Analytic continuation of local representations of symmetric spaces

We prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space (G, K, σ). The problem of passing from the Euclidean group to the Poincaré group appears first to have been addressed and solved this way by Klein and Landau. Let G be a Lie group, K a closed subgroup, and σ an involutive automorphism with K as fixed-point subgroup. If = + is the corresponding symmetric Lie algebra, we form ^* = + , and let G^* denote the simply connected Lie group with ^* as Lie algebra. We consider virtual representations π of G on a fixed complex Hilbert space , adopting the definitions due to J. Fröhlich, K. Osterwalder, and E. Seiler; in particular, π(g⁻¹) π(σ(g))^* (possibly unbounded operators) for g in a neighborhood of e in G. We prove that every such π continues analytically to a strongly continuous unitary representation of G^* on . Our theorem extends results due to Klein-Landau, Fröhlich et al., and others, earlier, for special cases. Previous results were known only for special (G, K, σ), and then only for certain π.     

The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.     

ON SYMMETRIC UNITS IN GROUP ALGEBRAS

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K. The anti-automorphism g → g ^?1 of G can be extended linearly to an anti-automorphism a → a ^* of KG. Let S _* (KG) = {x ∈ U(KG) | x ^* = x} be the set of all symmetric units of U(KG). We consider the following question: for which groups G and commutative rings K it is true that S _* (KG) is a subgroup in U(KG). We answer this question when either a) G is torsion and K is a commutative G-favourable integral domain of characteristic p≥ 0 or b) G is non-torsion nilpotent group and KG is semiprime.     

Equivariant K-Theory of Compact Connected Lie Groups Dedicated to Daniel Quillen on the Occasion of his Sixtieth Birthday

We compute the equivariant K-theory K _G ^* (G)for a compact connected Lie group Gsuch that ₁ (G)is torsion free (where Gacts on itself by conjugation). We prove that K _G ^* (G)is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a compact connected Lie group Gwith ₁ (G)torsion, namely PSU(3), and compute the corresponding equivariant K-theory.     

Hamiltonism and Partially Square Graphs

 Given a graph G, we define its partially square graph G ^* as the graph obtained by adding edges uv whenever the vertices u and v have a common neighbor x satisfying the condition N _G[x]⊆N _G[u]∪N _G [v], where N _G[x]=N _G(x)∪{x}. In particular, this condition is satisfied if x does not center a claw (an induced K _1,3). Obviously G⊆G ^*⊆G ², where G ² is the square of G. We prove that a k-connected graph (k≥2) G is hamiltonian if the independence number α(G ^*) of G ^* does not exceed k. If we replace G ^* by G we get a well known result of Chvátal and Erdo?s. If G is claw-free and G ^* is replaced by G ² then we obtain a result of Ainouche, Broersma and Veldman. Relationships between connectivity of G and independence number of G ^* for other hamiltonian properties are also given in this paper. Received: June 17, 1996 Revised: October 30, 1998     

Affine Hecke algebras and the Schubert calculus

Using a combinatorial approach that avoids geometry, this paper studies the structure of K_T(G/B), the T-equivariant K-theory of the generalized flag variety G/B. This ring has a natural basis (the double Grothendieck polynomials), where is the structure sheaf of the Schubert variety X_w. For rank two cases we compute the corresponding structure constants of the ring K_T(G/B) and, based on this data, make a positivity conjecture for general G which generalizes the theorems of M. Brion (for K(G/B)) and W. Graham (for H_T^*(G/B)). Let [X^λ]K_T(G/B) be the class of the homogeneous line bundle on G/B corresponding to the character of T indexed by λ. For general G we prove “Pieri–Chevalley formulas” for the products , , , and , where λ is dominant. By using the Chern character and comparing lowest degree terms the products which are computed in this paper also give results for the Grothendieck polynomials, double Schubert polynomials, and ordinary Schubert polynomials in, respectively K(G/B), H_T^*(G/B) and H^*(G/B).     

Enumerating Groups Acting Regularly on a d-Dimensional Cube

Let K be a field of characteristic p > 0, K* the multiplicative group of K and G = G _p  × B a finite group, where G _p is a p-group and B is a p′-group. Denote by K ^λ G a twisted group algebra of G over K with a 2-cocycle λ ∈Z ²(G, K*). In this article, we give necessary and sufficient conditions for K ^λ G to be of OTP representation type, in the sense that every indecomposable K ^λ G-module is isomorphic to the outer tensor product V#W of an indecomposable K ^λ G _p -module V and an irreducible K ^λ B-module W.     

Intersection Graphs and the Clique Operator

 Let P be a class of finite families of finite sets that satisfy a property P. We call ΩP the class of intersection graphs of families in P and CliqueP the class of graphs whose family of cliques is in P. We prove that a graph G is in ΩP if and only if there is a family of complete sets of G which covers all edges of G and whose dual family is in P. This result generalizes that of Gavril for circular-arc graphs and conduces those of Fulkerson-Gross, Gavril and Monma-Wei for interval graphs, chordal graphs, UV, DV and RDV graphs. Moreover, it leads to the characterization of Helly-graphs and dually chordal graphs as classes of intersection graphs. We prove that if P is closed under reductions, then CliqueP=Ω(P ^*∩H) (P ^*= Class of dual families of families in P). We find sufficient conditions for the Clique Operator, K, to map ΩP into ΩP ^*. These results generalize several known results for particular classes of intersection graphs. Furthermore, they lead to the Roberts-Spencer characterization for the image of K and the Bandelt-Prisner result on K-fixed classes. Received: August 18, 1997 Final version received: March 30, 1999     

Actions of Borel Subgroups on Homogeneous Spaces of Reductive Complex Lie Groups and Integrability

Let G be a real reductive Lie group, K its compact subgroup. Let A be the algebra of G-invariant real-analytic functions on T ^*(G/K) (with respect to the Poisson bracket) and let C be the center of A. Denote by 2(G,K) the maximal number of functionally independent functions from A\C. We prove that (G,K) is equal to the codimension (G,K) of maximal dimension orbits of the Borel subgroup BG ^C in the complex algebraic variety G ^C/K ^C. Moreover, if (G,K)=1, then all G-invariant Hamiltonian systems on T ^*(G/K) are integrable in the class of the integrals generated by the symmetry group G. We also discuss related questions in the geometry of the Borel group action.     

Morava K-Theories of p-Groups with Cyclic Maximal Subgroups and Other Related Groups

Let P be a non-Abelian finite p-group, p odd, with cyclic maximal subgroups, and let K(n)^*(–) denote the nth Morava K-theory at p. In this paper we determine the algebras K(n)^*(BP) and K(n)^*(BG) for all groups G with Sylow p-subgroups isomorphic to P, giving further evidence for the fact that Morava K-theory as an invariant of finite groups, is finer than ordinary modp cohomology. Mathematics Subject Classifications (2000): 55N20, 55N22.     

Characters of the infinite symplectic group—A Riesz ring approach

We classify both the finite and infinite characters of the inductive limit symplectic group G. An important feature of our technique is the systematic use of a multiplicative structure on an “ordered completion” of the K₀-group for the group C^*-algebra A of G. We also give explicit examples of the K-theory for certain primitive quotients of A.     

On a dual pair of spaces of smooth and generalized random variables

A dual pairG andG ^* of smooth and generalized random variables, respectively, over the white noise probability space is studied.G is constructed by norms involving exponentials of the Ornstein-Uhlenbeck operator,G ^* is its dual. Sufficient criteria are proved for when a function onL(ℝ) is theL-transform of an element inG orG ^*.     

On <Emphasis Type="Italic">c</Emphasis>*-normal subgroups in finite groups

A subgroup H of a finite group G is called a c*-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an S-quasinormal embedded subgroup of G. In this paper, the structure of a finite group G with some c*-normal maximal subgroups of Sylow subgroups is characterized and some known related results are generalized.     

Solution of the twisting problem for skew group algebras

LetK be any field of characteristicp>0 and letG be a finite group acting onK via a map τ. The skew group algebraK _τG may be nonsemisimple (precisely whenP|(H), H=Kert). In [1] necessary conditions were given for the existence of a class α∈H ²(G,K^*) which “twists” the skew group algebraK _τG into a semisimple crossed productK _τ ^αG . The “twisting problem” asks whether these conditions are sufficient. In [1] we showed that this is indeed so in many cases. In this paper we prove it in general. During the period of this research the second author was an Associate at the Center for Advanced Study, Urbana, Illinois.     

Commutative rings whose zero-divisor graph is a proper refinement of a star graph

A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G _c ^* be the subgraph of G induced on the vertex set V (G)\ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G = Γ(R), which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G _c ^* has at least two connected components. We prove that the diameter of the induced graph G _c ^* is two if Z(R)² ≠ {0}, Z(R)³ = {0} and G _c ^* is connected. We determine the structure of R which has two distinct nonadjacent vertices α, β ∈ Z(R)* \ {c} such that the ideal [N(α) ∩ N(β)]∪ {0} is generated by only one element of Z(R)*\{c}. We also completely determine the correspondence between commutative rings and finite complete graphs K _n with some end vertices adjacent to a single vertex of K _n .     

Continuous spectral decompositions of Abelian group actions on -algebras

Let G be a locally compact Abelian group. Following Ruy Exel, we view Fell bundles over the Pontrjagin dual group of G as continuous spectral decompositions of G-actions on C^*-algebras. We classify such spectral decompositions using certain dense subspaces related to Marc Rieffel's theory of square-integrability. There is a unique continuous spectral decomposition if the group acts properly on the primitive ideal space of the C^*-algebra. But there are also examples of group actions without or with several inequivalent spectral decompositions.     

Induced characters, Mackey analysis and primitive ideal spaces of nilpotent discrete groups

Let G be a nilpotent discrete group and Prim(C^*(G)) the primitive ideal space of the group C^*-algebra C^*(G). If G is either finitely generated or has absolutely idempotent characters, we are able to describe the hull-kernel topology on Prim(C^*(G)) in terms of a topology on a parametrizing space of subgroup-character pairs. For that purpose, we introduce and study induced traces and develop a Mackey machine for characters. We heavily exploit the fact that the groups under consideration have the property that every faithful character vanishes outside the finite conjugacy class subgroup.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

Local c*-Supplementation of Some Subgroups in Finite Groups

A subgroup H of a finite group G is called c*-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is S-quasinormally embedded in G. In this paper, we investigate the local c*-supplementation of maximal subgroups of some Sylow p-subgroup and present some sufficient and necessary conditions for a finite group to be p-nilpotent. As applications, we give some sufficient conditions for a finite group to be in a saturated formation.     

On self duality of pathwidth in polyhedral graph embeddings

Let G be a 3‐connected planar graph and G^* be its dual. We show that the pathwidth of G^* is at most 6 times the pathwidth of G. We prove this result by relating the pathwidth of a graph with the cut‐width of its medial graph and we extend it to bounded genus embeddings. We also show that there exist 3‐connected planar graphs such that the pathwidth of such a graph is at least 1.5 times the pathwidth of its dual. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 42–54, 2007