pinecone/core-2020-05-10-deduplication huggingface dataset:
The Structure of the Young Symmetrizers for Spin Representations of the Symmetric Group, II

AbstractThe first paper in this series established that the projective analogue of the Young symmetrizer recently introduced by Nazarov has a naturalPxQ-structure comparable with thepq-form of the classical symmetrizer. This second paper develops the theory on this decomposition further. A more efficient construction of the projective symmetrizer is presented for certain classes of partitions; this approach yields a “compact” expression for the intermediate factorxλin the projective decomposition
Reflection Groups on the Octave Hyperbolic Plane

AbstractFor two different integral formsKof the exceptional Jordan algebra we show that AutKis generated by octave reflections. These provide “geometric” examples of discrete reflection groups acting with finite covolume on the octave (or Cayley) hyperbolic plane OH2, the exceptional rank one symmetric space. (The isometry group of the plane is the exceptional Lie groupF4(−20).) Our groups are defined in terms of Coxeter's discrete subring K of the nonassociative division algebra O and we interpret them as the symmetry groups of “Lorentzian lattices” over K. We also show that the reflection group of the “hyperbolic cell” over K is the rotation subgroup of a particularrealreflection group acting onH8≅OH1. Part of our approach is the treatment of the Jordan algebra of matrices that are Hermitian with respect to any real symmetric matrix
Fake Congruence Modular Curves and Subgroups of the Modular Group

AbstractWe construct a special class of noncongruence modular subgroups and curves, analogous in some ways to the usual congruence ones. The subgroups are obtained via the Burau representation, and the associated quotient curves have a natural moduli space interpretation. In fact, they are reduced Hurwitz spaces corresponding to covers with 4 branch points and monodromy group equal to semi-direct products of a cyclic and an abelian group. Furthermore, they form a modular tower in the sense of Fried. We study representations on the cohomology of these fake congruence modular curves and also calculate the genera of certain quotient curves
Classes of Ultrasimplicial Lattice-Ordered Abelian Groups

AbstractA lattice-ordered abelian group is called ultrasimplicial iff every finite set of positive elements belongs to the monoid generated by some finite set of positiveZ-independent elements. This property originates from Elliott's classification of AFC*-algebras. Using fans and their desingularizations, it is proved that the ultrasimplicial property holds for everyn-generated archimedeanl-group whose maximall-ideals of ranknare dense. As a corollary we obtain simpler proofs of results, respectively by Elliott and by the present author, stating that totally ordered abelian groups, as well as freel-groups, are ultrasimplicial
Free Products of Units in Algebras I. Quaternion Algebras

AbstractLetAbe a quaternion algebra over a commutative unital ring. We find sufficient conditions for pairs of units ofAto generate a free group. Using the well-known isomorphism betweenSO(3,R) and the group of real quaternions of norm 1, we obtain free groups of rotations of the Euclidean 3-space. Specialization techniques allow us to find similar free subgroups in skew polynomial rings. A consequence is the following: letkGbe the group algebra of a residually (torsion-free nilpotent) groupGover a fieldkwhose characteristic is not 2. Ifxandyare any pair of noncommuting elements ofG, andc,d∈k* then 1+cxand 1+dygenerate a free subgroup of the Malcev–Neumann field of fractions ofkG
Transitive Permutation Groups with Bounded Movement Having Maximal Degree

AbstractLetGbe a transitive permutation group on a set Ω such thatGis not a 2-group and letmbe a positive integer. It was shown by the fourth author that if |Γg\Γ|≤mfor every subset Γ of Ω and allg∈G, then |Ω|≤⌊2mp/(p−1)⌋, wherepis the least odd prime dividing |G|. Ifp=3 the upper bound for |Ω| is 3m, and the groupsGattaining this bound were classified in the work of Gardiner, Mann, and the fourth author. Here we show that the groupsGattaining the bound forp≥5 satisfy one of the following: (a)G≔Zp⋊Z2a, |Ω|=p,m=(p−1)/2, where 2a|(p−1) for somea≥1; (b)G≔K⋊P, |Ω|=2sp,m=2s−1(p−1), where 1<2s<p,Kis a 2-group withp-orbits of length 2s, each element ofKmoves at most 2s(p−1) points of Ω, andP=Zpis fixed point free on Ω; (c)Gis ap-group. All groups in case (a) are examples. In case (b), there exist examples for everypwiths=1. In case (c), whereGis ap-group, we also prove that the exponent ofGis bounded in terms ofponly. Each transitive group of exponentpis an example, and it may be that these are the only examples in case (c)
Groups with a Bounded Number of Conjugacy Classes of Non-normal Subgroups

AbstractLet ν(G) denote the number of conjugacy classes of non-normal subgroups of a groupG. We prove that ifGis a finite group and ν(G)≠0, then there is a cyclic subgroupCof prime power order contained in the centre ofGsuch that the order ofG/Cis a product of at most ν(G)+1 primes. We also obtain a bound in the opposite direction, thus obtaining a criterion for a group to have a bounded number of conjugacy classes of non-normal subgroups. These results extend to infinite groups, with the subgroupCbeing the infinite Prüferp-group, but only whenGhas finitely many non-normal subgroups. This is to be expected because of the existence of monsters of the type constructed by S. V. Ivanov and A. Yu. Ol'shanskii. The structure of groups with infinitely many non-normal subgroups falling into finitely many conjugacy classes is also studied
For Which Pseudo Reflection Groups Are thep-adic Polynomial Invariants Again a Polynomial Algebra?

AbstractLetWbe a finite group acting on a latticeLover thep-adic integers Z∧p. The analysis of the ring of invariants of the associatedW-action on the algebra Z∧p[L] of polynomial functions onLis a classical question of invariant theory. Ifpis coprime to the order ofW, classical results show thatWis a pseudo reflection group, if and only if the ring of invariants is again polynomial. We analyze the situation for odd primes dividing the order ofWand, in particular, determine those pseudo reflection groups for which the ring of invariants Z∧p[L]Wis a polynomial algebra
Irreducible Representations of Witten's Deformations ofU(sl2)

AbstractUsing E. Witten's approach based on duality in conformal field theory, one can obtain a 7-parameter family of deformations ofU(sl2) (called the “Witten's Deformations”). Le Bruyn singled out a subfamily (called “conformalsl2algebras”) and studied their finite dimensional representations. Here we consider all of Witten's deformations. We compute the left spectrum of these algebras and thereby classify irreducible representations for generic values of parameters and finite dimensional representations in certain “roots of unity” cases
Extended Affine Weyl Groups

AbstractIn this paper we study the Weyl groups of reduced extended affine root systems, the root systems of extended affine Lie algebras. We start by describing the extended affine Weyl group as a semidirect product of a finite Weyl group and a Heisenberg-like normal subgroup. This provides a unique expression for the Weyl group elements (in terms of some naturally arisen transformations) which is crucial in the further study of extended affine Weyl groups. We use this to give a presentation, called a presentation by conjugation, for an important subclass of extended affine Weyl groups. Using a new notion, called the index which is an invariant of the extended affine root systems, we show that one of the important features of finite and affine root systems (related to Weyl group) holds for the class of extended affine root systems
P.I. Algebras with Hopf Algebra Actions

AbstractIfAis a p.i. algebra in characteristic zero with action from a finite-dimensional semisimple Hopf algebraH, thenAhas a nilpotentH-idealNsuch thatA/Nwill beH-verbally semiprime. EveryH-verbally semiprime algebra isH-p.i. equivalent to a direct sum ofH-verbally prime algebras. In the case of a finite group action or a grading by an abelian group, we show that the sum can be taken to be finite. In the case of an action by a finite cyclic groupG, we classify allG-p.i. algebras, up to equivalence. This paper generalizes the work of A. R. Kerner (1985,Math. USSR Izv.25)
Computing with Abelian Sections of Finitely Presented Groups

AbstractLetGbe a finitely presented group. This paper describes the theory and practice of a method for obtaining information about the finite and abelian-by-finite quotients ofG, which often allows computation about larger quotients of the group than has been possible by more traditional methods. The paper also describes the authors' software packagequotpic, which provides a user-friendly implementation of the algorithms involved in this method, as part of a complete range of facilities for computing various types of quotients of finitely presented groups
Classification of Orbit Closures of 4-Dimensional Complex Lie Algebras

AbstractLet Ln(C) be the variety of complexn-dimensional Lie algebras. The groupGLn(C) acts on it via change of basis. An orbitO(μ) under this action consists of all structures isomorphic to μ. The aim of this paper is to give a complete classification of orbit closures of 4-dimensional Lie algebras, i.e., determining all μ∈O(λ)where λ∈L4(C). Starting with a classification of complex Lie algebras of dimensionn≤4, we study the behavior of several Lie algebra invariants under degeneration, i.e., under transition to the orbit closure. As a corollary, we will show that all degenerations in L3(C) can be realized via a one-parameter subgroup, but this is not the case in L4(C)
Blocks of Defect Zero and Products of Elements of Orderp

AbstractSuppose thatGis a finite group and thatFis a field of characteristicp>0 which is a splitting field for all subgroups ofG. Lete0be the sum of the block idempotents of defect zero inFG, and let Ω be the set of solutions togp=1 inG. We show thate0=(Ω+)2, whenpis odd, ande0=(Ω+)3, whenp=2. In the latter case (Ω+)2=R+, whereRis the set of real elements of 2-defect zero. Soe0=Ω+R+=(R+)2. We also show thate0=Ω+Ω+4=(Ω+4)2, whenp=2, where Ω4is the set of solutions tog4=1. These results give us various criteria for the existence ofp-blocks of defect zero
On the Cohen–Macaulay Property of Modular Invariant Rings

AbstractIfVis a faithful module for a finite groupGover a field of characteristicp, then the ring of invariants need not be Cohen–Macaulay ifpdivides the order ofG. In this article the cohomology ofGis used to study the question of Cohen–Macaulayness of the invariant ring. One of the results is a classification of all groups for which the invariant ring with respect to the regular representation is Cohen–Macaulay. Moreover, it is proved that ifpdivides the order ofG, then the ring of vector invariants of sufficiently many copies ofVis not Cohen–Macaulay. A further result is that ifGis ap-group and the invariant ring is Cohen–Macaulay, thenGis a bireflection group, i.e., it is generated by elements which fix a subspace ofVof codimension at most 2
Multiplier Hopf Algebras of Discrete Type

AbstractIn this paper, we study regular multiplier Hopf algebras with cointegrals. They are a certain class of multiplier Hopf algebras, still sharing many nice properties with the (much smaller class of) finite-dimensional Hopf algebras. Recall that a multiplier Hopf algebra is a pair (A,Δ) whereAis an algebra over C, possibly without identity, and Δ is a comultiplication onA(a homomorphism ofAinto the multiplier algebraM(A⊗A) ofA⊗A) satisfying certain properties. The typical example is the algebraAof complex functions with finite support in a groupG, with pointwise multiplication and where the comultiplication is defined by (Δf)(p,q)=f(pq) wheneverf∈Aandp,q∈G. A left cointegral in a multiplier Hopf algebra is an elementh∈Asuch thatah=ϵ(a)hfor alla∈Awhere ϵ is the counit ofA. In the group example, this is the function that is 1 on the identity of the group and 0 everywhere else. In this paper, we show that cointegrals are unique (up to a constant) if they exist and that they are faithful. We also show that on a regular multiplier Hopf algebra with a left cointegral, there exists also a left integral. Recall that a left integral is a linear functional ϕ onAsuch that (ι⊗ϕ)Δ(a)=ϕ(a)1 where ι is the identity map (and where the equation is to be considered inM(A)). A multiplier Hopf algebra with cointegrals is therefore an algebraic quantum group of discrete type. We will also obtain different necessary and sufficient conditions on the algebraAfor a multiplier Hopf algebra (A,Δ) to have cointegrals (i.e., to be of discrete type). The algebras turn out to be Frobenius, quasi-Frobenius, and Kasch
Quantum Deformation of the Grassmannian Manifold

AbstractIn this paper we work out a deformation ofG(r,n), the grassmannian ofr-subspaces in a vector space of dimensionnover a fieldkof characteristic 0.G(r,n) is deformed as an homogeneous space forSLn(k), the special linear group ofkn; this means thatk[G(r,n)], the coordinate ring ofG(r,n), gets deformed together with with the coaction ofk[SLn], the coordinate ring ofSLn(k), on it. Our deformation comes together with a coaction of the corresponding deformation ofSLn(k). At the end we give an explicit presentation of the deformed grassmannian, in terms of generators and relations
Primitive Localizations of Group Algebras of Polycyclic-by-Finite Groups

AbstractLetGbe a polycyclic-by-finite group such that Δ(G) is torsion-free abelian andKa field. Denote bySa multiplicatively closed set of non-zero central elements ofK[G]; ifKis an absolute field assume thatScontains an element not inK. Our main result is when the localizationK[G]Sis a primitive ring. This turns out to be equivalent to the following three conditions: (1)A=K〈S,S−1〉 is aG-domain, (2) (Q(ZK[G]):Q(A)) is finite, and (3)J(K[G]S)=0. In caseGis not abelian-by-finite, condition (3) is not needed. An immediate consequence is the following. LetKbe a field; in caseKis an absolute field assume that Δ(G)≠1. ThenK[G]ZK[G]is a primitive ring. In the final section a class of examples is constructed
A Note on Words in Braid Monoids

AbstractThe purpose of this note is to give a self-contained (apart from simple facts about Coxeter groups) and we hope a bit shorter and more understandable account of some results of [C1,C2] on normal forms of braids which are themselves based on the papers [D1,T]. In particular a motivation was to give a proof of Proposition 5.1 that we use in [B-M]. Some proofs and results from Section 2 onwards seem to be new. I thank several people for improvements from earlier versions of the manuscript: M. Geck for pointing out some errors, F. Digne for pointing out that some results don't need the braid group to be of finite type, and J.-Y. Hée for suggesting (and providing) further improvements in that direction
Differential Hopf Algebras on Quantum Groups of Type A

AbstractLet A be a Hopf algebra and Γ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra Γ∧=Γ⊗/Jthat contains Γ as its first order part. Corresponding to the three choices of the idealJ, we distinguish the “universal” exterior algebra, the “second antisymmetrizer” exterior algebra, and Woronowicz' external algebra, respectively. Let Γ be one of theN2-dimensional bicovariant first order differential calculi on the quantum groupGLq(N) orSLq(N), and letqbe a transcendental complex number. For Woronowicz' external algebra we determine the dimension of the space of left-invariant and of bi-invariantk-forms, respectively. Bi-invariant forms are closed and represent different de Rham cohomology classes. The algebra of bi-invariant forms is graded anti-commutative. ForN≥3 the three differential Hopf algebras coincide. However, in case of the 4D±-calculi onSLq(2) the universal differential Hopf algebra is strictly larger than Woronowicz' external algebra. The bi-invariant 1-form is not closed
Graded, Selfinjective, and Koszul Algebras

AbstractIn this paper we continue our work on Koszul algebras initiated in earlier studies. The consideration about the existence of almost split sequences for Koszul modules appeared in our early work and only a partial answer is known. Koszul duality relates finite dimensional algebras of infinite global dimension with infinite dimensional algebras such that the injective dimension of the graded simples is finite. Two questions naturally arise: the existence of almost split sequences for infinite dimensional graded algebras and the relations between the global dimension of a graded algebra and the injective dimension of the graded simples. In the first part of the paper, we deal with the first matter; in the second, we consider the relations between the graded global dimension and the maximum of the injective dimensions of the graded simples, proving that in the noetherian case they coincide. In the third part, we consider finite dimensional graded selfinjective quiver algebras and prove that for such algebras all indecomposable projective modules have the same Lowey length. In the forth and last part of the paper, we specialize to selfinjective Koszul algebras and characterize their Yoneda algebras, proving they constitute non-commutative versions of the commutative regular algebras
Addition and Subtraction of Ideals

AbstractLetAbe a Noetherian ring,ĨandIbe comaximal ideals ofA, andPbe a projectiveA-module. “Addition” refers to being given a surjectionP↠Ĩand producing a surjectionP↠Ĩ∩I. This is useful, for example, whenPis free, to determine how many elements it takes to generate the idealĨ∩I. “Subtraction” refers to being givenP↠Ĩ∩Iand producing someP↠Ĩ. A major use of this is whenĨ=A, to show a projective module has a unimodular element. Here we extend certain addition and subtraction results of R. Sridharan (1995,J. Algebra176,947–958) and S. Mandal and R. Sridharan (1996,J. Math. Kyoto Univ.36,No. 3, 453–470). In both addition and subtraction, we weaken the hypotheses imposed onĨand in a suitable fashion remove the hypothesis thatPmust have trivial determinant. With certain restrictions, we also now allow dimA/I≤1, rather than htI=dimA. WhenAis an affine algebra over a fieldF, Mandal and Sridharan had subtraction results for whenIis the intersection of finitely many maximal ideals whose residue fields are quadratically closed, whenIhas this form (if charF≠2), or whenIis the maximal ideal of anF-rational point. In our generalization, we allowIto be the intersection of finitely many maximal ideals, all of whose residue fields are quadratically closed, except possibly one which instead defines anF-rational point. When charF≠2, we only need to requireIto have this form. In particular, we can subtract an ideal whose radical is the maximal ideal of anF-rational point; this has been used by S. M. Bhatwadekar and R. Sridharan (1998,Invent. Math.133,No. 1, 161–192) to construct a certain local complete intersection ideal which is not a complete intersection ideal
Modules over Crossed Products

AbstractJ. T. Stafford (1978,J. London Math. Soc. (2)18, 429–442) proved that any left ideal of the Weyl algebraAn(K) over a fieldKof characteristic 0 can be generated by two elements. In general, there is the problem of determining whether any left ideal of a Noetherian simple domain can be generated by two elements. In this work we show that this property holds for some crossed products of a simple ring with a supersolvable group and also for the tensor product of generalized Weyl algebras. We also prove that these rings are stably generated by 2 elements and that their finitely generated torsion left modules can be generated by two elements. Some results about stably 2-generated rings were found by V. A. Artamonov (1994,Math. Sb.185, No. 7, 3–12)
Arep-Algebras Having Cyclic Quadratic Extensions Necessarily Cyclic?

AbstractAlbert's candidate for a noncyclicp-algebra of prime degreepis actually seen to be ap-symbol (and thereby cyclic), as a consequence of calculations related to the corestriction down a quadratic extension. More generally, under many circumstances (which subsume the known casep=3), ap-algebra of degreepwhich becomes cyclic after a quadratic extension of the center already is already cyclic. However, there is a proposed counterexample to this assertion in general, which may be noncyclic algebra forp≥5
The Irreducible Brauer Characters of the Finite Special Linear Groups in Non-describing Characteristics

AbstractThe irreducible Brauer characters ofSLn(q) are investigated for primeslnot dividingq. They are described in terms of a set of ordinary characters ofSLn(q) whose reductions modulolare a generating set of the additive group of generalized Brauer characters and the decomposition numbers of this set. These decomposition numbers are described by combinatorial means in terms of the decomposition numbers ofGLn(q). The latter have been investigated in great detail the last 15 years and are known completely forn≤10
A Structure Theorem for Noetherian P.I. Rings with Global Dimension Two

AbstractLetAbe the Artin radical of a Noetherian ringRof global dimension two. We show thatA=ReRwhereeis an idempotent;Acontains a heredity chain of ideals and the global dimensions of the ringsR/AandeRecannot exceed two. Assume further thanRis a polynomial identity ring. LetPbe a minimal prime ideal ofR. ThenP=P2and the global dimension ofR/Pis also bounded by two. In particular, if the Krull dimension ofR/Pequals two for all minimal primesPthenRis a semiprime ring. In general, every clique of prime ideals inRis finite and in the affine caseRis a finite module over a commutative affine subring. Additionally, whenA=0, the ringRhas an Artinian quotient ring and we provide a structure theorem which shows thatRis obtained by a certain subidealizing process carried out on rings involving Dedekind prime rings and other homologically homogeneous rings
Interpolation by Integer-Valued Polynomials

AbstractLetRbe a Krull ring with quotient fieldKanda1,…,aninR. If and only if theaiare pairwise incongruent mod every height 1 prime ideal of infinite index inRdoes there exist for all valuesb1,…,bninRan interpolating integer-valued polynomial, i.e., anf∈K[x] withf(ai)=biandf(R)⊆R.IfSis an infinite subring of a discrete valuation ringRvwith quotient fieldKanda1,…,aninSare pairwise incongruent mod allMkv∩Sof infinite index inS, we also determine the minimald(depending on the distribution of theaiamong residue classes of the idealsMkv∩S) such that for allb1,…,bn∈Rvthere exists a polynomialf∈K[x] of degree at mostdwithf(ai)=biandf(S)⊆Rv
Complexity and Tor on a Complete Intersection

AbstractLet (R,m) be a complete intersection, that is, a local ring whose m-adic completion is the quotient of a regular local ring by a regular sequence. SupposeMis a finitely generatedR-module. It is known that the even and odd Betti sequences ofMare eventually given by polynomials of the same degreen; the complexity ofMis the nonnegative integern+1. We use this notion of complexity to study the vanishing of TorRi(M,N) for finitely generated modulesMandNover a complete intersectionR. We prove several theorems dealing with rigidity of Tor, which are generalizations and, in certain situations, improvements of known results. The main idea of these rigidity theorems is that the number of consecutive vanishing Tors required in the hypothesis of a rigidity theorem depends more on the minimum of the complexities ofMandNrather than on the codimension ofR. We give examples showing that this dependence is sharp. We also show that ifM⊗RNhas finite length, then, for sufficiently high indices, two consecutive vanishing Tors force the vanishing of all higher Tors