pinecone/core-2020-05-10-deduplication huggingface dataset:
Linking First Occurrence Polynomials over F2 by Steenrod Operations

AbstractIt is proved that for every irreducible representation L(λ) of the full matrix semigroup Mn(F2), the first occurrence of L(λ) as a composition factor in the polynomial algebra P=F2[x1,…,xn] is linked by a Steenrod operation to the first occurrence of L(λ) as a submodule in P. This Steenrod operation is given explicitly as the image of an admissible monomial in the Steenrod squares Sqr under the canonical anti-automorphism χ of the mod 2 Steenrod algebra A. The first occurrences of both kinds are also linked to higher degree occurrences of L(λ) by elements of the Milnor basis of A
Residual Finiteness of Outer Automorphism Groups of Certain Pinched 1-Relator Groups

AbstractE. K. Grossman (1974, J. London Math. Soc.2, 160–164) showed that outer automorphism groups of fundamental groups of closed orientable surfaces are residually finite. Here we generalize her result by showing that outer automorphism groups of generalized free products of two free groups amalgamating a maximal cyclic subgroup are residually finite. From this it follows that mapping class groups of closed orientable and nonorientable surfaces are residually finite. The latter answers a question raised by A. M. Gaglione
Equivalent Blocks of Finite General Linear Groups in Non-describing Characteristic

AbstractJ. Chuang, R. Kessar, and J. Rickard have proved Broué's Abelian defect group conjecture for many symmetric groups. We adapt the ideas of Kessar and Chuang towards finite general linear groups (represented over non-describing characteristic). We then describe Morita equivalences between certain p-blocks of GLn(q) with defect group Cpα×Cpα, as q varies (see Theorem 2). Here p and q are coprime. This generalizes work of S. Koshitani and M. Hyoue, who proved the same result for principal blocks of GLn(q) when p=3, α=1, in a different way
F-Rationality of Rees Algebras

AbstractIn this paper, using the notion of the tight integral closure, we will give a criterion for F-rationality of Rees algebras of m-primary ideals in a Cohen–Macaulay local ring. As its application, we prove the following results: (1) In dimension two, if A is F-rational and I is integrally closed, then the Rees algebra R(I) is F-rational. On the other hand, in higher dimensions, we construct many examples of Cohen–Macaulay, normal Rees algebras which are not F rational. (2) If both A and R(I) are F-rational, then so is the extended Rees algebra R′(I). (3) If R(I) is F-rational and a(G(I))≠−1, then A is F-rational.On the other hand, using resolution of singularities, we will prove that a two-dimensional rational singularity always admits F-rational Rees algebras. In particular, this theorem gives another way than that devised by Watanabe (1997, J. Pure Appl. Algebra122, 323–328) to construct counterexamples to the Boutot-type theorem for F-rational rings
Rees Algebras of F-regular Type

AbstractWe study the F-regularity of Rees algebras R(I)=A[It] in terms of the global F-regularity of the blowing-up X=ProjR(I) of SpecA. As it reads, global F-regularity is a global analog of strong F-regularity defined via splitting of Frobenius maps in prime characteristic, and these notions are extended to characteristic zero by reduction modulo p⪢0. We study in detail the case where (A,m) is a two-dimensional local ring and I is an m-primary ideal. In characteristic zero, the condition for R(I) to have F-regular type is described in terms of the dual graph of a resolution X̃ on which IOX̃ is invertible. We also prove some miscellaneous results concerning singularities of Rees algebras and extended Rees algebras of higher dimension
On Certain C-Test Words for Free Groups

AbstractLet Fm be a free group of a finite rank m≥2 and let Xi,Yj be elements in Fm. A non-empty word w(x1,…,xn) is called a C-test word in n letters for Fm if, whenever (X1,…,Xn)=w(Y1,…,Yn)≠1, the two n-typles (X1,…,Xn) and (Y1,…,Yn) are conjugate in Fm. In this paper we construct, for each n≥2, a C-test word vn(x1,…,xn) with the additional property that vn(X1,…,Xn)=1 if and only if the subgroup of Fm generated by X1,…,Xn is cyclic. Making use of such words vm(x1,…,xm) and vm+1(x1,…,xm+1), we provide a positive solution to the following problem raised by Shpilrain: There exist two elements u1,u2∈Fm such that every endomorphism ψ of Fm with non-cyclic image is completely determined by ψ(u1),ψ(u2)
Simple Jordan Color Algebras Arising from Associative Graded Algebras

AbstractIf R is a G-graded associative algebra, where G is an abelian group and ϵ is a bicharacter for G, then R also has the structure of a Jordan color algebra. In addition, if R is endowed with a color involution ∗, then the symmetric elements S under ∗ are also a Jordan color algebra. Generalizing results of I. N. Herstein, we examine the Jordan color structure of R and S. In particular, we show that if R is a graded-simple algebra, then both R and S are simple Jordan color algebras, except for some special cases which cannot arise in the ordinary case
Goldie Theory for Jordan Algebras

AbstractIt is shown that Zelmanov's version of Goldie's conditions still characterizes quadratic Jordan algebras having an artinian algebra of quotients which is nondegenerate. At the same time, Jordan versions of the main notions of the associative theory, such as those of the uniform ideal, uniform element, singular ideal, and uniform dimension, are studied. Moreover, it is proved that the nondegenerate unital Jordan algebras of finite capacity are precisely the algebras of quotients of nondegenerate Jordan algebras having the property that an inner ideal is essential if and only if it contains an injective element
Behavior of Test Ideals under Smooth and Étale Homomorphisms

AbstractWe investigate the behavior of the test ideal of an excellent reduced ring of prime characteristic under base change. It is shown that if h: A→D is a smooth homomorphism, then τAD=τD, assuming that all residue fields of A at maximal ideals are perfect and that formation of the test ideal commutes with localization. It is also shown that if h: (A,m)→D is a finite flat homomorphism of Gorenstein normal rings, étale in codimension 1, then τAD=τD. More generally, this last result holds under the assumption that the closed fiber of h: (A,m)→D is Gorenstein, provided one knows that the tight closure of zero and the finitistic tight closure of zero in the injective hulls of the residue fields of A and S are equal
Prime Power Graphs for Groups of Lie Type

AbstractWe associate a weighted graph Δ(G) to each finite simple group G of Lie type. We show that, with an explicit list of exceptions, Δ(G) determines G up to isomorphism, and for these exceptions, Δ(G) nevertheless determines the characteristic of G.This result was motivated by algorithmic considerations. We prove that for any finite simple group G of Lie type, input as a black-box group with an oracle to compute the orders of group elements, Δ(G) and the characteristic of G can be computed by a Monte Carlo algorithm in time polynomial in the input length. The characteristic is needed as part of the input in a previous constructive recognition algorithm for G
Equivariant Poincaré Polynomials and Counting Points over Finite Fields

AbstractSuppose a finite group acts as a group of automorphisms of a smooth complex algebraic variety which is defined over a number field. We show how, in certain circumstances, an equivariant comparison theorem in l-adic cohomology may be used to convert the computation of the graded character of the induced action on cohomology into questions about numbers of rational points of varieties over finite fields. This is carried through in three applications: first, for the symmetric group acting on the moduli space of n points on a genus zero curve; second, for a unitary reflection group acting on the complement of its reflecting hyperplanes; and third, for the symmetric group action on the space of configurations of points in any smooth variety which satisfies certain strong purity conditions
Rectangle Diagrams for the Lusztig Cones of Quantized Enveloping Algebras of Type A

AbstractLet U be the quantum group associated to a Lie algebra g of type An. The negative part U− of U has a canonical basis B defined by Lusztig and Kashiwara, with favorable properties. We show how the spanning vectors of the cones defined by Lusztig (1993, Israel Math. Conf. Proc.7, 117–132), when regarded as monomials in Kashiwara's root operators, can be described using a remarkable rectangle combinatorics. We use this to calculate the Lusztig parameters of the corresponding canonical basis elements, conjecturing that translates of these vectors span the simplicial regions of linearity of Lusztig's piecewise-linear function (1990, J. Amer. Math. Soc.3, 447–498, Sect. 2)
On a Class of Generalized Nilpotent Groups

AbstractWe explore the class B of generalized nilpotent groups in the universe c[formula] of all radical locally finite groups satisfying min-p for every prime p. We obtain that this class is the natural generalization of the class of finite nilpotent groups from the finite universe to the universe c[formula]. Moreover, the structure of B-groups is determined explicitly. It is also shown that B is a subgroup-closed c[formula]-formation and that in every c[formula]-group the Fitting subgroup is the unique maximal normal B-subgroup
Embeddings of Schur Functions into Types B/C/D

AbstractWe consider the problem of embedding the semi-ring of Schur-positive symmetric polynomials into its analogue for the classical types B/C/D. If we preserve highest weights and add the additional Lie-theoretic parity assumption that the weights in images of Schur functions lie in a single translate of the root lattice, there are exactly two solutions. These naturally extend the Kirillov–Reshetikhin decompositions of representations of symplectic and orthogonal quantum affine algebras Uq(ĝ)
Broué's Conjecture Holds for Principal 3-Blocks with Elementary Abelian Defect Group of Order 9

AbstractIn representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a finite group G has an abelian Sylow p-subgroup P, then the derived categories of the principal p-blocks of G and of the normalizer NG(P) of P in G are equivalent. We prove in this paper that Broué'Inage-kus conjecture holds for the principal 3-block of an arbitrary finite group G with an elementary abelian Sylow 3-subgroup P of order 9, by using initiated works for the case where G is simple, which are due to Puig, Okuyama, Waki, Miyachi, and the authors. The result depends on the classification of finite simple groups
Integral Hopf–Galois Structures on Degree p2 Extensions of p-adic Fields

AbstractLet L/K be a totally ramified, normal extension of p-adic fields of degree p2. We investigate the behavior of the valuation ring OL in the various Hopf–Galois structures on L/K. Specifically, we determine when OL is Hopf–Galois with respect to a Hopf order in the corresponding Hopf algebra. When this occurs, OL is necessarily a free module over this Hopf order. We also determine which Hopf orders can arise in this way. For cyclic extensions L/K of degree p2, L. N. Childs has shown, under certain restrictions on the ramification numbers, that if OL is Hopf–Galois with respect to a Hopf order in one of the Hopf–Galois structures on L/K, then the same is true in all p Hopf–Galois structures on L/K. We show that this no longer holds if the ramification conditions are relaxed, or if elementary abelian extensions of degree of p2 are considered. We illustrate our results with a special family of Kummer extensions, and with certain extensions arising from Lubin–Tate formal groups
The Linear Part of a Discontinuously Acting Euclidean Semigroup

AbstractLet Rn be a Euclidean space and let S be a Euclidean semigroup, i.e., a subsemigroup of the group of isometries of Rn. We say that a semigroup S acts discontinuously on Rn if the subset {s∈S:sK∩K≠∅} is finite for any compact set K of Rn. The main results of this work areTheorem.If S is a Euclidean semigroup which acts discontinuously on Rn, then the connected component of the closure of the linear part ℓ(S) of S is a reducible group.Corollary.Let S be a Euclidean semigroup acting discontinuously on Rn; then the linear part ℓ(S) of S is not dense in the orthogonal group O(n).These results are the first step in the proof of the followingMargulis' Conjecture.If S is a crystallographic Euclidean semigroup, then S is a group
Weakly Azumaya Algebras

AbstractWe define a class of algebras, weakly Azumaya algebras, which includes both Azumaya algebras and weak crossed products (cf. Haile [1982, J. Algebra74, 270–279; 1983, J. Algebra91, 521–539] and Haile et al. [1983, Amer. J. Math.105, No. 3, 689–814]). Just as with Azumaya algebras, these have a rank function whose values at localizations of the center are always a square. After a general description of these algebras, we specialize to the case where the center is a field F. The Jacobson radical need not be 0, and we prove a Wedderburn principal theorem for these algebras. Our class is closed under extension of scalars and under tensor products and yields an interesting monoid which generalizes the Brauer group. Our monoid is a union of groups, called stalks, in each of which the unit element is represented by an algebra called an idempotent algebra. The ideal structure of members of the same stalk is the same. A given stalk is not torsion, but the kernel of the restriction map to the algebraic closure is torsion modulo the idempotent algebras. At the end we consider low-dimensional examples in detail
Lie Algebras of CL Type

AbstractThe notion of a strongly nilpotent element of a Lie algebra is introduced. According to the existence or nonexistence of nontrivial strongly nilpotent elements, the simple modular Lie algebras are divided into two categories, CA type and CL type, which coincide with Lie algebras of generalized Cartan type and classical type, respectively, when the characteristic is greater than 7. Examples of nonclassical simple Lie algebras of CL type are given which all have affinities to the classical Lie algebras
Multivariate Interpolation and Standard Bases for Macaulay Modules

AbstractThe K-linear space of all n-linearly recursive functions (1.1) (=evaluated differential forms) for which a zero-dimensional ideal I⊂K[x1,…,xn] is the largest ideal which is contained in the kernel of all of them turns out to be the orthogonal K-space I⊥⊂(K[x1,…,xn])* of I and is known as Macaulay's inverse system of I. Making use of the antiderivative operatord, the whole space of all differential forms can be endowed with a structure of K[x1,…,xn]-module; with respect to finitely generated submodules of it (which we call Macaulay modules), we describe a dual analog of the Gröbner bases theory. The motivation for studying Macaulay modules has to be found mainly in multivariate interpolation problems and in the theory of polynomial bialgebras, though some application to algebraic geometry is not excluded
Chains of Differential Subvarieties in an Algebraic Variety

AbstractLet X be a δ-variety over some δ-field K. Denote by tdδ(X/K), or simply tdδ(X) if the ground field is understood, the δ-transcendental degree of K〈X〉 over K. Suppose tdδ(X)=d; Johnson [Comment. Math. Helv.44 (1969), 207–216] showed that there is an increasing chain of δ-subvarieties of length ωd in X. The question, also known as the Kolchin Catenary Problem, is: Given a point x∈X, is there an increasing chain of δ-subvarieties of length ωd starting at x? We will give an affirmative answer to this question if X is an algebraic variety
A Pieri-Type Formula for H*T(SLn(C)/B)

AbstractThe singular cohomology of the Grassmann variety of k-planes in Cn has a basis {sν} indexed by partitions. The classical Pieri formula is an explicit rule for determining the coefficients in the expansion of the cup products1m∪sλ=∑c1m,λμsμ,where 1m is a column of length m and s1m is the mth Chern class of the tautological bundle. Lascoux and Schutzenberger [C. R. Acad. Sci. Paris294 (1982), 447–450] formulated a generalization of Pieri's formula to the cohomology of the flag variety SLn(C)/B and briefly indicated an algebraic proof. (Manivel [Cours Spécialisés 3 (1998)] provides details of this proof.) A geometric proof was given by Sottile [Ann. Inst. Fourier (Grenoble) 46 (1996), 89–110]. In this paper we state and prove a generalization of this Pieri-type formula for the T-equivariant cohomology of the flag variety. We use the algebraic description of the T-equivariant cohomology of the flag variety due to Kostant and Kumar [Adv. Math.62 (1986), 187–237] and Arabia [Bull. Soc. Math. France117 (1989), 129–165], and our new formula exposes an equality of certain structure Constants in this algebra. Our proof is an induction based on the original idea in Lascoux and Schützenberger
Identities on Units of Algebraic Algebras

AbstractLet A be an algebraic algebra over an infinite field K and let U(A) be its group of units. We prove a stronger version of Hartley's conjecture for A, namely, if a Laurent polynomial identity (LPI, for short) f=0 is satisfied in U(A), then A satisfies a polynomial identity (PI). We also show that if A is non-commutative, then A is a PI-ring, provided f=0 is satisfied by the non-central units of A. In particular, A is locally finite and, thus, the Kurosh problem has a positive answer for K-algebras whose unit group is LPI. Moreover, f=0 holds in U(A) if and only if the same identity is satisfied in A. The last fact remains true for generalized Laurent polynomial identities, provided that A is locally finite
Curves of every genus with many points. I: Abelian and toric families

Let Nq(g) denote the maximal number of Fq-rational points on any curve of genus g over Fq. Ihara (for square q) and Serre (for general q) proved that lim supg→∞ Nq(g)/g > 0 for any fixed q. Here we prove limg→∞ Nq(g) = ∞. More precisely, we use abelian covers of ℙ1 to prove lim infg→∞ Nq(g)/(g/log g) > 0, and we use curves on toric surfaces to prove lim infg→∞ Nq(g)/g1/3 > 0; we also show that these results are the best possible that can be proved using these families of curves. © 2002 Elsevier Science (USA)
On the Cartan–Jacobson Theorem

AbstractThe well-known Cartan–Jacobson theorem claims that the Lie algebra of derivations of a Cayley algebra is central simple if the characteristic is not 2 or 3. In this paper we have studied these two cases, with the following results: if the characteristic is 2, the theorem is also true, but, if the characteristic is 3, the derivation algebra is not simple. We have also proved that in this last case, there is a unique nonzero proper seven-dimensional ideal, which is a central simple Lie algebra of type A2, and the quotient of the derivation algebra modulo this ideal turns out to be isomorphic, as a Lie algebra, to the ideal itself. The original motivation of this work was a series of computer-aided calculations which proved the simplicity of derivation algebras of Cayley algebras in the case of characteristic not 3. These computations also proved the existence of a unique nonzero proper ideal (which turns out to be seven-dimensional) in the algebra of derivations of split Cayley algebras in characteristic 3
Chain Conditions for Subgroups of Infinite Order or Index

AbstractA group G is said to satisfy max-∞ if each nonempty set of infinite subgroups of G has a maximal element. A group G is said to satisfy min-∞ if each nonempty set of subgroups of G with infinite index has at least one minimal element.Groups with max-∞ or min-∞ are the subject of this paper. Abelian, nilpotent, and solvable groups with max-∞ or min-∞ are examined in detail and structure theorems are given in each case. We then characterize the groups with max-∞ or min-∞ in the smallest class of groups containing all linear groups which is locally closed and closed with respect to the formation of ascending series with factors in the class
Almost Split Sequences for Comodules

AbstractLet Γ be a coalgebra over a field k. We introduce an operator Tr that takes a right quasi-finitely copresented Γ-comodule M to a left quasi-finitely copresented Γ-comodule Tr M. If M is indecomposable not injective and Tr M is finite-dimensional over k, we prove the existence of an almost split sequence 0→M→E→DTr M→0 in the category of all right Γ-comodules, where D= Homk( ,k). If Γ is right semiperfect and the embedding of each simple right comodule S into its injective envelope I(S) has the property that the socle of I(S)/S is finite-dimensional, the above almost split sequence exists for each finite-dimensional M, and DTr M is also finite-dimensional
The Blowup Closure of a Set of Ideals with Applications to TI Closure

AbstractWe introduce a new closure operation on sets of ideals in a commutative Noetherian ring of characteristic p, called the blowup closure. We develop the theory of this operation and prove that given a set of ideals in a Noetherian ring of characteristic p, under mild conditions, the tight integral closure of this set agrees with the blowup closure of certain extensions of those ideals in an extension of the original ring. We then use properties of blowup closure to settle open questions on tight integral closure posed by M. Hochster (J. Algebra230 (2000), 184–203). In particular, we show that under mild conditions on the ring, tight integral closure persists under ring maps and that it commutes with localization if and only if tight closure does
On Units of Twisted Group Algebras

AbstractWe study units of twisted group algebras. Let G be a finite group and K be a field of characteristic p>0. Assume that K is not algebraic over a finite field. We determine when units of Kt[G] do not contain any nonabelian free subgroup. We also discuss what will happen when G is locally finite. For twisted group algebras of locally finite groups over any infinite field of characteristic p>0, we characterize those twisted group algebras with units satisfying a group identity. Finally, we include a new characterization for twisted group algebras to satisfy a polynomial identity
Coxeter Elements and Kazhdan–Lusztig Cells

AbstractBy the correspondence between Coxeter elements of a Coxeter system (W,S,Γ) and the acyclic orientations of the Coxeter graph Γ, we study some properties of elements in the set C0(W). We show that when W is of finite, affine, or hyperbolic type, any w∈C0(W) satisfies w∼LRwJ with ℓ(wJ)=|J|=m(w) for some J⊂S. Now assume that W is of finite or affine type. We give an explicit description for all the distinguished involutions d of W with d∼Lw for some w∈E(W), which verifies a conjecture proposed in [Jian-yi Shi, Adv. Sci. China, Math.3 (1990), 79–98, Conjecture 8.10] in our case. We show that any left cell of W containing some element of C0(W) is left-connected, which verifies a conjecture of Lusztig [Ryoshi Hotta (Ed.), in “Problems from the Conference on Algebraic Groups and Representations held at Katata, August 29–September 3, 1983”] in our case
Double Centralizing Theorems for the Alternating Groups

AbstractLet V⊗n be the n-fold tensor product of a vector space V. Following I. Schur we consider the action of the symmetric group Sn on V⊗n by permuting coordinates. In the super (Z2 graded) case V=V0⊕V1, a ± sign is added. These actions give rise to the corresponding Schur algebras S(Sn,V). Here S(Sn,V) is compared with S(An,V), the Schur algebra corresponding to the alternating subgroup An⊂Sn. While in the classical (signless) case these two Schur algebras are the same for n large enough, it is proved that in the super case, where dim V0=dimV1,S(An,V) is isomorphic to the cross-product algebra S(An,V)≅S(Sn,V)⊗Z2