Injective Homomorphism Research Articles

Schur subalgebras II

Let p be a prime, and k an infinite field of characteristic p. In [2], the author and Martin re-proved a result of Henke [4] in which the Schur algebra S(2 ,d) over k is shown to embed in the Schur algebra S(2 ,r) for certain values of d and r, corresponding to certain self-similarity properties of the decomposition matrices for S(2 ,r) . We also constructed embeddings of S(2 ,r) in S(2 ,r p)for all r, reflecting further the structure of the decomposition matrices. Here, an embedding is not necessarily an injective homomorphism of algebras, but simply a linear injection preserving the multiplication rule. In this paper we continue to study such embeddings, and examine their consequences for decomposition numbers. Essential results concerning Schur algebras can be found in the books of Green [3] and Martin [6]; further results and notation are taken from [2]. In Section 2 we construct embeddings

Multiple Pattern Interpretations

A word w is obtained by an ordered n-pattern interpretation of a word x if there are n homomorphisms h1,h2,ċċċ, hn such that w=h1(x)h2(x)ċs hn(x). This ordered multiple pattern interpretation is naturally extended to languages. We show a strong relationship between the family of languages obtained by ordered multiple pattern interpretations of regular, linear, and context-free languages and the family of regular, linear, and context-free simple matrix languages. Concepts of ambiguity and inherent ambiguity of ordered multiple pattern interpretation are defined and it is shown that these properties are not decidable on the class of context-free languages. Then, we investigate arbitrary multiple pattern interpretations of the same classes of languages in the Chomsky hierarchy. We show that the classes of languages obtained in this way are recognizable in polynomial time provided that all components of the pattern interpretation are injective homomorphisms. We also present a series of open problems.

Partial covers of graphs

Given graphs G and H, a mapping f : V (G) → V (H) is a homomorphism if (f(u), f(v)) is an edge of H for every edge (u, v) of G. In this paper, we initiate the study of computational complexity of locally injective homomorphisms called partial covers of graphs. We motivate the study of partial covers by showing a correspondence to generalized (2,1)-colorings of graphs, the notion stemming from a practical problem of assigning frequencies to transmitters without interference. We compare the problems of deciding existence of partial covers and of full covers (locally bijective homomorphisms), which were previously studied.

ON THE TORELLI THEOREM FOR MINIMAL IMMERSIONS

International Journal of MathematicsVol. 10, No. 06, pp. 697-706 (1999) No AccessON THE TORELLI THEOREM FOR MINIMAL IMMERSIONSCLAUDIO AREZZOCLAUDIO AREZZOMathematics Institute, University of Warwick, Coventry CV4 7AL, UKThe author was supported by European Community Grant TRM N. ERBFMBICT 950079.https://doi.org/10.1142/S0129167X99000288Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetails Recommended Vol. 10, No. 06 Metrics History Received 5 August 1998 PDF download

On injective homomorphisms between Teichmüller modular groups I

In this paper and its sequel, we prove that injective homomorphisms between Teichmuller modular groups of compact orientable surfaces are necessarily isomorphisms, if an appropriately measured “size” of the surfaces in question differs by at most one. In particular, we establish the co-Hopfian property for modular groups of surfaces of positive genus.

Smooth representations of reductive p -ADIC groups: structure theory via types

Here, F denotes a non-Archimedean local field (with finite residue field) and G the group of F-points of a connected reductive algebraic group defined over F. Let R(G) denote the category of smooth, complex representations of G. Let B(G) be the set of pairs (L, σ), where L is an F-Levi subgroup of G and σ is an irreducible supercuspidal representation of L, taken modulo the equivalence relation generated by twisting with unramified quasi characters and G-conjugacy. To s ∈ B(G), one can attach a full (abelian) sub-category Rs(G) of R(G); the theory of the Bernstein centre shows that R(G) is the direct product of these Rs(G). The object of the paper is to give a general method for describing these factor categories via representations of compact open subgroups within a uniform framework. Fix s ∈ B(G). Let K be a compact open subgroup of G and ρ an irreducible smooth representation of K. The pair K, ρ is an s-type if it has the following property: an irreducible representation π of G contains ρ if and only if π in Rs(G). Let H(G, ρ) be the Hecke algebra of compactly supported ρ-spherical functions on G; if (K,ρ) is an s-type, then the category Rs(G) is canonically equivalent to the category H(G, ρ)-Mod of H(G, ρ-modules. Let M be a Levi subgroup of G; there is a canonical map B(M) to B(G). Take t ∈ B(M) with image s ∈ B(G). The choice of a parabolic subgroup of G with Levi component M gives functors of parabolic induction and Jacquet restriction connecting Rt(M) with Rs(G). We assume given a t-type (KM,ρM) in M; the paper concerns a general method of constructing from this data an s-type (K,ρ) in G. One thus obtains a description of these induction and restriction functors in terms of an injective ring homomorphism H(M,ρM) → H(G,ρ). The method applies in a wide variety of cases, and subsumes much previous work. Under further conditions, observed in certain particularly interesting cases, one can go some distance to describing H(G,ρ) explicitly. This enables one to isolate cases in which the map on Hecke algebras is an isomorphism, and this in turn implies powerful intertwining theorems for the types. 1991 Mathematics Subject Classification: 22E50.

Constructing essential laminations in 2-bridge knot surgered 3-manifolds

A nontrivial knot that can be drawn with only two relative maxima in the vertical direction is called a 2-bridge knot, and one that can be drawn on a torus is called a torus knot. Loosely speaking, a lamination in a manifold M is a foliation of M, except that it can have a nonempty open complement in M, and very loosely speaking, the lamination is essential if each leaf of it L is incompressible, i.e. inclusion of L into M induces an injective homomorphism from 1(L) into 1(M). Our main result is: Theorem 2. Every 3-manifold obtained by surgery on a nontorus 2-bridge knot admits essential laminations. Some immediate corollaries are that these manifolds are covered by R 3 , and have innite fundamental group. So Property P is true for non-torus 2-bridge knots, i.e. surgery on these knots never yields a homotopy 3-sphere, or a counterexample to the Poincare conjecture. We use general techniques which are not specic to 2-bridge knots to nd and explicitly construct these laminations. In trying to understand 3-manifolds with the hope of eventually classifying them, as with 2-manifolds, one approach that has turned out to be fruitful is to study objects of codimension one in them, more specically, incompressible surfaces, Reebless foliations, and essential laminations. For a 3-manifoldM containing an incompressible surface, with some extra hypotheses, Waldhausen proved: The universal cover of M is R 3 , the topology of M is determined by its fundamental group, and homotopic homeomorphisms of M are isotopic. Similar theorems for manifolds containing Reebless foliations were proven by Haefliger, Novikov, Palmeira, Rosenberg, and others. The essential lamination was developed in the 1980’s as a generalization of the incompressible surface and the Reebless foliation, which themselves qualify as essential laminations. In fact they are just extreme cases of essential laminations: At one end we have compact properly embedded surfaces,

Amalgamated sums of groups

Groups called amalgamated sums that arise as inductive limits of systems of groups and injective homomorphisms are studied. The problem is to find conditions under which the groups in the system do not collapse in the limit. Such a condition is given by J. Tits when certain subsystems are associated to buildings. This condition can be phrased to apply to certain systems of abstract groups and injective homomorphisms. It is shown to imply that no collapse occurs in the limit in a strong sense; namely the natural homomorphism of the amalgamated sum of any subsystem into the amalgamated sum of the full system is injective. This answers a question of S. J. Pride.

Algebraic and Turing Separability of Rings

LetFbe a countable field/ring. Then aweak presentationofFis an injective homomorphism fromFinto a field/ring whose universe is2such that all the field/ring operations are translated by total recursive functions. Given two recursive integral domainsR1andR2with quotient fieldsF1andF2respectively, we investigate under what circumstances there exists a weak presentation of the fieldF1F2such that the images ofR1andR2belong to two different recursively enumerable (r.e.) Turing degrees. In many cases we succeed in giving a completely algebraic necessary and sufficient condition for the “Turing separation” described above. More specifically, under some conditions, we can make the images ofR1andR2be of arbitrary r.e. degrees. The algebraic condition is a generalization of the notion of algebraic field separability. As a result of our investigation, we also show that3has an r.e. weak presentation as a ring which is not a weak presentation of4as a field.

Homomorphisms, ideals and commutativity in the Stone-Čech compactification of a discrete semigroup

In this paper we consider the Stone-Čech compactifications of discrete semigroups which are countable, commutative and cancellative. We show that, if S and T are two such semigroups, then any continuous injective homomorphism from T ∗ into S ∗ arises from a homomorphism mapping a cofinite subsemigroup of T into S. We also consider the structure of semiprincipal left ideals of βS, and show that any nonminimal ideal of this kind lies immediately above 2 c others for a certain order relationship on these ideals. We also show that it belongs to a reverse well-ordered chain of such ideals of type ω 1 ∗ , each maximal subject to being strictly less than all its predecessors in this ordering. We also show that each nonminimal idempotent of βS lies immediately above 2 c other idempotents for the order relation defined on idempotents by stating that α ⩽ β if α + β = β + α = α. Finally, if η is any nonminimal idempotent in βS, we show that the centre of the semigroup η + βS + η is contained in G + η, where G denotes the group generated by S.

Invariant Differential Operators in Symmetrical Spaces. II.Generalized Harish-Chandra Homomorphism

We construct in this article an injective homomorphism, γ, from the algebra of invariant differential operators on a general symmetric space in the field of invariant rational functions on the cotangent space at the origin. We conjecture that this homomorphism is an isomorphism into the invariant polynomial functions. This fact would prove a conjecture of Duflo. For describing γ, we introduce, in the first part, a class of symmetric spaces for which the symmetrisation is an isomorphism.

The adjoints to the derivative functor on species

As a direct consequence of the Kan Extension Theorem, the derivative functor, D, on (combinatorial) species, has both a left adjoint, M X , and a right adjoint, II. The functor M X can be described as “tensoring by X,” whereas the functor II is new. We show that there is an injective homomorphism, Card, from the rig (informally, a “ring without negatives”) of isomorphism classes of finitary species Species f x into the rig of functions N Mol under pointwise addition and multiplication, which “preserves” D and its adjoints. This, in effect, allows one to translate some of the calculations involving these functors to solving linear equations over the natural numbers.

Le groupe de Picard de l'anneau des polynômes à valeurs entières

We study the Picard group of the ring B of integer-valued polynomials over a noetherian one dimensional domain A with finite residue fields. This problem can be reduced to the case of a local ring A ; let m be the maximal ideal and A * the m-adic completion of A . There is a natural injective homomorphism from Pic ( B ) to the class group χ(A, π(A))/π(A) of locally constant functions from A to the group π(A) of principal fractional ideals of A . This homomorphism is an isomorphism if and only if the spectrum of B/mB is homeomorphic with A * and that is equivalent to the integrity of A * .

A Characterization of Ideals of C* -Algebras

AbstractLet A be a C*-algebra and let I be a C*-subalgebra of A. Denote by an extension of a state φ of B to a state of A. It is shown that I is an ideal of A if and only if there exists a homomorphism Q from A** onto I** such that Q is the identity map on I** and for every state φ on I. Furthermore it is also shown that I is an essential ideal of A if and only if there exists an injective homomorphism from A into the multiplier algebra of I which is the identity map on I.

A note on the coefficient rings of polynomial rings

LetA andB be two reduced commutative rings with finitely many minimal prime ideals. If the polynomial algebrasA[X 1 …X n ]=B[Y 1 …Y n ] whereX i ,Y iF are variables overA andB respectively, then there exists an injective ring homomorphism ϕ:A→B such thatB is finitely generated over ϕ(A).

On the diffeomorphism group of a reducible 3-manifold

The loop space of the configuration space of César de Sá-Rourke-Hendriks-Laudenbach is identified as a factor (up to homotopy) of a certain space of imbeddings of a punctured 3-cell B into the compact 3-manifold M. The restriction map from the diffeomorphism group of M to this space of imbeddings is shown to be a product fibration. As an application, it is proved that the imbedding Diff( Mrel Bmod ∂M)→Diff( Mmod ∂M) induces injective homomorphisms of homotopy groups.

On an alternative functional equation related to the Cauchy equation

We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, b∈B,b≠0,M a positive integer; find all functionsf:G →B such that for every (x, y) ∈G ×G the Cauchy differencef(x+y)−f(x)−f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG.

On the complete simulation of DOL schemes and locally catenative schemes

Let ( Σ, h ) be a DOL scheme where h is a homomorphism from Σ * to Σ * . ( Σ, h ) is said to completely simulate another DOL scheme (Δ, g ) if hf = fg and h n = fr} for some injective homomorphism f , some positive integer n , and some homomorphism r . The main results of this paper are: (1) Each complete simulation can be decomposed into a series of “primitive” simulations. (2) It is decidable whether or not there exists a standard locally catenative scheme which is simulated by a given DOL scheme.

TWO CLASSES OF HOMOMORPHISMS IN MODULE CATEGORIES

Abstract In a category R-Mod a homomorphism α:A → B is called projective if α factors through every epimorphism with B as image. Injective homomorphisms are defined dually. Some properties of such homomorphisms are derived, and it is shown that the hereditariness of the ring R is equivalent to some conditions which can be simply stated in terms of projective and injective homomorphisms.

A generalized weight for linear codes and a Witt-MacWilliams theorem

Let F be a finite field, H a subgroup of F ∗ of index ν, and α 1,…, α ν coset representatives. For each n-tuple u = ( u 1,…, u n ) ϵ F n define W H ( u) = ( w 1( u),…, w ν ( u)), where w m ( u) = #{ u i : u i ϵ α m H}. An H-monomial map on F n is an automorphism of F n whose matrix with respect to the co-ordinate basis is of the form P · D, where P is a permutation matrix and D is a diagonal matrix with non-zero entries from H. Suppose C is an ( n, k) code over F (that is, a k-dimensional subspace of F n ) and that ϕ: C → F n is an injective homomorphism which preserves W H in the sense that W H ( ϕ(u)) = W H (u) for all u ϵ C. We prove that ϕ may be extended to an H-monomial map on F n . This generalization of a theorem of MacWilliams on the (Hamming) equivalence of codes may be considered an analogue of the Witt theorem of metric vector space theory.