On critically connected digraphs

A digraph is called critically connected if it is connected, but the deletion of any vertex destroys the connectivity. We prove that every critically connected finite digraph has at least two vertices of outdegree one. As an application, we show that for n ≧ 2, there is no n-connected, non-complete, finite digraph such that the deletion of any n vertices results in a disconnected digraph.     

On graphs related to conjugacy classes of groups

LetG be a finite group. Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. In [2],[3] it is shown that the above bounds are best possible. The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at the Tel Aviv University under the supervision of Prof. Marcel Herzog.     

On finite reflexive homomorphism-homogeneous binary relational systems

A structure is called homogeneous if every isomorphism between finitely induced substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Nešet?il introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely induced substructures of the structure extends to an endomorphism of the structure.In this paper, we consider finite homomorphism-homogeneous relational systems with one reflexive binary relation. We show that for a large part of such relational systems (bidirectionally connected digraphs; a digraph is bidirectionally connected if each of its connected components can be traversed by ?-paths) the problem of deciding whether the system is homomorphism-homogeneous is coNP-complete. Consequently, for this class of relational systems there is no polynomially computable characterization (unless P=NP). On the other hand, in case of bidirectionally disconnected digraphs we present the full characterization. Our main result states that if a digraph is bidirectionally disconnected, then it is homomorphism-homogeneous if and only if it is either a finite homomorphism-homogeneous quasiorder, or an inflation of a homomorphism-homogeneous digraph with involution (a specific class of digraphs introduced later in the paper), or an inflation of a digraph whose only connected components are and .     

Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

 Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that the algebra of the differential operators on the Schwartz-space of τ which commute with τ is isomorphic to the algebra of H-invariant polynomials on the affine space . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in . We prove also that if is an ideal of , then the finite multiplicities of τ is equivalent to the fact that the algebra is commutative.     

Diagram automorphisms of quiver varieties

We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the ‘split-quotient quiver’ introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the quiver varieties of type A correspond to small self-dual representations, we show that the diagram involutions coincide with classical involutions of two-row Slodowy varieties. It follows that certain quiver varieties of type D are isomorphic to Slodowy varieties for orthogonal or symplectic Lie algebras.     

Representing groups by graphs with constant link and hypergraphs

A graph L is called a link graph if there is a graph G such that for each vertex of G its neighbors induce a subgraph isomorphic to L. Such a G is said to have constant link L. We prove that for any finite group Γ and any disconnected link graph L with at least three vertices there are infinitely many connected graphs G with constant link L and AutG ? Γ. We look at the analogous problem for connected link graphs, namely, link graphs that are paths or have disconnected complements. Furthermore we prove that for n, r ≥ 2, but not n = 2 = r, any finite group can be represented by infinitely many connected r-uniform, n-regular hypergraphs of arbitrarily large girth.     

Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

 Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that the algebra of the differential operators on the Schwartz-space of τ which commute with τ is isomorphic to the algebra of H-invariant polynomials on the affine space . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in . We prove also that if is an ideal of , then the finite multiplicities of τ is equivalent to the fact that the algebra is commutative. (Received 15 November 2000)     

Cartan decompositions and BGG-resolutions

Necessary and sufficient criteria are given for the existence of BGG-resolutions (finite resolutions of modules by finite direct sums of Weyl modules) for simple modules over quasi-hereditary algebras, which have strong exact Borel subalgebras and strong Δ-subalgebras. Our main technical tool is the existence of Cartan decompositions for these algebras. The results apply to simple objects in the BGG-categoryO of a finite-dimensional semisimple complex Lie algebra and to finite dimensional simple rational modules over simply connected semisimple algebraic groups.     

Trajectory nets connecting all critical points of a smooth function

The disconnected components of certain trajectory nets containing all critical points of a differentiable functionf can be connected by suitably chosen contour sets of a certain associated functiong. A recursive construction yields a locally 1-dimensional connected set Ω which contains all critical points. The possible ways of tracing this set numerically are discussed. This work was supported by the Deutsche Forschungsgemeinschaft.     

The topology at infinity of Coxeter groups and buildings

The connectivity at infinity of a finitely generated Coxeter group W is completely determined by topological properties of its nerve L (a finite simplicial complex). For example, W is simply connected at infinity if and only if L and the subcomplexes (where ranges over all simplices in L) are simply connected. This characterization extends to locally finite buildings. Received: May 3, 2001     

The up-down formula for nil-homogeneous spaces

One considers a multiflow on a nil-homogeneous space—that is, the action of a vector group A on a quotient N/H, where N is a simply connected nilpotent Lie group and H is a closed connected subgroup. If A is also a subgroup of N, then the corresponding unitary action of A on L²(N/H) is anup-down representation—a succession of induced and restricted representations. An explicit orbital formula for the direct integral decomposition of this unitary representation is obtained. An unexpectedly simple sufficient condition for finite multiplicity is derived as a consequence. Similarities to ergodic actions are indicated.Supported by NSF # DMS-90-02642.     

A tensor product of the Steinberg module and a certain simple kG(pr)-module

Let G be a connected, semisimple, and simply connected algebraic group defined and split over the finite field of order p, and let G(q) be the corresponding finite Chevalley or twisted group, where q = p^r. Recently, Anwar determines the direct sum decomposition of the tensor product of the rth Steinberg module and a simple G-module with a (p,r)-minuscule highest weight λ. In this paper, we determine that of the tensor product regarded as a module for G(q) under some weak assumptions for λ.     

A splitting theorem for manifolds of almost nonnegative Ricci curvature

Cheeger and Gromoll proved that a closed Riemannian manifold of nonnegative Ricci curvature is, up to a finite cover, diffeomorphic to a direct product of a simply connected manifold and a torus. In this paper, we extend this theorem to manifolds of almost nonnegative Ricci curvature.     

The representation dimension of domestic weakly symmetric algebras

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.     

Extremal Problems on Components and Loops in Graphs

The authors recently defined a new graph invariant denoted by Ω(G) only in terms of a given degree sequence which is also related to the Euler characteristic. It has many important combinatorial applications in graph theory and gives direct information compared to the better known Euler characteristic on the realizability, connectedness, cyclicness, components, chords, loops etc. Many similar classification problems can be solved by means of Ω. All graphs G so that Ω(G) ≤-4 are shown to be disconnected, and if Ω(G) ≥-2, then the graph is potentially connected. It is also shown that if the realization is a connected graph and Ω(G) =-2, then certainly the graph should be a tree.Similarly, it is shown that if the realization is a connected graph G and Ω(G) ≥ 0, then certainly the graph should be cyclic. Also, when Ω(G) ≤-4, the components of the disconnected graph could not all be cyclic and if all the components of G are cyclic, then Ω(G) ≥ 0. In this paper, we study an extremal problem regarding graphs. We find the maximum number of loops for three possible classes of graphs.We also state a result giving the maximum number of components amongst all possible realizations of a given degree sequence.     

Connectedness of number theoretical tilings (dual tilings generated by Pisot units of degree 3 and 4)

We study the connectedness of Pisot dual tilings. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4 which have infinitely many connected components. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, we give a complete classification of the β expansion of 1 for quartic Pisot units. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sur la connexité en géométrie analytique complexe locale

In [3] A. Grothendieck shows that a subscheme Y of a local and complete n-dimensional irreductible scheme, defined by p equations is connected in dimension n − p. This means that Y cannot be disconnected by removing a subspace of dimension < n − p. The aim of this paper is essentially to give a local analytic version of this theorem and a direct proof using only analytic geometry methods.