pinecone/core-2020-05-10-deduplication huggingface dataset:
Groups Satisfying Semigroup Laws, and Nilpotent-by-Burnside Varieties

AbstractWe investigate the structure of groups satisfying apositive law, that is, an identity of the formu≡v, whereuandvare positive words. The main question here is whether all such groups are nilpotent-by-finite exponent. We answer this question affirmatively for a large class C of groups including soluble and residually finite groups, showing that moreover the nilpotency class and the finite exponent in question are bounded solely in terms of the length of the positive law. It follows, in particular, that if a variety of groups is locally nilpotent-by-finite, then it must in fact be contained in the product of a nilpotent variety by a locally finite variety of finite exponent. We deduce various other corollaries, for instance, that a torsion-free, residually finite,n-Engel group is nilpotent of class bounded in terms ofn. We also consider incidentally a question of Bergman as to whether a positive law holding in a generating subsemigroup of a group must in fact be a law in the whole group, showing that it has an affirmative answer for soluble groups
Stability for an Abelian Category

AbstractThe main goal of the article is to give the definition of algebraic stability that would permit us to consider stability, not only for algebraic vector bundles or torsion-free coherent sheaves, but for the abelian category of coherent sheaves or for whatever abelian category. We present an axiomatic description of the algebraic stability on an abelian category and prove some general results. Then the stability for coherent sheaves on a projective variety is constructed which generalizes Gieseker stability. Stabilities for graded modules and for quiver representations are also discussed. The constructions could be used for other abelian categories as well
Parabolic Subgroups of Artin Groups

AbstractLet (A,Σ) be an Artin system. ForX⊆Σ, we denote byAXthe subgroup ofAgenerated byX. Such a group is called a parabolic subgroup ofA. We reprove Van der Lek's theorem: “a parabolic subgroup of an Artin group is an Artin group.” We give an algorithm which decides whether two parabolic subgroups of an Artin group are conjugate. LetAbe a finite type Artin group, and letAXbe a parabolic subgroup with connected associated Coxeter graph. The quasi-centralizer ofAXinAis the set of β inAsuch that βXβ−1=X. We prove that the commensurator ofAXinAis equal to the normalizer ofAXinA, and that this group is generated byAXand the quasi-centralizer ofAXinA. Moreover, ifAXis not of typeDl(l≥4 andleven), then this group is generated byAXand the centralizer ofAXinA
A Positive Characteristic Resolution Problem for SL(3,k)

AbstractIn 1975,1resolved an irreducible representation of a complex semisimple Lie algebra by Verma modules indexed by the Weyl group. This resolution is now commonly referred to as the Bernstein–Gel'fand–Gel'fand (or BGG) resolution. One consequence of the BGG resolution is a simple proof of the Weyl character formula. In this paper, we will describe an analogous resolution problem in positive characteristic: Is there a resolution of a highest weight irreducible representation (of a semisimple simply connected algebraic group over an algebraically closed field of positive characteristic) by restricted Verma modules? And if so, is it a generalization of the BGG resolution? This paper provides a complete answer to this problem for SL(3,k). Consequently, we are able to compute the formal character of the irreducible representation following a procedure similar to the BGG proof of the Weyl character formula
Finite Complete Intersection Algebras and the Completeness Radical

AbstractLetAbe a ring, and letBbe a finiteA-algebra. IfBis of the formA[X1,…,Xn]/(f1,…,fn) then we say thatBis a complete intersection overA. We show that such an algebra is projective as anA-module and Gorenstein as anA-algebra. Under the condition thatAis noetherian we show that the finiteA-algebras of the formA[[X1,…,Xm]]/(g1,…,gm) are exactly those complete intersectionsBoverAfor which one hasB=A·1+cr(A)·B. Here cr(A) denotes the largest ideal ofAwith respect to whichAis complete. This ideal, which we call the completeness radical ofA, satisfies the usual radical axioms
Small Cofinite Irreducibles

AbstractWe study approximately Gorenstein rings and modules which have small cofinite irreducibles (SCI). We give a criterion for a (not necessarily finitely generated) module over a local ring to have SCI via its Matlis dual. This is among other things applied to obtain Hochster's characterization of finitely generated modules over a complete local ring with SCI. This is done without using anything from the theory of excellent rings. Also the relation between cyclic purity and ordinary purity is investigated for modules. Finally we study how approximately Gorenstein rings behave with respect to flat homomorphisms and with respect to regular sequences. Essential use is made of certain flat modules related to infinite sequences of ring elements
Homological Aspects of Noetherian PI Hopf Algebras and Irreducible Modules of Maximal Dimension

AbstractWe prove that a Noetherian Hopf algebra of finite global dimension possesses further attractive homological properties, at least when it satisfies a polynomial identity. This applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity, as well as to classical enveloping algebras in positive characteristic. In all three cases we show that these algebras are Auslander-regular and Macaulay. We derive representation theoretic consequences concerning the coincidence of the non-Azumaya and singular loci for each of the above three classes of algebras
Some Hall Polynomials for Representation-Finite Trivial Extension Algebras

AbstractLetkbe a finite field and assume that Λ is a finite dimensional associativek-algebra with 1. Denote by modΛ the category of all finitely generated (right) Λ-modules and by indΛ the full subcategory in which every object is a representative of the isoclass of an indecomposable (right) Λ-module. We are interested in the existance of the Hall polynomial ϕMNLfor anL,M,N∈modΛ (for the definition, see977113or Section 1 below). In case Λ is directed,7has shown that Λ has Hall polynomials, and in case Λ is cyclic serial, the same result has also been obtained by4. It has been conjectured in8that any representation-finitek-algebra has Hall polynomials. In this investigation, we shall show that if Λ is a representation-finite trivial extension algebra, then, for anyL,M,N∈modΛ withNindecomposable, Λ has the Hall polynomials ϕMLNand ϕMNL. Using these Hall polynomials, we can naturally structure the free abelian group with a basis indΛ, denoted byK(modΛ), into a Lie algebra and the universal enveloping algebra ofK(modΛ)⊗ZQis just H(Λ)1⊗ZQ, where H(Λ)1is the degenerated Hall algebra of Λ (see Section 5 below)
The Algebra of Multiple Harmonic Series

AbstractRecently there has been much interest in multiple harmonic seriesζ(i1,i2,…,ik)=∑n1>n2>···>nk≥11n1i1n2i2···nkik(which converge when the exponentsijare positive integers andi1>1), also known as multiple zeta values or Euler/Zagier sums. Starting with the noncommutative polynomial algebraQ〈x,y〉, we define a second multiplication which is commutative and associative, and call the resulting structure the harmonic algebra h. As a graded commutative algebra, h turns out to be a free polynomial algebra with the number of generators in degreengiven by the Witt formula for the numberN(n) of basis elements of degreenin the free Lie algebra on two generators. Multiple harmonic series can be thought of as images under a map ζ:h0→Rwhich is a homomorphism with respect to the commutative multiplication, where h0is an appropriate subalgebra of h. If we calli1+···+ikthe weight of the series ζ(i1,…,ik), then (forn>1) there are at mostN(n) series of weightnthat are irreducible in the sense that they are not sums of rational multiples of products of multiple harmonic series of lower weight. In fact, our approach gives an explicit set of “algebraically” irreducible multiple harmonic series. We also show that there is a subalgebra h0⊂h1⊂h related to the shuffle algebra which contains the algebra S of symmetric functions: in fact, ζ maps the elements of S∩h0to algebraic combinations of zeta values ζ(i), for integeri≥2. The map ζ is not injective; we show how several results about multiple harmonic series can be recast as statements about the kernel of ζ, and propose some conjectures on the structure of the algebra h0/kerζ
An Upper Bound for the Length of a Finite-Dimensional Algebra

AbstractLetFbe a field, and letAbe a finite-dimensionalF-algebra. Writed=dimFA, and letebe the largest degree of the minimal polynomial for anya∈A. Define the functionf(d,e)=e2d/(e−1)+1/4+e/2−2. We prove that, ifSis any finite generating set forAas anF-algebra, the words inSof length less thanf(d,e) spanAas anF-vector space. In the special case ofn-by-nmatrices, this bound becomesf(n2,n)=n2n2/(n−1)+1/4+n/2−2∈O(n3/2). This is a substantial improvement over previous bounds, which have all beenO(n2). We also prove that, for particular setsSof matrices, the bound can be sharpened to one that is linear inn. As an application of these results, we reprove a theorem of Small, Stafford, and Warfield about semiprime affineF-algebras
Amalgamated Free Products of Inverse Semigroups

AbstractThis paper is the first of two papers devoted to the study of amalgamated free products of inverse semigroups. The subject of our second paper is the structure of amalgamated free products while this one is concerned with offering concrete descriptions of normal forms. We define a lower bounded amalgam of inverse semigroups and present a procedure by which the Schützenberger automata of the amalgamated free product can be constructed. Those automata which can be such a Schützenberger automaton are characterized, yielding a canonical form for the amalgamated free product. Our proof makes use of the graph-theoretic ideas developed by Jones, Margolis, Meakin, and Stephen for presentations of inverse semigroups
Upper Bounds for the Number of Conjugacy Classes of a Finite Group

AbstractFor a finite groupG, letk(G) denote the number of conjugacy classes ofG. We prove that a simple group of Lie type of untwisted ranklover the field ofqelements has at most (6q)lconjugacy classes. Using this estimate we show that for completely reducible subgroupsGofGL(n,q) we havek(G)≤q10n, confirming a conjecture of Kovács and Robinson. For finite groupsGwithF*(G) ap-group we prove thatk(G)≤(cp)awherepais the order of a Sylowp-subgroup ofGandcis a constant. For groups withOp(G)=1we obtain thatk(G)≤|G|p′. This latter result confirms a conjecture of Iranzo, Navarro, and Monasor. We also improve various earlier results concerning conjugacy classes of permutation groups and linear groups. As a by-product we show that any finite groupGhas a soluble subgroupSand a nilpotent subgroupNsuch thatk(G)≤|S| andk(G)≤|N|3
Noetherian Rings between a Semilocal Domain and Its Completion

AbstractWe consider a general technique for constructing local Noetherian integral domains. LetRbe a semilocal Noetherian domain with Jacobson radicalmand field of fractionsK. Letybe a nonzero element ofmand letR* be the (y)-adic completion ofR. For elements τ1,…,τs∈yR* algebraically independent overK, we obtain a necessary and sufficient condition forA≔K(τ1,…,τs)∩R* to be simultaneously Noetherian and a directed union of localized polynomial rings insvariables overR. We specify conditions in order that excellence be preserved, and we use the construction to obtain a non-Noetherian ringAof the formK(τ)∩R* which is a directed union of localized polynomial rings overR
Irreducible Affine Varieties over a Free Group II. Systems in Triangular Quasi-quadratic Form and Description of Residually Free Groups

AbstractWe shall prove the conjecture of Myasnikov and Remeslennikov [4] which states that a finitely generated group is fully residually free (every finite set of nontrivial elements has nontrivial images under some homomorphism into a free group) if and only if it is embeddable in the Lyndon's exponential groupFZ[x], which is theZ[x]-completion of the free group. HereZ[x] is the ring of polynomials of one variable with integer coefficients. Historically, Lyndon's attempts to solve Tarski's famous problem concerning the elementary equivalence of free groups of different ranks led him to introduceFZ[x].An ∃-free group is a groupGsuch that the class of ∃-formulas, true inG, is the same as the class of ∃-formulas, true in a nonabelian free group. A finitely generated group is ∃-free if and only if it is fully residually free [22]. Our result gives an algebraic description of ∃-free groups.We shall give an algorithm to represent a solution set of an arbitrary system of equations overFas a union of finite number of irreducible components in the Zariski topology onFn. The solution set for every system is contained in the solution set of a finite number of systems in triangular form with quadratic words as leading terms. The possibility of such a decomposition for a solution set was conjectured by Razborov in [20] and also by Rips.We shall give a description of systems of equations determining irreducible components using methods developed in [13,19]; it is possible to find some of these methods in [18]. We are thankful to E. Rips for attracting our attention to these techniques
The Cohomology of the Regular Semisimple Variety

AbstractWe use the equivariant cohomology of hyperplane complements and their toral counterparts to give formulae for the Poincaré polynomials of the varieties of regular semisimple elements of a reductive complex Lie group or Lie algebra. As a result, we obtain vanishing theorems for certain of the Betti numbers. Similar methods, usingl-adic cohomology, may be used to compute numbers of rational points of the varieties over the finite field Fq. In the classical cases, one obtains, both for the Poincaré polynomials and for the numbers of rational points, polynomials which exhibit certain regularity conditions as the dimension increases. This regularity may be interpreted in terms of functional equations satisfied by certain power series, or in terms of the representation theory of the Weyl group
Hadamard Difference Sets in Nonabelian 2-Groups with High Exponent

AbstractNontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Hadamard difference sets. In the abelian case, a group of order 22t+2has a difference set if and only if the exponent of the group is less than or equal to 2t+2. In a previous work (R. A. Liebler and K. W. Smith,in“Coding Theory, Design Theory, Group Theory: Proc. of the Marshall Hall Conf.,” Wiley, New York, 1992), the authors constructed a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 24t+2with exponent 23t+2. Thus a nonabelian 2-groupGwith a Hadamard difference set can have exponent |G|3/4asymptotically. Previously the highest known exponent of a nonabelian 2-group with a Hadamard difference set was |G|1/2asymptotically. We use representation theory to prove that the group has a difference set
Symmetry Groups of Boolean Functions and Constructions of Permutation Groups

AbstractIn this paper we deal with the symmetry groupS(f) of a boolean functionfonn-variables, that is, the set of all permutations onnelements which leavefinvariant. The main problem is that of concrete representation: which permutation groups onnelements can be represented asG=S(f) for somen-ary boolean functionf. Following P. Clote and E. Kranakis [6] we consider, more generally, groups represented in such a way byk-valued boolean functions (i.e., functions on a two-element set withkpossible values) and call such permutation groupsk-representable (k≥2). The starting point of this paper is a false statement in one of the theorems of [6] that everyk-representable permutation group is 2-representable for allk≥2. We show that there exists a 3-representable permutation group that is not 2-representable. A natural question arises whether there are other examples like this. This question turns out to be not easy and leads us to considering general constructions of permutation groups and investigating their properties. The main result of the present paper may be summarized as follows: using known groups and “standard” constructions no other example like that above can be constructed. Nevertheless, we conjecture that there arek+1-representable permutation groups that are notk-representable for allk≥2. If this is true, then this would open an interesting avenue toward investigating and classifying finite permutation groups
Applications of Frobenius Algebras to Representation Theory of Schur Algebras

AbstractA Schur algebra is a subalgebra of the group algebraRGassociated to a partition ofG, whereGis a finite group andRis a commutative ring. For two classes of Schur algebras we study the relationship between indecomposable modules over the Schur algebra and overRG, but we discuss this problem in a more general context. Further we develop a character theory for Schur algebras; in particular, we express primitive central idempotents in terms of trace functions and we derive orthogonality relations for trace functions. These results are also presented in a more general context, namely for Frobenius algebras over rings. Moreover, we focus on class functions on Schur algebras
Homogeneous Locally Compact Groups

AbstractWe determine all locally compact abelian groups with the property that the group of all topological automorphisms acts transitively on the set of nontrivial elements. Such groups are called homogeneous. The connected ones are the additive groups of finite-dimensional vector spaces over the real numbers. The compact ones are the (not necessarily finite) powers of cyclic groups of prime order. Actually, the commutativity hypothesis is needed only in the remaining cases: the disconnected torsion-free homogeneous abelian locally compact groups are the divisible hulls of powers of the group ofp-adic integers; and the homogeneous abelian locally compact torsion groups are the products of (compact) powers of cyclic groups and discrete elementary abelian groups. A characterization of additive groups of vector spaces of finite dimension over locally compact fields is obtained
A Classification of Fully Residually Free Groups of Rank Three or Less

AbstractA groupGisfully residually freeprovided to every finite setS⊂G\{1} of non-trivial elements ofGthere is a free groupFSand an epimorphismhS:G→FSsuch thathS(g)≠1 for allg∈S. Ifnis a positive integer, then a groupGisn-freeprovided every subgroup ofGgenerated bynor fewer distinct elements is free. Our main result shows that a fully residually free group of rank at most 3 is either abelian, free, or a free rank one extension of centralizers of a rank two free group. To prove this we prove that every 2-free, fully residually free group is actually 3-free. There are fully residually free groups which are not 2-free and there are 3-free, fully residually free groups which are not 4-free
Unification in Commutative Semigroups

AbstractUnification is one of the basic concepts of automated theorem proving. It concerns such questions as finding solutions of finite sets of equations, determining if every solution comes from a most general solution, and if so, determining how many most general solutions are needed to generate all solutions. These solutions given in terms of substitutions are called, more formally, unifiers. The unification type of a variery (equational class) of algebras is defined according to the cardinality or existence of minimal complete sets of most general unifiers. Of particular interest, from a computational point of view, are varieties of groups and semigroups. So far the problem has been considered mainly for particular varieties. In this paper we determine unification types for all varieties of commutative semigroups. In particular, we prove that for commutative semigroups the unification problem is solvable in the very strong sense that there is an algorithm which for any two finite sets Σ1and Σ2of semigroup equations produces the minimal complete set of the most general unifiers of Σ1over the variety of commutative semigroups generated by Σ2. It seems that this is the first so general decidability result in the area
On Modular Homology in the Boolean Algebra, II

AbstractLetRbe an associative ring with identity and Ω ann-element set. Fork≤nconsider theR-moduleMkwithk-element subsets of Ω as basis. Ther-step inclusion map∂r:Mk→Mk−ris the linear map defined on this basis through where the Γiare the (k−r)-element subsets of Δ. Form<rone obtains chains of inclusion maps which have interesting homological properties ifRhas characteristicp>0. V. B. Mnukhin and J. Siemons (J. Combin. Theory74, 1996 287–300;J. Algebra179, 1995, 191–199) introduced the notion ofp-homologyto examine such sequences whenr=1 and here we continue this investigation whenris a power ofp. We show that any section of M not containing certainmiddle termsisp-exact and we determine the homology modules for such middle terms. Numerous infinite families of irreducible modules for the symmetric groups arise in this fashion. Among these thesemi-simple inductive systemsdiscussed by A. Kleshchev (J. Algebra181, 1996, 584–592) appear and in the special casep=5 we obtain theFibonacci representationsof A. J. E. Ryba (J. Algebra170, 1994, 678–686). There are also applications to permutation groups of order co-prime top, resulting inEuler–Poincaréequations for the number of orbits on subsets of such groups
Topological Morita Equivalences Induced by Ideals Generated by Dense Idempotents

AbstractGiven two complete right linearly topologized rings (R,ρ) and (S,σ), and a bimoduleRBSendowed with a complete topology β, in such a way that (BS,β)∈CLT-(S,σ) and there be a continuous ring homomorphism (R,ρ)→CEnduS(B,β), we define a functor −⊗̂RB:CLT-(R,ρ)→CLT-(S,σ) which is left adjoint to the functor CHomuS((B,β),−):CLT-(S,σ)→CLT-(R,ρ). Then we consider the particular case in which (S,σ)=eRewith its induced topology, whereeis a dense idempotent ofR(that is,ReRis dense in (R,ρ)). Under these hypotheses we show that the pair of functors −⊗̂RRe:CLT-(R,ρ)→CLT-(S,σ) and −⊗̂SeR:CLT-(S,σ)→CLT-(R,ρ) is an equivalence of categories. As an application of this result, we re-obtain a theorem of Xu, Shum, and Turner-Smith on similarities between infinite matrix subrings
Testing Modules of Groups of Even Order for Simplicity

AbstractIn this paper we exhibit an intimate relationship between the simplicity of anFG-moduleVof a finite groupGgenerated by a noncentral involutiont≠1 and some other elementuinGand the socles of the eigenspacesEλfor λ in {−1,1} of the involutiontconsidered asFC-modules, whereC=CG(t) is the centralizer oftinG. In fact the two main results yield new simplicity criteria forFG-modulesVover fieldsFwith odd or even characteristicp, respectively. Forp≠2 the result follows from a general module theoretic simplicity test proved in the first section of the paper. It builds on ideas of the Meat-axe algorithm of previous work. We show the practicability of our new tests by examples
Galois Correspondences for Hopf Bigalois Extensions

AbstractLetHbe a Hopf algebra with bijective antipode over a commutative ringk. A rightH-Galois extension ofkis a rightH-comodule algebraAsuch thatk=AcoHand a certain canonical mapA⊗A→A⊗His a bijection. We investigate Galois connections for Hopf–Galois extensions that can be formulated with the help of an additional Hopf algebraLover whichAis also a leftL-Galois extension ofk, and anL-H-bicomodule—such an additional Hopf algebra always exists and is unique up to isomorphism. The Galois connection between quotient coalgebras and left modules ofLandH-subcomodule algebras ofAinduces a bijection between those quotients over whichLis faithfully coflat, and those subalgebras over whichAis faithfully flat. As a consequence, the lattices of Hopf subalgebras ofLandHover which these are faithfully flat are isomorphic