Root Sublattices and Torus-Restriction of Prehomogeneous Vector Spaces Associated with Dynkin Quivers

For a Dynkin quiver Γ with r vertices, a subset S of the vertices of Γ, and an r-tuple d = (d⁽¹⁾, d⁽²⁾,…, d^(r)) of positive integers, we define a “torus-restricted” representation (G_S, R _d (Γ)) in natural way. Here we put G_S = G₁ × G₂ × … ×G_r, where each G_i is either SL(d⁽ⁱ⁾) or GL(d⁽ⁱ⁾) according to S containing i or not. In this paper, for a prescribed torus-restriction S, we give a necessary and sufficient condition on d that R _d (Γ) has only finitely many G_S-orbits. This can be paraphrased as a condition whether or not d is contained in a certain lattice spanned by positive roots of Γ. We also discuss the prehomogeneity of (G_S, R _d (Γ)).     

Ergodic Theory and Maximal Abelian Subalgebras of the Hyperfinite Factor

Let T be a free ergodic measure-preserving action of an abelian group G on (X,μ). The crossed product algebra R_T=L^∞(X,μ)? G has two distinguished masas, the image C_T of L^∞(X,μ) and the algebra S_T generated by the image of G. We conjecture that conjugacy of the singular masas S_T⁽¹⁾ and S_T⁽²⁾ for weakly mixing actions T⁽¹⁾ and T⁽²⁾ of different groups implies that the groups are isomorphic and the actions are conjugate with respect to this isomorphism. Our main result supporting this conjecture is that the conclusion is true under the additional assumption that the isomorphism γ : R_T⁽¹⁾→R_T⁽²⁾ such that γ(S_T⁽¹⁾)=S_T⁽²⁾ has the property that the Cartan subalgebras γ(C_T⁽¹⁾) and C_T⁽²⁾ of R_T⁽²⁾ are inner conjugate. We discuss a stronger conjecture about the structure of the automorphism group Aut(R_T,S_T), and a weaker one about entropy as a conjugacy invariant. We study also the Pukanszky and some related invariants of S_T, and show that they have a simple interpretation in terms of the spectral theory of the action T. It follows that essentially all values of the Pukanszky invariant are realized by the masas S_T, and there exist non-conjugate singular masas with the same Pukanszky invariant.     

On subgraph number independence in trees

For finite graphs F and G, let N_F(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. If $\text{F}$ and $\text{G}$ are families of graphs, it is natural to ask then whether or not the quantities N_F(G), F∈$\text{F}$, are linearly independent when G is restricted to $\text{G}$. For example, if $\text{F}$ = {K₁, K₂} (where K_n denotes the complete graph on n vertices) and $\text{F}$ is the family of all (finite) trees, then of course N_K1(T) ? N_K2(T) = 1 for all T∈$\text{F}$. Slightly less trivially, if $\text{F}$ = {S_n: n = 1, 2, 3,…} (where S_n denotes the star on n edges) and $\text{G}$ again is the family of all trees, then Σ_n=1^∞(?1)ⁿ⁺¹N_Sn(T)=1 for all T∈$\text{G}$. It is proved that such a linear dependence can never occur if $\text{F}$ is finite, no F∈$\text{F}$ has an isolated point, and $\text{G}$ contains all trees. This result has important applications in recent work of L. Lovász and one of the authors (Graham and Lovász, to appear).     

Techniques of computations of Dolbeault cohomology of solvmanifolds

We consider semi-direct products ${\mathbb{C}^{n}\ltimes_{\phi}N}$ of Lie groups with lattices Γ such that N are nilpotent Lie groups with left-invariant complex structures. We compute the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ by using the Dolbeaut cohomology of the Lie algebras of the direct product ${\mathbb{C}^{n}\times N}$ . As a corollary of this computation, we can compute the Dolbeault cohomology H ^p,q (G/Γ) of G/Γ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices Γ.     

A Spanning Tree with High Degree Vertices

Let G be a connected graph, let ${X \subset V(G)}$ and let f be a mapping from X to {2, 3, . . .}. Kaneko and Yoshimoto (Inf Process Lett 73:163–165, 2000) conjectured that if |N _G (S) ? X| ≥ f (S) ? 2|S| + ω _G (S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d _T (x) ≥ f (x) for all ${x \in X}$ . In this paper, we show a result with a stronger assumption than this conjecture; if |N _G (S) ? X| ≥ f (S) ? 2|S| + α(S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d _T (x) ≥ f (x) for all ${x \in X}$ .     

Homeomorphic measures on stationary Bratteli diagrams

We study the set S of ergodic probability Borel measures on stationary non-simple Bratteli diagrams which are invariant with respect to the tail equivalence relation R. Equivalently, the set S is formed by ergodic probability measures invariant with respect to aperiodic substitution dynamical systems. The paper is devoted to the classification of measures μ from S with respect to a homeomorphism. The properties of the clopen values set S(μ) are studied. It is shown that for every measure μ∈S there exists a subgroup G⊂R such that S(μ)=G∩[0,1]. A criterion of goodness is proved for such measures. Based on this result, the measures from S are classified up to a homeomorphism. We prove that for every good measure μ∈S there exist countably many measures {μi}_i∈N⊂S such that the measures μ and μi are homeomorphic but the tail equivalence relations on the corresponding Bratteli diagrams are not orbit equivalent.     

Hecke Algebras of Group Extensions

Abstract

We describe the Hecke algebra ?(Γ,Γ₀) of a Hecke pair (Γ,Γ₀) in terms of the Hecke pair (N,Γ₀) where N is a normal subgroup of Γ containing Γ₀. To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S ? Γ/N satisfies S ^?1 S = Γ/N, we show that ? (Γ,Γ₀) is the twisted crossed product of ? (N,Γ₀) by S. This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products.     

Critical Independent Sets and König–Egerváry Graphs

A set S of vertices is independent or stable in a graph G, and we write S ∈ Ind (G), if no two vertices from S are adjacent, and α(G) is the cardinality of an independent set of maximum size, while core(G) denotes the intersection of all maximum independent sets. G is called a König–Egerváry graph if its order equals α(G) + μ(G), where μ(G) denotes the size of a maximum matching. The number def (G) = | V(G) | ?2μ(G) is the deficiency of G. The number \({d(G)=\max\{\left\vert S\right\vert -\left\vert N(S)\right\vert :S\in\mathrm{Ind}(G)\}}\) is the critical difference of G. An independent set A is critical if \({\left\vert A\right\vert -\left\vert N(A)\right\vert =d(G)}\) , where N(S) is the neighborhood of S, and α _c (G) denotes the maximum size of a critical independent set. Larson (Eur J Comb 32:294–300, 2011) demonstrated that G is a König–Egerváry graph if and only if there exists a maximum independent set that is also critical, i.e., α _c (G) = α(G). In this paper we prove that: (i) \({d(G)=\left \vert \mathrm{core}(G) \right \vert -\left \vert N (\mathrm{core}(G))\right\vert =\alpha(G)-\mu(G)=def \left(G\right)}\) holds for every König–Egerváry graph G; (ii) G is König–Egerváry graph if and only if each maximum independent set of G is critical.     

On the genus of generalized unit and unitary Cayley graphs of a commutative ring

Let R be a commutative ring, U(R) be the set of all unit elements of R, G be a multiplicative subgroup of U(R) and S be a non-empty subset of G such that S ^?1={s ^?1:?s∈S}?S. In [16], K. Khashyarmanesh et al. defined a graph of R, denoted by Γ(R,G,S), which generalizes both unit and unitary Cayley graphs of R. In this paper, we derive several bounds for the genus of Γ(R,U(R),S). Moreover, we characterize all commutative Artinian rings R for which the genus of Γ(R,U(R),S) is one. This leads to the characterization of all commutative Artinian rings whose unit and unitary Cayley graphs have genus one.     

Solution of the Hamiltonian problem for self-complementary graphs

A graph G is said to be highly constricted if there exists a nonempty subset S of vertices such that (i) G ? S has more than |S| components, (ii) S induces the complete graph, and (iii) for every u ∈ S and v ? S, we have d_G(u) > d_G(v), where d_G(u) denotes the degree of u in G. In this paper it is shown that a non-hamiltonian self-complementary graph G of order p is highly constricted, unless p = 4N and G is a particular graph $\text{G}{}^1\text{(4N)}$. It is also proved that if G is a self-complementary graph of order p(≥8) and π its degree sequence, then G is pancyclic if π has a realization with a hamiltonian cycle, and G has a 2-factor if π has a realization with a 2-factor, unless p = 4N and $\text{G = G}{}^1\text{(4N)}$.     

On the geometry of flat pseudo-Riemannian homogeneous spaces

Let M = ? _s ⁿ /Γ be a complete flat pseudo-Riemannian homogeneous manifold, Γ ? Iso(? _s ⁿ ) its fundamental group and G the Zariski closure of Γ in Iso(? _s ⁿ ). We show that the G-orbits in ? _s ⁿ are affine subspaces and affinely diffeomorphic to G endowed with the (0)-connection. If the restriction of the pseudo-scalar product on ? _s ⁿ to the G-orbits is nondegenerate, then M has abelian linear holonomy. If additionally G is not abelian, then G contains a certain subgroup of dimension 6. In particular, for non-abelian G, orbits with non-degenerate metric can appear only if dim G ≥ 6. Moreover, we show that ? _s ⁿ is a trivial algebraic principal bundle G → M → ? ^n?k . As a consquence, M is a trivial smooth bundle G/Γ → M → ? ^n?k with compact fiber G/Γ.     

The fundamental constituents of iteration digraphs of finite commutative rings

For a finite commutative ring R and a positive integer k ? 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = a ^k . Let R = R ₁ ⊕ … ⊕ R _s , where s > 1 and R _i is a finite commutative local ring for i ∈ {1, …, s}. Let N be a subset of {R ₁, …, R _s } (it is possible that N is the empty set \(\not 0\) ). We define the fundamental constituents G _N ^* (R, k) of G(R, k) induced by the vertices which are of the form {(a ₁, …, a _s ) ∈ R: a _i ∈ D(R _i ) if R _i ∈ N, otherwise a _i ∈ U(R _i ), i = 1, …, s}, where U(R) denotes the unit group of R and D(R) denotes the zero-divisor set of R. We investigate the structure of G* _N (R, k) and state some conditions for the trees attached to cycle vertices in distinct fundamental constituents to be isomorphic.     

Minimality of horospherical flows

IfN is the nilpotent constituent of an Iwasawa decomposition of the semi-simple groupG (finite center and no compact factors), it is proved thatN acts minimally onG/Γ for every uniform lattice Γ ?G, generalizing theorems of Hedlund and L. Greenberg.     

Continuity of Operators Intertwining with Convolution Operators

Let G be a locally compact abelian group, let μ be a bounded complex-valued Borel measure on G, and let T_μ be the corresponding convolution operator on L¹(G). Let X be a Banach space and let S be a continuous linear operator on X. Then we show that every linear operator Φ: X→L¹(G) such that ΦS=T_μΦ is continuous if and only if the pair (S,T_μ) has no critical eigenvalue.     

Alternating Sign Multibump Solutions of Nonlinear Elliptic Equations in Expanding Tubular Domains

Let Γ denote a smooth simple curve in ? ^N , N ≥ 2, possibly with boundary. Let Ω _R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx | x ∈ Γ??Γ}. Consider the superlinear problem ? Δu + λu = f(u) on the domains Ω _R , as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions.     

Maximal Crossed Product Orders Over Discrete Valuation Rings

The problem of determining when a (classical) crossed product T = S ^f ?G of a finite group G over a discrete valuation ring S is a maximal order, was answered in the 1960s for the case where S is tamely ramified over the subring of invariants S ^G . The answer was given in terms of the conductor subgroup (with respect to f) of the inertia. In this article we solve this problem in general when S/S ^G is residually separable. We show that the maximal order property entails a restrictive structure on the subcrossed product graded by the inertia subgroup. In particular, the inertia is abelian. Using this structure, one is able to extend the notion of the conductor. As in the tame case, the order of the conductor is equal to the number of maximal two-sided ideals of T and hence to the number of maximal orders containing T in its quotient ring. Consequently, T is a maximal order if and only if the conductor subgroup is trivial.     

Permutation-operator inequalities on certain symmetry classes of tensors

If A∈T(m, N), the real-valued N-linear functions on E_m, and σ∈S_N, the symmetric group on {…,N}, then we define the permutation operator P_σ: T(m, N) → T(m, N) such that P_σ(A)(x₁,x₂,…,x_N = A(x_σ(1),x_σ(2),…, x_σ(N)). Suppose Σ^q_i=1n_i = N, where the n_i are positive integers. In this paper we present a condition on σ that is sufficient to guarantee that 〈P_σ(A₁?A₂???A_q),A₁?A₂?? ? A_q〉 ? 0 for A_i∈S(m, n_i), where S(m, n_i) denotes the subspace of T(m, n_i) consisting of all the fully symmetric members of T(m, n_i). Also we present a broad generalization of the Neuberger identity which is sometimes useful in answering questions of the type described below. Suppose G and H are subgroups of S_N. We let T_G(m, N) denote all A∈T(m, N) such that P_σ(A) = A for all σ∈G. We define the symmetrizer $\text{S}$_G: T(m, N)→T_G(m,N) such that $\text{S}{}_{\mathrm{G}}\text{(A) = 1/|G|Σ}{}_{\mathrm{\sigma \in G}}\text{P}{}_{\mathrm{\sigma }}\text{(A)}$. Suppose H is a subgroup of G and A∈T_H(m, N). Clearly $\text{6}\text{S}{}_{\mathrm{<mtext>G</mtext>}}\text{6(A) 6? 6A6.}$ We are interested in the reverse type of comparison. In particular, if D is a suitably chosen subset of T_H(m,N), then can we explicitly present a constant C>0 such that $\text{6}\text{S}{}_{\mathrm{G}}\text{(A)6?C6A6}$ for all A∈D?     

Limits of translates of divergent geodesics and integral points on one-sheeted hyperboloids

For any non-uniform lattice Γ in SL₂(?), we describe the limit distribution of orthogonal translates of a divergent geodesic in Γ\SL₂(?). As an application, for a quadratic form Q of signature (2, 1), a lattice Γ in its isometry group, and v ₀ ∈ ?³ with Q(v ₀) > 0, we compute the asymptotic (with a logarithmic error term) of the number of points in a discrete orbit v ₀Γ of norm at most T, when the stabilizer of v ₀ in Γ is finite. Our result in particular implies that for any non-zero integer d, the smoothed count for the number of integral binary quadratic forms with discriminant d ² and with coefficients bounded by T is asymptotic to c · T log T + O(T).     

On Unicyclic Graphs with Uniquely Restricted Maximum Matchings

A graph is called unicyclic if it owns only one cycle. A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. Clearly, μ _r (G) ≤ μ(G), where μ _r (G) denotes the size of a maximum uniquely restricted matching, while μ(G) equals the matching number of G. In this paper we study unicyclic bipartite graphs enjoying μ _r (G) = μ(G). In particular, we characterize unicyclic bipartite graphs having only uniquely restricted maximum matchings. Finally, we present some polynomial time algorithms recognizing unicyclic bipartite graphs with (only) uniquely restricted maximum matchings.     

New results related to a conjecture of Moore

Let Γ be a group, Γ′ be a subgroup of Γ of finite index, and R be a ring with identity. Assume that M is an RΓ-module whose restriction to RΓ′ is projective. Moore’s conjecture: Assume that, for all ${x \in (\Gamma-\Gamma^{\prime})}$ , either there is an integer n such that ${1 \neq x^{n} \in \Gamma^{\prime}}$ or x has finite order and is invertible in R. Then M is also projective over RΓ. In this paper, we consider an analogue of this conjecture for injective modules. It turns out that the validity of the conjecture for injective modules implies the validity of it on projective and flat modules. It is also shown that the conjecture for injective modules is true whenever Γ belongs to Kropholler’s hierarchy ${{\bf LH}\mathfrak{F}}$ . In addition, assume that M is an RΓ-module whose restriction to RΓ′ is Gorenstein projective (resp. injective), it is proved that M is Gorenstein projective (resp. injective) over RΓ whenever Γ′ is a subgroup of Γ of finite index.