear lattices. However, it is not known whether the Arguesian identities of higher orders have such latticial extensions. In this paper, we give an affirmative answer to the above question in the congruence variety of Abelian groups. We prove that every Arguesian identity, regardless of the order, can be systematically translated into a lattice inequality holding in the congruence variety of Abelian groups. In particular, such a lattice inequality holds in the lattices of subspaces of vector spaces, which are characteristic-free and independent of dimensions. As a consequence, many classical theorems of projective geometry, including Desargues, Bricard, Fontené, and their higher dimensional generalizations, can be extended to lattice inequalities in the general projective spaces, with the variables representing subspaces of arbitrary dimensions
Borel Measure Extensions of Measures Defined on Sub-σ-Algebras

AbstractWe develop a new approach to the measure extension problem, based on nonstandard analysis. The class of thick topological spaces, which includes all locally compact and all K-analytic spaces, is introduced in this paper, and measure extension results of the following type are obtained: If (X, T) is a regular, Lindelöf, and thick space, A⊂σ[T] is a σ-algebra, and ν is a finite measure on A, inner regular with respect to the closed sets in A, then ν has a Radon extension. The methods developed here allow us to improve on previously known extension results
Semicanonical Bases Arising From Enveloping Algebras

AbstractLet U+ be the plus part of the enveloping algebra of a Kac–Moody Lie algebra g with a symmetric Cartan datum. In [L1] we have defined a canonical basis of U+ under the assumption that the Cartan datum is of finite type; this was later generalized to Cartan data of possibly infinite type in [K, L3]. The basis of U+ is obtained from a canonical basis of the quantized version of U+ by specializing the quantum parameter to 1. In [L4] we have constructed a basis of U+ in terms of constructible functions on a Lagrangian variety, assuming that the Cartan datum is of affine type. (The same construction applies for Cartan data of finite type.) This basis will be called the semicanonical basis ofU+. In this paper we extend the definition of semicanonical basis to include the case where the Cartan datum is not necessarily of affine or finite type. While the semicanonical basis is not necessarily the same as the canonical basis, we will show that the semicanonical basis has a number of properties in common with the canonical basis: compatibility with various filtrations of U+, compatibility with the canonical antiautomorphism of U+
On Unbounded p-Summable Fredholm Modules

AbstractWe prove that odd unbounded p-summable Fredholm modules are also bounded p-summable Fredholm modules (this is the odd counterpart of a result of A. Connes for the case of even Fredholm modules). The approach we use is via estimates of the form ‖φ(D)−φ(D0)‖Lp(M,τ)⩽C·‖D−D0‖1/2, where φ(t)=t(1+t2)−1/2, D0=D*0 is an unbounded linear operator affiliated with a semifinite von Neumann algebra M, D−D0 is a bounded self-adjoint linear operator from M and (1+D20)−1/2∈Lp(M,τ), where Lp(M,τ) is a non-commutative Lp-space associated with M. It follows from our results that if p∈(1,∞), then φ(D)−φ(D0) belongs to the space Lp(M,τ)
Representable Multicategories

AbstractWe introduce the notion of representable multicategory, which stands in the same relation to that of monoidal category as fibration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe represent ability in elementary terms via universal arrows. We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2-category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a several object version of representable multicategories
A Coherent Approach to Pseudomonads

AbstractThe formal theory of monads can be developed in any 2-category, but when it comes to pseudomonads, one is forced to move from 2-categories to Gray-categories (semistrict 3-categories). The first steps in developing a formal theory of pseudomonads have been taken by F. Marmolejo, and here we continue that program. We exhibit a Gray-category Psm such that a Gray-functor from Psm to a Gray-category A is precisely a pseudomonad in A; this may be viewed as a complete coherence result for pseudomonads. We then describe the pseudoalgebras for a pseudomonad, the morphisms of pseudoalgebras, and so on, in terms of a weighted limit in the sense of Gray-enriched category theory. We also exhibit a Gray-category Psa such that a Gray-functor from Psa to A is precisely a pseudoadjunction in A, show that every pseudoadjunction induces a pseudomonad, and that every pseudomonad is induced by a pseudoadjunction, provided that A admits the limits referred to above. Finally we define a Gray-category PSM(A) of pseudomonads in A and show that it contains A as a full reflective subcategory, which is coreflective if and only if A admits these same limits
Stably Embedded Surfaces of Bounded Integral Curvature

AbstractWe study the effect of simultaneous bounds on the local L1 norms of the second fundamental form and of the Gauss curvature on the geometry of surfaces Σ embedded in a Riemannian manifold M. Such bounds are natural since (together with an area bound) they amount to a local bound on the area of the manifold of unit normals to Σ, living in the sphere bundle of M. The main technical point is that, given a sequence of surfaces Σi with uniform local bounds of this type converging in the Hausdorff metric topology to a surface Σ0, the length space structures of the Σi converge to that of Σ0 (i.e., distances do not collapse). It follows that under suitable local topological conditions the surface Σ0 is a manifold of bounded integral curvature (MBC), in the sense of Alexandrov, under all smooth complete changes of the metric on M. In particular the boundary of a convex set with nonempty interior in any smooth complete M3 is MBC. Our method is to construct local coordinates for smoothly embedded surfaces Σ satisfying W1, BV bounds based solely on curvature integrals. In the process we demonstrate a regularity property for cut loci of surfaces
A Note on Maximal Hierarchies

AbstractGiven a (finite or infinite) set X, a collection C⊆P(X) of subsets of X is called a hierarchy if it satisfies the condition “C1, C2∈C⇒C1∩C2∈{∅,C1,C2}.” In this note, we characterize maximal hierarchies as set systems that contain the empty set, the full set, and all one-element sets, and—in addition—satisfy either one of the following two requirements: (1) They are ample and finitary hierarchies, i.e., they are hierarchies that satisfy the two conditions (a) “C1, C2∈C and #{C∈C∣C1⊆C⊆C2}=2⇒C2−C1∈C” and (b) “∪C′∈C as well as ∩C′∈C holds for every non-empty chain C′ contained in C.” (2) They are minimal among all ample and finitary set systems that contain the empty set, the full set, and all one-element sets
Smith-Type Inequalities for a Polytope with a Solvable Group of Symmetries

AbstractThe object of this paper is to obtain a set of inequalities relating the face numbers of different orbit types of a simplicial polytope P with a finite solvable group G of linear symmetries. It is assumed that (1) for each subgroup H of G, the fixed point set PH is a subpolytope of P, and (2) the toric variety X(P) associated to P is nonsingular. The action of G on P induces an action on X(P), and we describe a set of Smith-type inequalities between the Betti numbers of X(P)H, where H ranges through the set of subgroups of G. By relating each X(P)H with X(PH), we then express these inequalities in terms of the face numbers of the different orbit types of P and the rank of fixed point sets of certain compact tori. This rank is determined explicitly when G is abelian. Moreover, assumption (2) is removed for a polytope of dimension 2
The Douady Space of a Complex Surface

AbstractWe prove that a standard realization of the direct image complex via the so-called Douady–Barlet (Hilbert-Chow in the algebraic case) morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi-isomorphism of complexes. This is a more precise form of a special case of the decomposition theorems of Beilinson, Bernstein, Deligne, Gabber, and M. Saito of which the proof we present is independent; in addition it is elementary and transparent and does not use either perverse sheaves, the methods of positive characteristic, nor Saito's theory of Mixed Hodge Modules. The proof hinges on the special case of the bi-disk in the complex affine plane where we make explicit use of a construction of Nakajima's and of the corresponding representation-theoretic interpretation foreseen by Vafa and Witten. Some consequences of the decomposition theorem: the Göttsche Formula holds for complex surfaces; interpretation of the rational cohomologies of Douady spaces as a kind of Fock space; new proofs of results of Briançon and Ellingsrud and Stromme on punctual Hilbert schemes; and computation of the mixed Hodge structure of the Douady spaces in the Kähler case. We also derive a natural connection with Equivariant K-Theory for which, in the case of algebraic surfaces, Bezrukavnikov and Ginzburg have proposed a different approach
Zeta Functions of Finite Graphs and Coverings, Part II

AbstractGalois theory for normal unramified coverings of finite irregular graphs (which may have multiedges and loops) is developed. Using Galois theory we provide a construction of intermediate coverings which generalizes the classical Cayley and Schreier graph constructions. Three different analogues of Artin L-functions are attached to these coverings. These three types are based on vertex variables, edge variables, and path variables. Analogues of all the standard Artin L-functions results for number fields are proved here for all three types of L-functions. In particular, we obtain factorization formulas for the zeta functions introduced in Part I as a product of L-functions. It is shown that the path L-functions, which depend only on the rank of the graph, can be specialized to give the edge L-functions, and these in turn can be specialized to give the vertex L-functions. The method of Bass is used to show that Ihara type quadratic formulas hold for vertex L-functions. Finally, we use the theory to give examples of two regular graphs (without multiple edges or loops) having the same vertex zeta functions. These graphs are also isospectral but not isomorphic
On the Quasi-Heredity of Birman–Wenzl Algebras

AbstractIn this paper we consider the Birman–Wenzl algebras over an arbitrary field and prove that they are cellular in the sense of Graham and Lehrer. Furthermore, we determine for which parameters the Birman–Wenzl algebras are quasi-hereditary. So the general theory of cellular algebras and quasi-hereditary algebras applies to Birman–Wenzl algebras. As a consequence, we can determine all irreducible representations of the Birman–Wenzl algebras by linear algebra methods. We prove also that the new Hecke algebras induced from Birman–Wenzl algebras are Frobenius over a field (but not always cellular)
On Cocommutative Hopf Algebras of Finite Representation Type

AbstractLet G be a finite algebraic group, defined over an algebraically closed field k of characteristic p>0. Such a group decomposes into a semidirect product G=G0×Gred with a constant group Gred and a normal infinitesimal subgroup G0. If the principal block B0(G) of the group algebra H(G) has finite representation type, then both constituents have the same property, with at least one of them being semisimple. We determine the structure of the infinitesimal constituent G0 up to the classification of V-uniserial groups
Categories of Representations of Coalgebroids

AbstractFundamental results of Tannaka duality include the reconstruction of a coalgebra in the category of vector spaces from its category of representations equipped with the forgetful functor, and the characterization of those categories equipped with a functor into the category of vector spaces which are equivalent to the category of representations of some coalgebra. This paper generalizes these results by replacing the category of vector spaces by an arbitrary monoidal category and replacing coalgebras by their several-object analogue. Applications include providing sufficient conditions on a small R-linear category to ensure that its category of presheaves is equivalent to the category of representations of some R-coalgebra. In particular we construct an R-coalgebra whose category of representations is equivalent to the category of chain complexes over R
The Minimal Primitive Spectrum of the Enveloping Algebra of the Lie Superalgebra osp(1, 2l)

AbstractA well-known theorem of Duflo, the “annihilation theorem,” claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is centrally generated. For the Lie superalgebra osp(1, 2l), this result does not hold. In this article, we introduce a “correct” analogue of the centre for which the annihilation theorem does hold in the case osp(1, 2l). This substitute of the centre is the centralizer of the even part of the enveloping algebra. This algebra shares some nice properties with the centre. As a consequence of the annihilation theorem we obtain the description of the minimal primitive spectrum of the enveloping algebra of the Lie superalgebra osp(1, 2l). We also deduce a criterium for a osp(1, 2l)-Verma module to be a direct sum of sp(2l)-Verma modules
Counting Lattice Points of Rational Polyhedra

AbstractThe generating function F(P)=∑α∈P∩ZNxα for a rational polytope P carries all essential information of P. In this paper we show that for any positive integer n, the generating function F(P, n) of nP={nx:x∈P} can be written as F(P, n)=∑α∈APα(n)xnα, where A is the set of all vertices of P and each Pα(n) is a certain periodic function of n. The Ehrhart reciprocity law follows automatically from the above formula. We also present a formula for the coefficients of Ehrhart polynomials in terms of elementary symmetric functions
Some New Results on Eigenvectors via Dimension, Diameter, and Ricci Curvature

AbstractWe generalise for a general symmetric elliptic operator the different notions of dimension, diameter, and Ricci curvature, which coincide with the usual notions in the case of the Laplace–Beltrami operators on Riemannian manifolds. If λ1 denotes the spectral gap, that is the first nonzero eigenvalue, we investigate in this paper the best lower bound on λ1 one can obtain under an upper bound on the dimension, an upper bound on the diameter, and a lower bound of the Ricci curvature. Two cases are known: namely if the Ricci curvature is bounded below by a constant R>0, then λ1⩾nR/(n−1), and this estimate is sharp for the n-dimensional spheres (Lichnerowicz's bound). If the Ricci curvature is bounded below by zero, then Zhong–Yang's estimate asserts that λ1⩾π2d2, where d is an upper bound on the diameter. This estimate is sharp for the 1-dimensional torus. In the general case, many interesting estimates have been obtained. This paper provides a general optimal comparison result for λ1 which unifies and sharpens Lichnerowicz and Zhong–Yang's estimates, together with other comparison results concerning the range of the associated eigenfunctions and their derivatives
Ribbon Operators and Hall–Littlewood Symmetric Functions

AbstractGiven a partition λ=(λ1, λ2, …, λk), let λrc=(λ2−1, λ3−1, …, λk−1). It is easily seen that the diagram λ/λrc is connected and has no 2×2 subdiagrams, we shall call it a ribbon. To each ribbon R, we associate a symmetric function operator SR. We may define the major index of a ribbon maj(R) to be the major index of any permutation that fits the ribbon. This paper is concerned with the operator Hq1k=∑Rqmaj(R)SR where the sum is over all 2k−1 ribbons of size k. We show here that Hq1k has truly remarkable properties, in particular, it is a Rodriguez operator that adds a column to the Hall–Littlewood symmetric functions. We believe that some of the tools we introduce here to prove our results should also be of independent interest and may be useful for establishing further symmetric function identities
Free Stochastic Measures via Noncrossing Partitions

AbstractWe consider free multiple stochastic measures in the combinatorial framework of the lattice of all diagonals of an n-dimensional space. In this free case, one can restrict the analysis to only the noncrossing diagonals. We give definitions of what free multiple stochastic measures are, and calculate them for the free Poisson and free compound Poisson processes. We also derive general combinatorial Itô-type relationships between free stochastic measures of different orders. These allow us to calculate, for example, free Poisson–Charlier polynomials, which are the orthogonal polynomials with respect to the free Poisson measure
Computations of Quandle Cocycle Invariants of Knotted Curves and Surfaces

AbstractState-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors (Quandle cohomology and state-sum invariants of knotted curves and surfaces, preprint). In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed
Entropy and Dyadic Equivalence of Random Walks on a Random Scenery

AbstractFor any 1–1 measure-preserving map T of a probability space, consider the [T, T−1] endomorphism and the corresponding decreasing sequence of σ-algebras. We demonstrate that if the decreasing sequence of σ-algebras generated by [T, T−1] and [S, S−1] are isomorphic, then T and S must have equal entropies. As a consequence, if the [T, T−1] endomorphism is isomorphic to the [S, S−1] endomorphism, then the entropy of T is equal to the entropy of S. Central to this is a relationship between Feldman's f metric (1976, Israel J. Math.24, 16–38) and Vershik's v metric (1970, Dokl. Akad. Nauk SSSR193, 748–751)
Hedgehogs and Zonoids

AbstractLet R be a continuous real function on the unit sphere Sn of (n+1)-dimensional Euclidean space Rn+1. We prove that the maph:Sn→R,p↦∫Sn|〈p, q〉|R(q)dσ(q),where 〈·, ·〉 is the standard inner product and σ the spherical Lebesgue measure, is of class C2. It follows that the boundaries of zonoids (resp. generalized zonoids), whose generating measure have a continuous density with respect to σ, can be considered as hedgehogs (envelopes parametrized by their Gauss map) with a C2 support function. We deduce a local property for such zonoids. We give a formula for the curvature function of the hedgehog defined by h and we deduce a necessary and sufficient condition for h being the support function of a convex body of class C2+. We define projection hedgehogs (resp. mixed projection hedgehogs) and interpret their support functions in terms of n-dimensional volume (resp. mixed volume). Finally, we consider the extension of the classical Minkowski problem to hedgehogs
The Descriptive Complexity of Brownian Motion

AbstractA continuous function x on the unit interval is a generic Brownian motion when every probabilistic event which holds almost surely with respect to the Wiener measure is reflected in x, provided that the event has a suitably effective description. We show that a generic one-dimensional Brownian motion can be computed from an infinite binary string which is complex in the sense of Kolmogorov–Chaitin. Conversely, one can construct a Kolmogorov–Chaitin random string from the values at the rational numbers of a generic Brownian motion. In this way, we construct a recursive isomorphism between encoded versions of generic Brownian motions and Kolmogorov–Chaitin random reals