pinecone/core-2020-05-10-deduplication huggingface dataset:
Characteristic Polynomials of Subspace Arrangements and Finite Fields

AbstractLet A be any subspace arrangement in Rndefined over the integers and let Fqdenote the finite field withqelements. Letqbe a large prime. We prove that the characteristic polynomialχ(A, q) of A counts the number of points in Fnqthat do not lie in any of the subspaces ofA, viewed as subsets of Fnq. This observation, which generalizes a theorem of Blass and Sagan about subarrangements of the Bnarrangement, reduces the computation ofχ(A, q) to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years. The basic idea has its origins in the work of Crapo and Rota (1970). We find new classes of hyperplane arrangements whose characteristic polynomials have simple form and very often factor completely over the nonnegative integers
The Crossing Number of a Graph on a Compact 2-Manifold

AbstractWe introduce a general framework to estimate the crossing number of a graph on a compact 2-manifold in terms of the crossing number of the complete graph of the same size on the same manifold. The bounds are tight within a constant multiplicative factor for many graphs, including hypercubes, some complete bipartite graphs and random graphs. We determine the crossing number of a complete graph, and hence of many other graphs, on a compact 2-manifold up to a polylog genus multiplicative factor. We give estimations for related Turán numbers
Plethystic Formulas for Macdonaldq, t-Kostka Coefficients

AbstractThis work is concerned with the Macdonaldq, t-analogue of the Kostka matrix. This matrix relates the two parameter Macdonald basis {Pμ(x; q, t)} to the modified Schur basis {Sλ[X(1−t)]}. The entries in this matrix, which have come to be denoted byKλ, μ(q, t), have been conjectured by Macdonald to be polynomials inq, twith positive integral coefficients. Our main result here is an algorithm for the construction of explicit formulas for theKλ, μ(q, t). It is shown that this algorithm yields expressions which are polynomials with integer coefficients. Recent work of J. Remmel shows that the resulting formulas also yield positivity of the coefficients for a wide variety of entries in the Macdonaldq, t-Kostka matrix. We also obtain in this manner new explicit expressions for the Macdonald polynomials
Invariants of Finite Groups over Fields of Characteristicp

AbstractLetVdenote a finite-dimensionalKvector space and letGdenote a finite group ofK-linear automorphisms ofV. LetVmdenote the direct sum ofmcopies ofVand letGact on the symmetric algebraK[Vm] ofVmby the diagonal action onVm. A result of Noether implies that, if charK=0, thenK[Vm]Gcan be generated as aK-algebra by polynomials whose degrees are ⩽|G|, no matter how largemis. This paper proves that this result no longer holds when the characteristic ofKdivides |G|. More precisely, it is proved in this case that there is a positive numberα, depending only on |G| and charK, such that every set ofK-algebra generators ofK[Vm]Gcontains a generator whose degree is ⩾αm
Explicit Generators of the Invariants of Finite Groups

AbstractLetRdenote a commutative (and associative) ring with 1 and letAdenote a finitely generated commutativeR-algebra. LetGdenote a finite group ofR-algebra automorphisms ofA. In the case thatRis a field of characteristic 0, Noether constructed a finite set ofR-algebra generators of the invariants ofG. This paper proves that the same construction produces a set of generators of the invariants ofGwhen |G|! is invertible inR. Generators of the invariants ofGare also explicitly described in the case thatGis solvable and |G| is invertible inR
Convexity and Steinitz's Exchange Property

Abstract“Convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz's exchange property. This includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave functions on the integral base polytope of submodular systems. It is shown that a functionωhas the Steinitz exchange property if and only if it can be extended to a concave functionωsuch that the maximizers of (ω+any linear function) form an integral base polytope. A Frenchel-type min–max theorem and discrete separation theorems are established which imply, as immediate consequences, Frank's discrete separation theorem for submodular functions, Edmonds' intersection theorem, Fujishige's Fenchel-type min–max theorem for submodular functions, and also Frank's weight splitting theorem for weighted matroid intersection
Vector Invariants ofU2(Fp) : A Proof of a Conjecture of Richman

AbstractIn this paper we prove a conjecture of D. R. Richman concerning the vector invariants of the groupU2(Fp). LetVbe a vector space of dimension 2 with basisx,yover the fieldFpand letFp[x, y] be the symmetric algebra ofVoverFp. Ifσdenotes a generator ofU2(Fp) then we may assumeσ(x)=xandσ(y)=x+y. LetAnbe the symmetric algebra ofV⊕n. We obtain an automorphism ofAnof order p by using the diagonal action ofσextended to the whole ofAn. The subalgebra of polynomials left invariant by this action is called the ring of vector invariants ofU2(Fp). Richman conjectured that these rings of invariants have certain sets of generators and gave a proof in the casep=2. We prove his conjecture for all primes
A Branching Random Walk Model for Diffusion–Reaction–Convection

AbstractA discrete stochastic model is introduced for populations which are diffusing, interacting, and drifting on an integer lattice in a finite-dimensional space. The model is used to construct a Markov process whose countable state space is a set of location maps, assigning individuals to their positions. The expected population sizes and population densities satisfy partial difference approximations to the non-linear partial differential equations of diffusion–reaction–convection. Constructive convergence estimates give convergence results for the random process and its ensemble average
A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman–Wenzl, and Type A Iwahori–Hecke Algebras

AbstractWe show how the ribbon Hopf algebra structure on the Drinfel'd–Jimbo quantum groups of Types A, B, C, and D can be used to derive formulas giving explicit realizations of the irreducible representations of the Iwahori–Hecke algebras of type A and the Birman–Wenzl algebras. We use this derivation to give explicit realizations of the irreducible representations of the Brauer algebras as well. The derivation is accomplished by way of a combination of techniques from operator algebras, quantum groups, and the theory of 3-manifold invariants. Although our applications are in the cases of the quantum groups of Types A, B, C, and D, most of the aspects of our approach apply in the general setting of ribbon Hopf algebras
Möbius Functions of Lattices

AbstractWe introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any finite lattice. We then show how this result can be used to compute and combinatorially explain the Möbius function in various examples including non-crossing set partitions, shuffle posets, and integer partitions in dominance order. Next we present a generalization of Stanley's theorem that the characteristic polynomial of a semimodular supersolvable lattice factors over the integers. We also give some applications of this second main theorem, including the Tamari lattices
Higher-Dimensional Algebra II. 2-Hilbert Spaces

AbstractA 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an Abelian category enriched over Hilb with a ∗-structure, conjugate-linear on the hom-sets, satisfying ⦠fg, h⦔=⦠g, f*h⦔=⦠f, hg*⦔. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which we call 2-H*-algbebras, braided 2-H*-algebras, and symmetric 2-H*-algbebras, and we describe the relation between these and tangles in two, three, and four dimensions, respectively. We prove a generalized Doplicher–Roberts theorem starting that every symmetric 2-H*-algebra is equivalent to the category Rep(G) of continuous unitary finite-dimensional representations of some compact supergroupoidG. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform whenGis a compact Abelian group. Finally, we characterize Rep(G) by its universal properties whenGis a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2-H*-algebra on one even object of dimensionn
A Combinatorial Description of Knotted Surfaces and Their Isotopies

AbstractWe discuss the diagrammatic theory of knot isotopies in dimension 4. We project a knotted surface to a three-dimensional space and arrange the surface to have generic singularities upon further projection to a plane. We examine the singularities in this plane as an isotopy is performed, and give a finite set of local moves to the singular set that can be used to connect any two isotopic knottings. We show how the notion of projections of isotopies can be used to give a combinatoric description of knotted surfaces that is sufficient for categorical applications. In this description, knotted surfaces are presented as sequences of words in symbols, and there is a complete list of moves among such sequences that relate the symbolic representations of isotopic knotted surfaces
Tori Invariant under an Involutorial Automorphism II

AbstractThe geometry of the orbits of a minimal parabolick-subgroup acting on a symmetrick-variety is essential in several areas, but its main importance is in the study of the representations associated with these symmetrick-varieties (see for example [5, 6, 20, and 31]). Up to an action of the restricted Weyl group ofG, these orbits can be characterized by theHk-conjugacy classes of maximalk-split tori, which are stable underk-involutionθassociated with the symmetrick-variety. HereHis a openk-subgroup of the fixed point group ofθ. This is the second in a series of papers in which we characterize and classify theHk-conjugacy classes of maximalk-split tori. The first paper in this series dealt with the case of algebraically closed fields. In this paper we lay the foundation for a characterization and classification for the case of nonalgebraically closed fields. This includes a partial classification in the cases, where the base field is the real numbers, p-adic numbers, finite fields, and number fields
A Convexity Principle for Interacting Gases

AbstractA new set of inequalities is introduced, based on a novel but natural interpolation between Borel probability measures onRd. Using these estimates in lieu of convexity or rearrangement inequalities, the existence and uniqueness problems are solved for a family of attracting gas models. In these models, the gas interacts with itself through a force which increases with distance and is governed by an equation of stateP=P(ϱ) relating pressure to density.P(ϱ)/ϱ>(d−1)/dis assumed non-decreasing for ad-dimensional gas. By showing that the internal and potential energies for the system are convex functions of the interpolation parameter, an energy minimizing state—unique up to translation—is proven to exist. The concavity established for ¶ρt¶−p/dqas a function oft∈[0,1] generalizes the Brunn–Minkowski inequality from sets to measures
Analytic Discs Attached to Conormal Bundles of CR-Manifolds

AbstractLetMbe a CR-submanifold ofX=Cn,T*MXthe conormal bundle toMinX,zo(resp.po) a point ofM(resp.T*MX:≔defT*MX\{0}) withπ(po)=zo,Aa small analytic disc,C1up to the boundary, attached toMthroughzo,A* aliftofAtoT*XthroughpoattachedT*MX. If the Levi-form ofMhas a constant number of negative eigenvalues in a neighborhood ofp0then we show that in factAis contained inMandA* inT*MX. WhenMis a hypersurface and it is pseudoconvex (i.e., the number of its negative Levi-eigenvalues is identically 0), then the inclusionA⊂Mis classical and follows from the Hopf Lemma applied to a plurisubharmonic exhaustion function as in [D-F]. Note that in this case the existence of a liftA*⊂T*MXis automatic according to [B-F]. 1980 Math. Subject Classification (1985 Revision): 32 F
Pointwise Ergodic Theorems for Radial Averages on the Heisenberg Group

AbstractLetH=Hn=Cn×R denote the Heisenberg group, and letσrdenote the normalized Lebesgue measure on the sphere {(z, 0):|z|=r}. Let (X, B, m) be a standard Borel probability space on whichHacts measurably and ergodically by measure preserving transformations, and letπ(σr) denote the operator canonically associated withσronLp(X). We prove maximal and pointwise ergodic theorems inLp, for radial averagesσron the Heisenberg groupHn,n>1. The results are best possible for actions of the reduced Heisenberg group. The method of proof is to use the spectral theory of the Banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods, in particular Littlewood–Paley–Stein square-functions and analytic interpolation
Cells in Affine Weyl Groups and Tensor Categories

AbstractIn this paper we construct a tensor (or monoidal) category for any two-sided cell in a finite or affine Weyl group. The isomorphism classes of simple objects of this category correspond to the elements of the two-sided cell; the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold, which imitates geometrically the definition of the ringJgiven in G. Lusztig (J. Algebra109, 1987, 536–548). Similarly, we construct, for any left cell, a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse (a right cell). This contains as a special case the category of equivariant perverse sheaves on an affine grassmannian, which, as a consequence of G. Lusztig (Astérisque101–102(1983), 208–209) is a tensor category for convolution (without truncation), closely related to the tensor category of representations of the group dual, in the sense of Langlands, to the group which gives rise to the affine grassmannian. But our construction can be specialized in other ways, and it gives a (conjectural) realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the Langlands dual group
Subspace Arrangements over Finite Fields: Cohomological and Enumerative Aspects

AbstractThe enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the ℓ-adic cohomology of the arrangement (as a variety). We describe the eigenvalues of the Frobenius map acting on this cohomology, which corresponds to a finer decomposition of the zeta function. The ℓ-adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data. Finally, it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases
Integral Geometry in Minkowski Spaces

AbstractWe investigate the possibility of extending classical integral–geometric results, involving lower-dimensional areas, from Euclidean space to Minkowski spaces (finite-dimensional Banach spaces). Of the two natural notions of area in a Minkowski space, due respectively to Busemann and to Holmes and Thompson, the latter turns out to be the more tractable one. For the Holmes–Thompson area, we derive a translative intersection formula and, in the class of hypermetric Minkowski spaces, full analogues of the Crofton formulae for rectifiable sets and for convex bodies. For the Busemannk-area, we give a short proof of the fact that it coincides, fork-rectifiable sets, with thek-dimensional Hausdorff measure induced by the Minkowski metric
The 2-Blocks of the Covering Groups of the Symmetric Groups

AbstractLetŜnbe a double cover of the finite symmetric groupSnof degreen, i.e.,Ŝnhas a central involutionzsuch thatŜn/⦠z⦔≃Sn. An irreducible character ofŜnis calledordinaryorspinaccording to whether it haszin its kernel or not.The purpose of this paper is to determine the distribution of the spin characters ofŜninto 2-blocks. The methods applied here are essentially different from those applied to previous questions of this type. We also discuss some consequences of our main result for the decomposition numbers. An analogue of James' well-known result for the decomposition numbers of the symmetric groups is proved, providing also a generalization of a theorem of Benson [Ben, Theorem 1.2]. In Section 1 we present the background for our results and give some preliminaries. In Section 2 we give an explicit formula for the number of spin characters in a 2-block. We also prove a result about the weight of a block containing a given non-self-associate spin character which will be important for the proof of our theorem on the 2-block distribution of spin characters. Section 3 presents some fundamental combinatorial concepts used in Sections 4 and 5. The theorem concerning the spin characters in a given 2-block is proved in Section 4, and in Section 5 we present our results on the decomposition numbers
Balanced3ϕ2Summation Theorems forU(n) Basic Hypergeometric Series

AbstractIn this paper we begin the theory and application of theU(n+1) generalization of the classical Bailey Transform and Bailey Lemma. We work in the setting of multiple basic hypergeometric series very-well-poised on unitary groupsU(n+1). TheU(n+1) Bailey Transform is obtained from a suitably modifiedU(n+1) terminating very-well-poised4ϕ3summation theorem and termwise transformations. It is then interpreted as a matrix inversion result for two infinite, lower- triangular matrices. This provides a higher-dimensional generalization of Andrews' matrix inversion formulation of the Bailey Transform. As in the classical case, the concept of aU(n+1) Bailey Pair is introduced, and then inverted. ThisU(n+1) inversion enables us to deriveU(n+1) terminating balanced3ϕ2summations directly from suitableU(n+1) terminating very-well-poised6ϕ5summations, and vice-versa. These pairs ofU(n+1) “dual” identities extend the classical one-variable case of Andrews. Special limiting cases of theU(n+1) terminating balanced3ϕ2summation yieldU(n+1)q-Gauss summations, severalU(n+1)q-Chu-Vandermonde summations,U(n+1)q-binomial theorems, and aU(n+1) Cauchy identity. Many other consequences of theU(n+1)6ϕ2and3ϕ2summations, and theU(n+1) Bailey Transform are further developed in our subsequent papers. These include a derivation of theU(n+1) Bailey Lemma, several terminatingU(n+1)q-Whipple transformations,U(n+1)10ϕ9transformations, and nonterminatingU(n+1)q-Whipple transformations. The classical case of all this work, corresponding toA1or equivalentlyU(2), contains a substantial amount of the theory and application of one-variable basic hypergeometric series
Generalized Bibasic Hypergeometric Series and TheirU(n) Extensions

AbstractWe survey summation theorems for generalized bibasic hypergeometric series found recently by Chu and perfected by Macdonald. These contain arbitrary sequences of parameters, and generalize bibasic summation theorems found by Gosper, Gasper, and Rahman. They also contain a host of matrix inversions, for infinite lower-triangular matrices, including those found by Gould, Hsu, Carlitz, and Krattenthaler. Further special cases of Krattenthaler's theorem are Andrews' matrix formulation of the Bailey Transform and Bressoud's and Gasper's inversions. We extend the telescoping arguments used for bibasic summation theorems to provideU(n+1) extensions of Gosper's and Gasper's bibasic summation theorems. Special cases includeU(n+1) extensions of Carlitz's and Gasper's matrix inversions. Further specializations correspond to theU(n+1) Bailey Transform andU(n+1) Bressoud matrix inversion found previously by Milne. Applications includeU(n+1) transformation and expansion formulas, and aU(n+1) extension of aq-analogue of Abel's Binomial Theorem. Thisq-analogue is different from those found by Jackson and Johnson
The UniversalR-Matrix, Burau Representation, and the Melvin–Morton Expansion of the Colored Jones Polynomial

AbstractP. Melvin and H. Morton [9] studied the expansion of the colored Jones polynomial of a knot in powers ofq−1 and color. They conjectured an upper bound on the power of color versus the power of −1. They also conjectured that the bounding line in their expansion generated the inverse Alexander–Conway polynomial. These conjectures were proved by D. Bar-Natan and S. Garoufalidis [1]. We have conjectured [12] that other ‘lines' in the Melvin–Morton expansion are generated by rational functions with integer coefficients whose denominators are powers of the Alexander–Conway polynomial. Here we prove this conjecture by using theR-matrix formula for the colored Jones polynomial and presenting the universalR-matrix as a ‘perturbed' Burau matrix
Cauchy's Integral Theorem on a Finitely Generated, Real, Commutative, and Associative Algebra

AbstractLetR[α]=R[α1,α2,…,αn] (whereα1=1) be a real, unitary, finitely generated, commutative, and associative algebra. We consider functionsf(z)=f(∑ni=1xiαi) which mapR[α]n={∑ni=1aiαi|1⩽i⩽n,ai∈R} intoR[α]=finite dimensional subspaces of {∑k∈Nn0bkαk|bk∈R,k=(k1,…,kn)∈Nn0} whereN0={0,1,2,…}. We impose a total order on an algorithmically defined basisBforR[α]. The resulting algebra and ordered basis will be written as (R[α],<). We then use this basis to define a norm ‖·‖ on (R[α],<). Continuous functions, differentiable functions, and the concept of Riemann integration will then be defined and discussed in this new setting. We then show that ∫γf(z)dz=0 whenf(z) is a continuous and differentiable function defined in a simply connected regionG⊂R[α]n⊂(R[α],<) containing the closed pathγ
Subunit Balls for Symbols of Pseudodifferential Operators

AbstractIn this work we shall study a definition of subunit ball for non-negative symbols of sub-elliptic pseudodifferential operators, extending in phase-space the one given by Stein, Nagel, and Wainger in the differential-operator case. Using microlocal methods introduced by Fefferman and Phong, we prove that these balls can be straightened, by means of a canonical transformation, to contain and be contained in boxes of certain sizes, which we give in terms of the size of the symbol. After microlocalizing the symbol, in Section 3 we define classes of subunit symbols and study some of their basic properties. Then we define the subunit ball. In the last section the main structure theorems, in the (n+n)-dimensional elliptic case and in the (1+1)- and (2+2)-dimensional nonelliptic–nondegenerate cases are stated and proved
Tits Systems in a Class of Kac–Moody Steinberg Groups

AbstractThe construction of the Kac–Moody generalizations of the Steinberg groups has not included the infinite dimensional versions of the great Lie algebras. This is because the construction has assumed the existence of representations with dominant integral highest weights, an assumption which does not hold for those Lie algebras. An extension of the definition of the Weyl group to the adjoint of the Cartan subalgebra permits the construction of the Steinberg groups with Tits systems for that case