pinecone/core-2020-05-10-deduplication huggingface dataset:
Actions of Algebraic Groups on the Spectrum of Rational Ideals

AbstractLetkbe an algebraically closed field andGa linear algebraic group overkacting rationally on ak-algebraV. Generalizing work of Moeglin and Rentschler in characteristic zero, we study the action ofGon the spectrum of rational ideals ofV. The main result is the following. Suppose thatVis semiprime left Goldie. LetLbe aG-stable commutative semisimple subalgebra of the total ring of fractionsQ(V) ofVsuch thatLG=k·1L. This occurs, for example, if the zero ideal ofVisG-rational andLis the center ofQ(V). Then there is, for some closed subgroupHofG, aG-equivariant embedding ν ofLintoQ(G/H) (the algebra of rational functions onG/H) such thatQ(G/H) is purely inseparable over ν(L). This has applications to the closure of the orbit of a rational ideal
Closing the Relation Gap by Direct Product Stabilization

AbstractThe generation gap of a groupGis the difference between the minimal number of generators ofGand the rank of the augmentation ideal. The relation gap of a presentationF/Nis the difference between the minimal number of elements that generateNas a normal subgroup and the minimal number ofG-module generators of the relation moduleN/[N,N]. We show that ifGis a finitely presented group then there existsnsuch thatG×∏i=1nZp,Zpbeing the cyclic group of orderp, has zero generation and zero relation gap. We apply this result to questions concerning the efficiency of finite groups
Stability of Cohomology via Double-Coset Products and Suborbit Diagrams

AbstractStability of subgroup cohomology is typically defined by a condition on individual double cosets, though use of the natural multiplication among them can reduce the amount of direct checking needed. The study of geometries for many finite groups, notably the sporadic simple groups, has frequently included the determination of suborbit diagrams, from which the needed multiplication coefficients are immediately available. The technique is applied to simplify certain calculations in the recent literature on the cohomology of the sporadic groups
Galois Groups of Unramified Covers of Projective Curves in Characteristicp

AbstractWe show several results concerning the finite groups that occur as Galois groups of unramified covers of projective curves in characteristicp. In particular, we prove that every finite group withggenerators occurs over some curve of genusg. This implies, for example, that every finite simple group occurs in genus 2. By similar methods, we obtain several other families of groups which occur in genus 2. In addition, we show that if a groupGoccurs over some curve of genusg, then it must occur over “almost all” curves of genusgor greater. The results are obtained using formal patching
Module Categories with Infinite Radical Cube Zero

AbstractLetAbe an artin algebra and let modAdenote the category of finitely generated rightA-modules. We denote by rad(modA) the ideal in modAgenerated by all non-invertible morphisms between indecomposable modules in modA. The infinite radical rad∞(modA) of modAis the intersection of all powers radi(modA),i≥1, of rad(modA). It is known thatAis representation-finite if and only if (rad∞(modA))2=0. In this paper we study the representation-infinite algebrasAsuch that (rad∞(modA))3=0. Examples of such algebras include all representation-infinite tilted algebras of Euclidean type
Bass–Serre Theory for Groupoids and the Structure of Full Regular Semigroup Amalgams

AbstractT. E. Hall proved in 1978 that if [S1,S2;U] is an amalgam of regular semigroups in whichS1∩S2=Uis a full regular subsemigroup ofS1andS2(i.e.,S1,S2, andUhave the same set of idempotents), then the amalgam is strongly embeddable in a regular semigroupSthat containsS1,S2, andUas full regular subsemigroups. In this case the inductive structure of the amalgamated free produceS1∗US2was studied by Nambooripad and Pastijn in 1989, using Ordman's results from 1971 on amalgams of groupoids. In the present paper we show how these results may be combined with techniques from Bass–Serre theory to elucidate the structure of the maximal subgroups ofS1∗US2. This is accomplished by first studying the appropriate analogue of the Bass–Serre theory for groupoids and applying this to the study of the maximal subgroups ofS1∗US2. The resulting graphs of groups are arbitrary bipartite graphs of groups. This has several interesting consequences. For example ifS1andS2are combinatorial, then the maximal subgroups ofS1∗US2are free groups. Finite inverse semigroups may be decomposed in non-trivial ways as amalgams of inverse semigroups
Hereditary Torsion Theory Counter Equivalences

AbstractOne of the main ingredients of a tilting theory, a generalization of Morita equivalence betweenR-Mod andS-Mod that has evolved from the work of S. Brenner and M. C. R. Butler (Lecture Notes in Math.,8321980, 103–170) and D. Happel and C. M. Ringel (Trans. Amer. Math. Soc.,2741982, 399–443), is a pair of category equivalences T⇆E and F⇆S between the members of torsion theories (T,F) inR-Mod and (S,E) inS-Mod. Recently, we began to investigate this phenomenon as a further generalization of Morita equivalence, in particular showing that such equivalences are induced by pairs of bimodules (R. R. Colby and K. R. Fuller,Comm. Algebra,231995, 4833–4849). Here we are concerned with conditions under which the torsion theories are hereditary. We begin with a characterization of the bimodules, and we apply it to determine just when one or both of the torsion theories are hereditary, to obtain examples, and to investigate global dimensions and Grothendieck groups ofRandS. We assume the notation and conventions of F. W. Anderson and K. R. Fuller (1992, “Rings and Categories of Modules,” 2nd ed., Springer-Verlag, New York) and B. Stenström (1970, “Rings of Quotients,” Springer-Verlag, New York/Heidelberg/Berlin). In addition all “subcategories” are full subcategories of modules and all “functors” are additive functors
Valuations on Extensions of Weyl Skew Fields

AbstractLetk⊂Kbe a finitely generated field extension of transcendence degree 1. Consider the corresponding extension D1(k)⊂D1(K) of skew fields of the first Weyl algebra. We show that the D1(k)-valuations of D1(K) are in one-to-one correspondence with thek-valuations ofK. The intersection of the D1(k)-valuation rings of D1(K) isnotD1(k) as one would expect but a principal ideal domain such that its quotient division ring is D1(K). IfDis a positive divisor, then L(D)={f∈D1(K):div(f)+D≥0} is a finite-dimensional vector space over D1(k) whose dimension equals the degree ofD
Eventually H-Related Sets and Systems of Equations over Finite Semigroups and Rings

AbstractI prove that given a finite semigroup or finite associative ringSand a system Σ of equations of the formax=borxa=b, wherea,b∈S,xis an unknown, it is algorithmically impossible to decide whether or not Σ is solvable overS, that is, whether or not there exists a bigger semigroup or ring (resp. finite semigroup, finite ring)T>Ssuch that Σ has a solution inT. The proof employs the unsolvability of the uniform word problem in the case of groups (Novikov) and in the class of finite groups (Slobodskoii) and the so-called split systems
Free Lie Algebras, Generalized Witt Formula, and the Denominator Identity

AbstractLet Γ be a countable abelian semigroup satisfying a suitable finiteness condition, and letL=⊕α∈ΓLαbe the free Lie algebra generated by a Γ-graded vector spaceVoverC. In this paper, from the denominator identity, we derive a dimension formula for the homogeneous subspaces of the free Lie algebraL=⊕α∈ΓLαand discuss numerous applications of our dimension formula to various interesting cases. Our dimension formula will be expressed in terms of the Witt partition functions. Various expressions of the Witt partition functions will give rise to a number of interesting combinatorial identities. As a special case, we obtain a recursive relation for the coefficients of the elliptic modular functionj
Reduced Weighted Complete Intersection and Derivations

AbstractLetA=F[x1,…,xn]/(f1,…,fn) be a zero-dimensional weighted complete intersection (charF=0). We prove a general result on the (homogeneous) derivations ofA. In particular we deduce that in the “generic” case when F[x1,…,xn]/(f1,…,fn−1) is reduced (deg(fi)≤deg(fn),i=1,…,n−1) there are no nontrivial derivations of strictly negative degree. If moreover all the weights of the variablesxiare even it follows that the Serre spectral sequence of any orientable fibrationF↪E→B, withH*F=Aas graded algebras, collapses at theE2-term, thus verifying a conjecture of Halperin. We also discuss several examples
Relative Projectivity and Ideals in Cohomology Rings

AbstractIn this paper we explore one aspect of the relationship between group cohomology and representation theory. For a finite groupGand a fieldkin characteristicp>0, ideals in the cohomology ringH*(G,k) can sometimes be characterized by exact sequences ofkG-modules in much the same way that elements of ExtkGare classically represented by exact sequences. Furthermore the maps on the stable category which represent the exact sequences have functorial properties which mimic the structure of the ideas in the ring. The natural setting for this study is the relative homological algebra for the module category, relative to the subcategory generated by a single module using the tensor product operation
Simple Groups, Probabilistic Methods, and a Conjecture of Kantor and Lubotzky

AbstractWe prove that a randomly chosen involution and a randomly chosen additional element of a finite simple groupGgenerateGwith probability →1 as |G|→∞. This confirms a conjecture of Kantor and Lubotzky. Applications and related results are derived. For example, we show that, except for the Suzuki groups and finitely many possible other exceptions, all finite simple groups can be generated by two elements, one of which has order 3. We also obtain sharp estimates on the probabilityP(G) of generating a finite simple group of exceptional Lie typeGby two randomly chosen elements. This complements analogous estimates of Babai and Kantor for alternating and classical groups. Denoting bym(G) the minimal index of a proper subgroup of a finite simple groupG, we conclude, in particular, thatm(G)(1−P(G)) is bounded between two positive absolute constants
Hecke Algebras of TypeAwithq=−1

AbstractIn this paper we study the decomposition matrices of the Hecke algebras of typeAwithq=−1 over a field of characteristic 0. We give explicit formulae for the columns of the decomposition matrices indexed by all 2-regular partitions with 1 or 2 parts and an algorithm for calculating the columns of the decomposition matrix indexed by partitions with 3 parts. Combining these results we find all of the rows of the decomposition matrices which are indexed by partitions with at most four parts. All this is accomplished by means of a more general theory which begins by showing that the decomposition numbers in the columns of the decomposition matrices indexed by 2-regular partitions with “enormous 2-cores” are Littlewood–Richardson coefficients
Triangular Decomposition of the Composition Algebra of the Kronecker Algebra

AbstractLetAbe a finite-dimensional hereditary algebra over a finite field, and letH(A) andC(A) be, respectively, the Ringel–Hall algebra and the composition algebra ofA. Definerdto be the element ∑[M]∈H(A), where [M] runs over the isomorphism classes of the regularA-modules with dimension vectord. We prove thatrdand the exceptionalA-modules all lie inC(A). LetKbe the Kronecker algebra,P(resp.I) the subalgebra ofC(K) generated by the preprojective (resp. preinjective)K-modules, andTthe subalgebra generated byr(n,n)forn≥0. Then we prove thatC(K)=P·T·Iand thenTis just the subalgebra ofC(K) generated by all regular elements
Graded Polynomial Identities of the Jordan Superalgebra of a Bilinear Form

AbstractLetWsp,2q=W0(p)⊕W1(2q)be a direct sum of two vector spaces of dimensionpand 2q, respectively, over a fieldkof characteristic zero,p=2,3,…,∞;q=1,2,…,∞; and let 〈x,y〉 be a nondegenerate bilinear form onWsp,2qwhich is symmetric onW0(p)and skew-symmetric onW1(2q)and such that 〈W0(p),W1(2q)〉=〈W1(2q),W0(p)〉=0. Then the vector spaceBsp,2q=k⊕Wsp,2qequipped with the product (α+v)(β+u)=(αβ+〈v,u〉)+(αu+βv), α,β∈k,v,u∈Wsp,2q, is a simple Jordan superalgebra, withk+W0(p)as its even component andW1(2q)as the odd one. In this paper explicit finite bases for graded polynomial identities of the series of Jordan superalgebrasBsp,2q,p=2,3,…,∞;q=1,2,…,∞; are found
On Equivalences between Blocks of Group Algebras: Reduction to the Simple Components

AbstractA conjecture of Michel Broué states that ifDis an abelian Sylowp-subgroup of a finite groupG, andH=NG(D), then the principal blocks ofGandHare Rickard equivalent. The structure of groups with abelian Sylowp-subgroups, as determined by P. Fong and M. E. Harris, raises the following question: Assuming that Broué's conjecture holds for the simple components ofG, under what conditions does it hold forGitself? Due to the structure ofG, this problem requires mainly the lifting of Rickard complexes top′-extensions of the simple components and the construction of complexes over wreath products. We give here these reduction steps, which may be regarded as a “Clifford theory” of tilting complexes
Finite Nonsolvable Groups in Which Only Two Nonlinear Irreducible Characters Have Equal Degrees

AbstractY. Berkovichet al.[Proc. Amer. Math. Soc.115(1992), 955–959] classified finite groups in which the degrees of the nonlinear irreducible characters are distinct. Theorem 24.7 from [Y. Berkovich,J. Algebra184(1996), 584–603] contains the classification of solvable groups in which only two nonlinear irreducible characters have equal degrees (D1-groups). In this paper we obtain the classification of nonsolvableD1-groups, completing the classification ofD1-groups. Our proof depends on the classification of finite simple groups. The results of the important paper [Illinois J. Math.33, No. 1 (1988), 103–131] on rational simple groups play a key role as well
Translative Isomorphisms and Left-Symmetric Structures onW(m,n)

AbstractLetW≔W(m,n) be the general Lie algebra of Cartan type and A≔A(m,n) the corresponding divided power algebra on whichWacts. Translative isomorphisms, i.e., isomorphisms ofWintoW∔gl(m,A) of the formA↦A+ϕ(A),A∈W, ϕ(A)∈gl(m,A), are investigated. The mixed product theorems of the author (1986,Sci. Sinica Ser. A29, No. 6, 570–581) are generalized. Under certain conditions, translative isomorphisms induce left-symmetric structures onW. Classes of left-symmetric structures onWare constructed, among which are restricted left-symmetric structures on the restricted Lie algebraW(m,1)
Class Semigroups of Prüfer Domains

AbstractThe class semigroup of a commutative integral domainRis the semigroupS(R) of the isomorphism classes of the nonzero ideals ofRwith the operation induced by multiplication. The aim of this paper is to characterize the Prüfer domainsRsuch that the semigroupS(R) is a Clifford semigroup, namely a disjoint union of groups each one associated to an idempotent of the semigroup. We find a connection between this problem and the following local invertibility property: an idealIofRis invertible if and only if every localization ofIat a maximal ideal ofRis invertible. We consider the (#) property, introduced in 1967 for Prüfer domainsR, stating that if Δ1and Δ2are two distinct sets of maximal ideals ofR, then ∩{RM|M∈Δ1}≠∩{RM|M∈Δ2}. LetCbe the class of Prüfer domains satisfying the separation property (#) or with the property that each localization at a maximal ideal if finite-dimensional. We prove that, ifRbelongs toC, then the local invertibility property holds onRif and only if every nonzero element ofRis contained only in a finite number of maximal ideals ofR. Moreover ifRbelongs toC, thenS(R) is a Clifford semigroup if and only if every nonzero element ofRis contained only in a finite number of maximal ideals ofR
Primitive Jordan Pairs and Triple Systems

AbstractIn this paper we give a characterization of primitivity of Jordan pairs and triple systems in terms of their local algebras. As a consequence of that local characterization we extend to Jordan pairs and triple systems most of the known results about primitive Jordan algebras. In particular, we describe primitive Jordan pairs and triple systems over an arbitrary ring of scalars in the sense of “The Structure of Primitive Quadratic Jordan Algebras” by J. A. Anquela, T. Cortés, and F. Montaner (1995,J. Algebra172,530–553, 5.1)
Orbital Integrals on Symmetric Spaces and Spherical Characters

AbstractIn the first section of this paper we obtain an asymptotic expansion near semi-simple elements of orbital integrals μx(f) ofCc∞-functionsfon symmetric spacesG/H. HereGis a reductivep-adic group, andHis the group of fixed points of an involution σ onG. This extends the germ expansion of J. Shalika and M.-F. Vigneras in the group case.The main part of the paper studies examples of groupsGwith involution σ, which have the property that the spherical characters associated with their spherical admissible representations are not identically zero on the regular set ofG/H. These includeG=GL(n+m),H=GL(n)×GL(m) forn=m=1 or 2, andn=1,m≥3. More general results had been obtained by J. Sekiguchi in the case of real symmetric spaces, generalizing Harish-Chandra's work in the group case, over archimedean and non-archimedean fields. Our interest is in thep-adic case. There the techniques are entirely different from Sekiguchi's. In fact we use the recent work of J. Hakim and C. Rader and S. Rallis who showed that the spherical character is smooth on the regular set, and has asymptotic expansion in terms of Fourier transforms of invariant distributions on the nilpotent cone, as found by Harish-Chandra in the group case.Our study of the nonvanishing of some spherical characters uses a construction of an explicit basis of the space of invariant distributions on the nilpotent cone. This is done on regularizing spherical orbital integrals, and taking suitable linear combinations. This local work is motivated by concrete applications to the theory of Deligne-Kazhdan lifting of spherical automorphic representations. In some other examples, concerningG=GL(3n) andH=GL(n)×GL(2n), andG=O(3,2),H=O(2,2), we explicitly construct invariant distributions on the nilpotent cone which are equal to their Fourier transform. Such examples do not exist in Harish-Chandra's group case.In the last two sections, following Harish-Chandra's simple proof in the group case, we show that μx(f) is locally constant on the regular set ofx, uniformly inf, in some cases. Following D. Kazhdan's proof of his density theorem we show that anfwhich annihilates all spherical characters has μx(f)=0 on the regular elliptic set
A Description of the Brauer–Severi Scheme of Trace Rings of Generic Matrices

AbstractHere we use invariant theory to describe the Brauer–Severi scheme of the fibers of trace rings of generic matrices with an algebraically closed base field of characteristic zero when the trace ring is viewed as a sheaf of algebras over the variety of matrix invariants. Using this approach, we first prove that the Brauer–Severi scheme of a trace ring is isomorphic to ProjQ, for a graded ringQwhose generators we describe in the first section. This description also has a relevant interpretation over base fields of arbitrary characteristic. In the second section of this paper we show that the Brauer–Severi scheme of the fiber of a trace ring over a point that is not too degenerate will have smooth irreducible components meeting transversally and describe these irreducible components as Brauer–Severi schemes of certain algebras
Minimal Noncryptic e-Varieties of Regular Semigroups

AbstractWe describe all minimal noncryptic e-varieties of regular semigroups, thus generalising earlier results by Rasin and Reilly that dealt with the completely regular and the inverse cases, respectively. As corollaries, we prove that an e-variety of regular semigroups is cryptic if and only if its intersections with the variety of all completely regular semigroups and the variety of all inverse semigroups are cryptic. We also find an equational characterization of group-bound cryptic varieties; this generalises some recent results by Doyle and Yeh
The Elements of the Orthogonal Group Ωn(V) as Products of Commutators of Symmetries

AbstractWe will assume throughout thatFis a field of characteristic charF≠2 and thatVis a non-degenerate quadratic space overFof finite dimension dimV=n. The orthogonal group ofVis denotedOn(V) and Ωn(V) is its commutator subgroup. John Hsia, in reference to the classical fact that every element of Ωn(V) is a product of commutators of symmetries, asked the following very basic question: ForFa local field, does there exist a boundk, depending only onn, such that every element of Ωn(V) is a product ofksuch commutators or fewer? A corollary of the results of this article answers this question completely for a non-dyadicF: Every element in Ωn(V) is a product of [n/2] such commutators, except for a “handful” of elements which can be listed (all are certain types of involutions whenn≥6), where [n/2]+1 factors are required. (See Theorem 4.) Of course, Hsia's question can be asked for anyF, and most of the analysis is carried out in the more general context
A Criterion for the Conjugacy Separability of Amalgamated Free Products of Conjugacy Separable Groups

AbstractWe first prove a criterion for the conjugacy separability of generalized free products of two conjugacy separable groups amalgamating a cyclic subgroup. Applying this result, we show that tree products of a finite number of conjugacy separable, residually finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups are conjugacy separable. From this we derive that tree products of finitely generated torsion-free nilpotent groups, free groups, or surface groups amalgamating cyclic subgroups are conjugacy separable
Monomial Modular Representations and Construction of the Held Group

AbstractMonomial representations of familiar finite groups over finite fields are used to construct (infinite) semi-direct products of free products of cyclic groups by groups of monomial automorphisms. Finite homomorphic images of theseprogenitorsin which the actions on the group of automorphisms and on the cyclic components are faithful are sought. The smallest non-trivial images of this type are often sporadic simple groups. The technique is demonstrated by three examples over the fieldsZ3,Z5, andZ7, which produce the Mathieu groupM11, the unitary groupU3(5):2, and the Held group, respectively
Cohomology of Schematic Algebras

AbstractLetRbe a positively graded Noetheriank-algebra and let ProjR=(R,κ+)−grbe the quotient category of the category of gradedR-modulesR−grmodulo those of finite length. IfRis schematic (cf. [F. Van Oystaeyen and L. Willaert,J. Pure Appl. Algebra104(1995), 109–122]) and has finite global dimension, we show that the cohomology groups ofR(as defined in [M. Artin and J. J. Zhang,Adv. in Math.109(1994), 228–287]) are finite dimensional. Furthermore, we introduce Čech cohomology groups for schematic algebras and prove that they coincide with the usual graded cohomology groups. These generalized Čech cohomology groups are still manageable; we illustrate this with an example
Group Actions of Some Subgroups of Parabolic Subgroups

AbstractLetKbe the finite field withqelements,Vann-dimensional vector space overK,Pthe parabolic subgroup ofGL(V) associated to the flagV≔V0⊃V1⊃…⊃Vr={0},Uthe unipotent radical ofP. Suppose thatGis a subgroup ofGL(V) such that (i)U⊂G⊂P, and (ii) Image {G→GL(V0/V1)×GL(V1/V2)×…×GL(Vr−1/Vr)}=G1×G2×…×Gr, whereGiis a subgroup ofGL(Vi−1/Vi) for 1≤i≤r.THEOREM(i)If K[Vi−1/Vi]Giis a polynomial ring(resp. a complete intersection, a Gorenstein ring, a Cohen–Macaulay ring)for1≤i≤r,so is K[V]G. (ii)If K[Vi−1/Vi]is a free module over K[Vi−1/Vi]Gifor1≤i≤r,so is K[V]over K[V]G. (iii)If K(Vi−1/Vi)Giis rational over K for1≤i≤r,so is K(V)Gover K