pinecone/core-2020-05-10-deduplication huggingface dataset:
A Note on Rationality of Representations of Profinite Groups

AbstractLet G be a profinite group. The purpose of this note is to construct subfieldsFof the field of complex numbers over which a given finite dimensional complex continuous linear representationDof G is realizable in the following sense: There is a one-dimensional representation λ of G such that λ⊗Dis isomorphic to a linear representation which is realizable overF. It turns out that certain invariants related to the torsion part of the maximal profinite abelian quotient Gabof G play a key role
Syntactic and Rees Indices of Subsemigroups

AbstractWe define two different notions of index for subsemigroups of semigroups: the (right) syntactic index and the Rees index. We investigate the relationships between them and with the group index. In particular, we show that the syntactic index is a generalisation of both the group index and the Rees index. We use this fact to prove further similarities between the group index and the Rees index. It turns out, however, that very few of the nice properties of the group index are inherited by the syntactic index
Complex Representations of Finite Monoids II. Highest Weight Categories and Quivers

AbstractIn this paper we continue our study of complex representations of finite monoids. We begin by showing that the complex algebra of a finite regular monoid is a quasi-hereditary algebra and we identify the standard and costandard modules. We define the concept of a monoid quiver and compute it in terms of the group characters of the standard and costandard modules. We use our results to determine the blocks of the complex algebra of the full transformation semigroup. We show that there are only two blocks when the degree ≠3. We also show that when the degree ≥5, the complex algebra of the full transformation semigroup is not of finite representation type, answering negatively a conjecture of Ponizovskii
The Structure of the Young Symmetrizers for Spin Representations of the Symmetric Group, I

AbstractThis paper is the first in a series of three papers on the Young symmetrizers for the spin representations of the symmetric group. In this opening paper, it is shown that the projective analogue of the Young symmetrizer recently introduced by Nazarov has a structure resembling thepλqλ-form exhibited by the classical symmetrizer. In contrast to the classical case, our decomposition of the projective symmetrizer requires an additional intermediate factorxλ. This factor will be studied in greater detail in the later papers
On Infinite Goldie Dimension

AbstractTwo elementsxandyof a partially ordered setPare said to bedisjointif there is noz∈Psuch thatz≤xandz≤y. Denote by δ(P) the supremum of the cardinals κ such thatPcontains a subset of pairwise disjoint elements with cardinal number κ. P. Erdös and A. Tarski (Ann. of Math.44, 1943, 315–329) proved that, unless δ(P) is weakly inaccessible,Pcontains a subset of pairwise disjoint elements with cardinal number δ(P). J. Dauns and L. Fuchs (J. Algebra115, 1988, 297–302) defined theGoldie dimensionof a moduleM, denoted by GdM, as the supremum of all cardinals κ such thatMcontains the direct sum of κ nonzero submodules. They proved that, unless GdMis weakly inaccessible,Mcontains a direct sum of GdMsubmodules. In this paper, a unified proof of these two results is given. It is also shown that similar results hold in the context of modular lattices and abelian categories
Scale Functions on Linear Groups Over Local Skew Fields

AbstractLetGbe a general or special linear group over a local skew field. ThenGis a totally disconnected, locally compact group, to which G. Willis (Math. Ann.300, 1994, 341–363) associates its scale functions:G→N. We computeson the subset of diagonalizable matrices. We also consider the projective situation, and we discuss scale functions on groups of upper triangular matrices. The latter can be computed completely, provided that the underlying field is commutative. The data computed suffice to determine the modular functions on the groups considered, and they facilitate an easy proof of the fact that general or special linear groups over local skew fields with distinct modules cannot be isomorphic as topological groups
Primitive Superalgebras with Superinvolution

AbstractOur main purpose is to provide for primitive associative superalgebras a structure theory analogous to that for algebras [5,6,10] and to classify primitive superrings with superinvolution having a minimal one-sided superideal. We were led to this problem by our work on finite dimensional central simple Jordan superalgebras over fields of characteristic not 2 [9] (see also [7]). Of course, just as symmetric elements give rise to Jordan superalgebras, skewsymmetric elements give rise to Lie superalgebras [8,4]. The results and methods are closely related to those of structure theory of associative rings and central simple associative algebras with involution [5, Chap. I;6, Chaps. II, III;1, Chap. X;10, Chap. 2]. Some of the results have been announced in [13]
Mackey Functors with Chain Conditions

AbstractLetAbe a Green functor for a finite groupGand letMbe a leftA-module.Mis callednoetherian (artinian)if the lattice of all submodules ofMsatisfies ACC (respectively DCC).Mis calledtotally noetherian (totally artinian)ifMXis noetherian (respectively artinian) for every finiteG-setX. In this paper it is shown that the totally noetherian (totally artinian) leftA-modules behave in the same way as the classical noetherian (artinian) modules. The left noetherian (left artinian) Green functors are less well behaved but some properties from classical algebra still hold. It is shown that Cohen's theorem holds in this setting. Our main result shows that if a commutative Green functorAis artinian, then the ringsA(H)/Jac(A(H)) are semisimple artinian for all subgroupsHofG
The Set of Dominance-Minimal Roots

AbstractIf α and β are positive roots in the root system of a Coxeter groupW, we say that α dominates β ifwβ is negative wheneverwα is negative forw∈W. We say that α is elementary or dominance-minimal, if it does not dominate any β≠α. It is shown by the author and R. B. Howlett (Math. Ann.296, 1993, 179–190) that the set E of dominance-minimal roots is finite if and only ifWhas finite rank; this is used to show thatWis automatic. To limit the size of the relevant automata, and possibly facilitate other Coxeter group algorithms, we give an explicit description of the set of elementary roots
Transfer Maps and Virtual Projectivity

AbstractWhenGis a finite group andHis a subgroup ofG, the concept of relativeH-projectivity is of fundamental importance in modular representation theory. Unfortunately, in some ways the properties of relativelyH-projective modules are not as closely related to other ideas of representation theory as one might hope. For example, the variety of a module bears only a weak connection to relative projectivity. The purpose of this paper is to introduce and study a related concept known asvirtual H-projectivitythat seems to be much more closely related to the theory of varieties. The study is actually carried out in the more general setting of virtual projectivity with respect to a finitely generated moduleW, and the main tool is a transfer map TrWthat generalizes the usual transfer map TrGHwith respect to a subgroupH
Generalizations of Principally Injective Rings

AbstractA ringRis said to be rightP-injective if every homomorphism of a principal right ideal toRis given by left multiplication by an element ofR. This is equivalent to saying thatlr(a)=Rafor everya∈R, wherelandrare the left and right annihilators, respectively. We generalize this to only requiring that for each 0≠a∈R,lr(a) containsRaas a direct summand. Such rings are called rightAP-injective rings. Even more generally, if for each 0≠a∈Rthere exists ann>0 withan≠0 such thatRanis not small inlr(an),Rwill be called a rightQGP-injective ring. Among the results for rightQGP-injective rings we are able to show that the radical is contained in the right singular ideal and is the singular ideal with a mild additional assumption. We show that the right socle is contained in the left socle for semiperfect rightQGP-injective rings. We give a decomposition of a rightQGP-injective ring, with one additional assumption, into a semisimple ring and a ring with square zero right socle. In the third section we explore, among other things, matrix rings which areAP-injective, giving necessary and sufficient conditions for a matrix ring to be anAP-injective ring
On the Intersection of Double Cosets in Free Groups, with an Application to Amalgamated Products

AbstractIt is shown that ifH,Kare any finitely generated subgroups of a free groupFandUis any cyclic subgroup ofF, then any intersectionHg1U∩Kg2Uof double cosets contains only a finite number of double cosets (H∩K)gU, and an explicit upper bound for this number is given in terms of the ranks ofHandKand a generator ofU. This result is then applied to the intersection of finitely generated subgroupsH,Kof a free product with amalgamation [formula] withAfree andUmaximal cyclic inA. Under the assumption thatHandKintersect all conjugates ofUtrivially, an upper estimate is established for the “Karrass–Solitar rank” ofH∩Kin terms of the KS-ranks ofHandK, a generator ofU, and [equation] Here theKarrass–Solitar rankof [formula] is defined to be the size of a natural set of generating subgroups ofH, afforded by the Karrass–Solitar subgroup theorem for amalgamated products [formula]
A General Theory of Canonical Induction Formulae

AbstractSo far there exist three independent constructions of two different canonical versions of Brauer's induction theorem for complex characters due to V. Snaith, P. Symonds, and the author. “Canonical” in this context means functorial with respect to restrictions along group homomorphisms. In this article we axiomatize the situation in which the above canonical induction formulae are constructed. Mackey functors and related structures arise in this way naturally as a convenient language. This approach allows us to construct canonical induction formulae for arbitrary Mackey functors. In particular we obtain canonical induction formulae for the Brauer character ring, the group of projective characters, the ring of trivial source modules, and the ring of linear source modules. In most cases, it is not difficult to construct such formulae over the rational numbers. A much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers. We give a sufficient condition in the general framework of Mackey functors for a canonical induction formula to be integral. As an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings, as, for example, the span of linear characters in the case of the canonical Brauer induction formula. This will be used in a subsequent article in the case of Adams operations and Chern classes
Actions of Algebraic Groups on the Spectrum of Rational Ideals, II

AbstractWe study rational actions of a linear algebraic groupGon an algebraV, and the induced actions on Rat(V), the spectrum of rational ideals ofV(a subset of Spec(V) which often includes all primitive ideals). This work extends results of Moeglin and Rentschler to prime characteristic, often also relaxing their finiteness assumptions onV. In particular, we relate properties of a rational idealJand itsorb, the ideal (J:G)=⋂γ∈Gγ(J). The rational ideals ofVcontaining the orb ofJare precisely those in the Zariski-closureXof the orbit ofJin Rat(V). TheG-stratumofJconsists of all rational ideals inXwhose orbit is dense inX(i.e., whose orb is equal to the orb ofJ). We show that theG-stratum of a rational ideal consists of exactly oneG-orbit, and that rational ideals are maximal in their strata in a strong sense. These results are useful for studying prime and primitive spectra of certain algebras, cf. recent work by Goodearl and Letzter. We further show that the orbit ofJis open in its closure in Rat(V), provided thatJis locally closed. Among other results, we show that the semiprime ideal (J:G) is Goldie, and we relate the uniform and Gelfand–Kirillov dimensions ofV/JandV/(J:G)
Laplace Operator and Polynomial Invariants

AbstractLetAbe a finite dimensional simple algebra (not necessarily associative) over the field of complex numbersC, and letGdenote the automorphism group Aut(A). Suppose thatAhas a symmetric nondegenerate associativeG-invariant bilinear form 〈x,y〉 and a compact real form, i.e., a subalgebraBoverRof dimension dimRB=dimCA, whereAis equal to the span ofBoverCand the restriction of 〈x,y〉 toBis positive definite. We describe generators of the algebra of polynomialG-invariants of a system of several vectors fromAin terms of 〈x,y〉 and Laplace operators. In particular, we give generators of the algebra of polynomial invariants of the adjoint representation of a simple linear algebraic group of any exceptional type ≠E6. As a consequence, we get the First Main Theorem on matrix invariants, invariants of minimal representation ofG2andF4
Compact Subgroups of Linear Algebraic Groups

AbstractThe general problem underlying this article is to give a qualitative classification of all compact subgroups Γ⊂GLn(F), whereFis a local field andnis arbitrary. It is natural to ask whether Γ is an open compact subgroup ofH(E), whereHis a linear algebraic group over a closed subfieldE⊂F. We show that Γ indeed has this form, up to finite index and a finite number of abelian subquotients. When Γ is Zariski dense in a connected semisimple group, we give a precise openness result for the closure of the commutator group of Γ. In the case char(F)=0 the answers have long been known by results of Chevalley and Weyl. The motivation for this work comes from the positive characteristic case, where such results are needed to study Galois representations associated to function fields. We also derive openness results over a finite number of local fields
E-Algebraic Functions over Fields of Positive Characteristic—An Analogue of Differentially Algebraic Functions

AbstractA function (or a power series)fis called differentially algebraic if it satisfies a differential equation of the formP(x,y,y′,…,y(n))=0, wherePis a nontrivial polynomial. This notion is usually defined only over fields of characteristic zero and is not so significant over fields of characteristicp>0 asf(p)≡0. For a formal power series over a perfect fieldKof positive characteristic we shall define an analogue of the concept of a differentially algebraic power series. We shall show that these series together with ordinary addition and multiplication of series form a field ΓKwith some natural properties. We also show that ΓKis not closed under the Hadamard product operation
Integral Representations of Cyclic Groups of Orderp2

AbstractLetpbe any prime number,Gbe the cyclic group of orderp2, and Λ≔RGbe the group algebra ofGover a Dedekind domainRsuch thatpRis a maximal ideal inRand bothR[T]/Φp(T) andR[T]/Φp2(T) are Dedekind domains also, where Φn(T) is thenth cyclotomic polynomial. We shall provide a full list of indecomposable Λ lattices constructed in an explicit way. A complete set of invariants of these indecomposable Λ lattices will be determined also, under an additional assumption onR. The above results were obtained by Reiner [Pacific J. Math.78(1978), 467–501] whenR=Z
Composition Algebras of Affine Type

AbstractThe aim of this paper is to study the structure of the composition algebras of affine type. It turns out that they have a triangular decomposition P⊗T⊗I corresponding to the division of the indecomposables into the preprojectives, the regulars, and the preinjectives. By the recent Ringel–Green theorem the composition algebra can be twisted in order to obtain the positive partU+of the Drinfeld–Jimbo quantized enveloping algebraU=U−⊗U0⊗U+of the corresponding Kac–Moody algebra, and one obtains a corresponding triangular decomposition forU+, in particular, a natural basis ofU+in terms ofA-representations
Ideal Theory of Right Cones and Associated Rings

AbstractRight cones are semigroups with a left cancellation law such that for any two elementsa,bthere exists an elementcwithb=acora=bc. Valuation rings, cones of ordered or left ordered groups, semigroups of ordinal numbers, and Hjelmslev rings are examples. The ideal theory of these semigroups is described in terms of prime and completely prime ideals, and a classification of prime segments is given that can be used to solve a problem raised by Skornyakov. The Archimedean case can be dealt with in a satisfactory way with the help of Hölder's theorem. Right cones of rank 1 are classified. We then consider the problem of constructing for a given right coneHa right chain ringRwith the same right ideal and ideal structure asH
Linkage by Generically Gorenstein, Cohen–Macaulay Ideals

AbstractIn this paper, we study linkage by a wider class of ideals than the complete intersections. We are most interested in how the Cohen–Macaulay property behaves along this more general notion of linkage. In particular, if idealsAandBare linked by a generically Gorenstein Cohen–Macaulay idealI, and ifAis a Cohen–Macaulay ideal, we give a criterion forBto be a Cohen–Macaulay ideal. WhenR/Bis not Cohen–Macaulay, we can give in many cases an easy description of the non–Cohen–Macaulay locus ofR/B, and also a criterion forR/Bto have almost maximal depth
Classification of the Hopf Galois Structures on Prime Power Radical Extensions

AbstractLetpbe an odd prime andna positive integer and letkbe a field of characteristic zero. LetK=k(w) withwpn=a∈kwhereais such that [K:k]=pnand letrdenote the largest integer between 0 andnsuch thatK∩k(ζpr)=k(ζpr), where ζprdenotes a primitiveprth root of unity. The extensionK/kis separable, but not necessarily normal and, by Greither and Pareigis, isH-Galois withHaK̃-Hopf algebra form of a group ringkNwhereK̃is the normal closure ofK/k.His said to bealmost classicalifN<Gal(K̃/k). The result is that ifr<nthen there areprHopf Galois structures onK/kfor which the associated groupNis cyclic of orderpn. Of these,pmin(r,n−r)are almost classical and the rest are non-almost classical. Whenr=n, there arepn−1H-Galois structures for whichN≅Cpnof which only one is almost classical. Finally, we show that these are the only structures possible. That is, for this class of extensions,N mustbe cyclic
Test Polynomials for Automorphisms of Polynomial and Free Associative Algebras

AbstractIn this paper we consider test polynomials in the polynomial algebra and the free associative algebra. A test polynomial is defined by the following property: every endomorphism which fixes the polynomial is an automorphism. We construct families of test polynomials for the polynomial algebra and the free associative algebra and show how different techniques may be used in the investigation of test polynomials. We also introduce the notion of a test vector space and determine all test vector spaces of the free associative algebra
The Minimal Free Resolution of the Migliore–Peterson Rings in the Case That the Reflexive Sheaf Has Even Rank

AbstractLetRbe a commutative noetherian ring and ϕ:F→Gbe a homomorphism of freeR-modules where rankF=fand rankG=g. Fix an elementbg+1∈⋀g+1Fand a generator ωG*for ⋀gG*. The module action of ⋀·F* on ⋀·Fproduces the elementb1=[(⋀gϕ*)(ωG*)](bg+1) inF. LetJdenote the image ofb1:F*→R. Assume that gradeJ=f−g, which is the largest grade possible and is attained in the generic case. The idealJmay be interpreted as the defining ideal of the degeneracy locus of a regular section of a rankf−greflexive sheaf. It may also be interpreted as the order ideal of an element in a second syzygy module of rankf−g. Also,Jmay be interpreted as the defining ideal for the symmetric algebra of a module of projective dimension two. Migliore and Peterson have studied the idealJunm, which is the unmixed part ofJ. Under geometric hypotheses, they have shown thatR/Junmis a Cohen–Macaulay ring and they have resolved this ring. Furthermore, iff−gis odd, thenJunmis a Gorenstein ideal and is not equal toJ. On the other hand, iff−gis even, thenJunm=J. In the present paper, we produce the resolution ofR/Jby freeR-modules in the case thatf−gis even and (f−g−2)! is a unit inR. Our resolution is minimal whenever the data are local or homogeneous. Our resolution is built from the differential graded algebra (⋀·F*〈X1,…,Xg〉,d), where the restriction ofdto ⋀·F* is the Koszul complex associated tob1:F*→Rand the degree two divided power variablesX1,…,Xghave been adjointed in order to kill the cycles ϕ*(G*)⊆⋀1F*. The acyclicity lemma is used to prove exactness. Ifg=1, then the idealJis equal to the Huneke–Ulrich almost complete intersection idealI1(yX), whereyis a 1×fmatrix andXis anf×falternating matrix. The resolution of this ideal is already known
A Characterization of the Block Lie Algebra and Itsq-Forms in Characteristic 0

AbstractThe Block algebraLreferred to here is the Lie algebra over a fieldFof characteristic 0 with basis {er,s∣(r,s)∈Z×Z\{(0,0)}} and subject to the commutation relations [eh,k,er,s]=(rk−sh)eh+r,k+s. Let 0,1≠q∈F. Theq-formL(q) ofLis the Lie algebra with basis {er,s(q)∣(r,s)∈Z×Z\{(0,0)}} and subject to the commutation relations [eh,k(q),er,s(q)]=(qrk−qsh)eh+r,k+s(q). We study here aZ×Z-graded Lie algebra[formula] withA0,0=0 and dimAi,j≤1 for all (i,j)∈Z×Z\{(0,0)}. In this paper, we show thatAis necessarily isomorphic toLorL′(q)=[L(q),L(q)] for someq∈FifAsatisfies a certain nondegeneracy condition
The Support of an Irreducible Lie Algebra Representation

AbstractWe present an explicit description of the h-support suppMof any irreducible h-locally finite g-moduleM, where g is any finite-dimensional Lie algebra and h is an arbitrary nilpotent Lie subalgebra of g. If h contains a Cartan subalgebra of the semi-simple part of g, we reformulate the description of suppMin terms of a latticeL̂Mand of the convex hullSMof suppM. When g is reductive it is known that suppMis nothing but the intersection ofSMwith the root lattice 〈Δ〉 shifted by an arbitrary element of suppM. Our general description is similar, but the root lattice 〈Δ〉 must be replaced by a certain sublatticeL̂M, and suppMmay now have “holes” near the boundary ofSM. The paper is concluded by a detailed discussion of examples
Algebraic Theories of Quasivarieties

AbstractAnalogously to the fact that Lawvere's algebraic theories of (finitary) varieties are precisely the small categories with finite products, we prove that (i) algebraic theories of many-sorted quasivarieties are precisely the small, left exact categories with enough regular injectives and (ii) algebraic theories of many-sorted Horn classes are precisely the small left exact categories with enough M-injectives, where M is a class of monomorphisms closed under finite products and containing all regular monomorphisms. We also present a Gabriel–Ulmer-type duality theory for quasivarieties and Horn classes
Branching of Modular Representations of the Alternating Groups

AbstractBased on Kleshchev's branching theorems for thep-modular irreducible representations of the symmetric group and on the recent proof of the Mullineux Conjecture, we investigate in this article the corresponding branching problem for thep-modular irreducible representations of the alternating groupAn. We obtain information on the socle of the restrictions of suchAn-representations toAn−1as well as on the multiplicities of certain composition factors; furthermore, irreducibleAn-representations with irreducible restrictions toAn−1are studied