Two stochastic models of a random walk in the U(n)-spherical duals of U(n + 1)

The random walk to be considered takes place in the δ-spherical dual of the group U(n + 1), for a fixed finite dimensional irreducible representation δ of U(n). The transition matrix comes from the three-term recursion relation satisfied by a sequence of matrix valued orthogonal polynomials built up from the irreducible spherical functions of type δ of SU(n + 1). One of the stochastic models is an urn model and the other is a Young diagram model.     

Degenerate series representations of the universal covering group of SU(2, 2)

We prove a reducibility criterion for certain families of representations induced from irreducible finite dimensional representations of the 11-dimensional parabolic subgroup of the universal covering group of SU(2, 2). If an induced representation is reducible and can be considered as a representation of SU(2, 2) as well, we compute the number of composition factors.     

The spectral picture and the closure of the similarity orbit of strongly irreducible operators

An operator on a complex, separable, infinite dimensional Hilbert space is strongly irreducible if it does not commute with any nontrivial idempotent. This article answers the following questions of D. A. Herrero: (i) Given an operatorT with connected spectrum, can we find a strongly irreducible operatorL such that they have same spectral picture? (ii) When we use a sequence of irreducible operators to approximateT, can the approximation be the “most economic”? i.e., does there exist a strongly irreducible operatorL such thatT ∈S(L) ^? (the closure of the similarity orbit ofL)? It is shown that the answer for the two questions is yes.     

On the first cohomology of cocompact arithmetic groups

Results of Matsushima and Raghunathan imply that the first cohomology of a cocompact irreducible lattice in a semisimple Lie groupG, with coefficients in an irreducible finite dimensional representation ofG, vanishes unless the Lie group isSO(n, 1) orSU(n, 1) and the highest weight of the representation is an integral multiple of that of the standard representation. We show here that every cocompact arithmetic lattice inSO(n, 1) contains a subgroup of finite index whose first cohomology is non-zero when the representation is one of the exceptional types mentioned above.     

Embeddings of minimal non-abelian p-groups

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. For a given finite group G, let p(G) denote the minimal degree of a faithful representation of G by permutation matrices, and let c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices. See [4]. It is easy to see that c(G) is a lower bound for p(G). Behravesh [H. Behravesh, The minimal degree of a faithful quasi-permutation representation of an abelian group, Glasg. Math. J. 39 (1) (1997) 51-57] determined c(G) for every finite abelian group G and also [H. Behravesh, Quasi-permutation representations of p-groups of class 2, J. Lond. Math. Soc. (2) 55 (2) (1997) 251-260] gave the algorithm of c(G) for each finite group G. In this paper, we first improve this algorithm and then determine c(G) and p(G) for an arbitrary minimal non-abelian p-group G.     

<Emphasis Type="Italic">K</Emphasis>-finite matrix elements of irreducible Harish-Chandra modules are hypergeometric functions

We show that each K-finite matrix element of an irreducible infinite-dimensional representation of a semisimple Lie group can be obtained from spherical functions by a finite collection of operations. In particular, each matrix element admits a finite expression via the Heckman-Opdam hypergeometric functions.     

Multiplicity formulas for finite dimensional and generalized principal series representations

The article presents two results. (1) Let a be a reductive Lie algebra over ℂ and let b be a reductive subalgebra of a. The first result gives the formula for multiplicity with which a finite dimensional irreducible representation of b appears in a given finite dimensional irreducible representation of a in a general situation. This generalizes a known theorem due to Kostant in a special case. (2) LetG be a connected real semisimple Lie group andK a maximal compact subgroup ofG. The second result is a formula for multiplicity with which an irreducible representation ofK occurs in a generalized representation ofG arising not necessarily from fundamental Cartan subgroup ofG. This generalizes a result due to Enright and Wallach in a fundamental case.     

Dimension of the crown Skn

In 1941, Dushnik and Miller introduced the concept of the dimension of a poset (X, P) as the minimum number of linear extensions of P whose intersection is exactly P. Although Dilworth has given a formula for the dimension of distributive lattices, the general problem of determining the dimension of a poset is quite difficult. An equally difficult problem is to classify those posets which are dimension irreducible, i.e., those posets for which the removal of any point lowers the dimension. In this paper, we construct for each n≥3, k≥0, a poset, called a crown and denoted S^k_n, for which the dimension is given by the formula $\text{2?(n+k)}\text{(k+2)}$. Furthermore, for each t≥3, we show that there are infinitely many crowns which are irreducible and have dimension t. We then demonstrate a method of combining a collection of irreducible crowns to form an irreducible poset whose dimension is the sum of the crowns in the collection. Finally, we construct some infinite crowns possessing combinatorial properties similar to finite crowns.     

Finite-dimensional Representations for a Class of Generalized Intersection Matrix Lie Algebras

In this paper, the authors study a class of generalized intersection matrix Lie algebras gim(M_n), and prove that its every finite-dimensional semisimple quotient is of type M(n, a, c, d). Particularly, any finite dimensional irreducible gim(M_n) module must be an irreducible module of Lie algebra of type M(n, a, c, d) and any finite dimensional irreducible module of Lie algebra of type M(n, a, c, d) must be an irreducible module of gim(M_n).     

On the topology and the geometry of SO(3)-manifolds

Consider the non-standard embedding of SO(3) into SO(5) given by the five-dimensional irreducible representation of SO(3), henceforth called SO(3)_ir. In this note, we study the topology and the differential geometry of five-dimensional Riemannian manifolds carrying such an SO(3)_ir structure, i.e., with a reduction of the frame bundle to SO(3)_ir.     

Irreducible highest weight representations of the simple n-Lie algebra

A. Dzhumadil??daev classified all irreducible finite dimensional representations of the simple n-Lie algebra. Using a slightly different approach, we obtain in this paper a complete classification of all irreducible, highest weight modules, including the infinite-dimensional ones. As a corollary we find all primitive ideals of the universal enveloping algebra of this simple n-Lie algebra.     

Brauer characters with cyclotomic field of values

It has been shown in an earlier paper [G. Navarro, Pham Huu Tiep, Rational Brauer characters, Math. Ann. 335 (2006) 675-686] that, for any odd prime p, every finite group of even order has a non-trivial rational-valued irreducible p-Brauer character. For p=2 this statement is no longer true. In this paper we determine the possible non-abelian composition factors of finite groups without non-trivial rational-valued irreducible 2-Brauer characters. We also prove that, if p≠q are primes, then any finite group of order divisible by q has a non-trivial irreducible p-Brauer character with values in the cyclotomic field Q(exp(2πi/q)).     

SLEs as boundaries of clusters of Brownian loops

In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain clusters of Brownian loops (of the clusters in a Brownian loop soup). For small densities c of loops, we show that the outer boundaries of the clusters created by the Brownian loop soup are SLE_κ-type curves where κ∈(8/3,4] and c related by the usual relation c=(3κ?8)(6?κ)/2κ (i.e., c corresponds to the central charge of the model). This gives (for any Riemann surface) a simple construction of a natural countable family of random disjoint SLE_κ loops, that behaves “nicely” under perturbation of the surface and is related to various aspects of conformal field theory and representation theory. To cite this article: W. Werner, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Polynomial representations of the symplectic groups

The irreducible finite dimensional representations of the symplectic groups are realized as polynomials on the irreducible representation spaces of the corresponding general linear groups. It is shown that the number of times an irreducible representation of a maximal symplectic subgroup occurs in a given representation of a symplectic group, is related to the betweenness conditions of representations of the corresponding general linear groups. Using this relation, it is shown how to construct polynomial bases for the irreducible representation spaces of the symplectic groups in which the basis labels come from the representations of the symplectic subgroup chain, and the multiplicity labels come from representations of the odd dimensional general linear groups, as well as from subgroups. The irreducible representations of Sp(4) are worked out completely, and several examples from Sp(6) are given.     

The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras

We address two problems with the structure and representation theory of finite W-algebras associated with general linear Lie algebras. Finite W-algebras can be defined using either Kostant's Whittaker modules or a quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of finite W-algebras. The second main result is a parameterization of finite families of irreducible Gelfand-Tsetlin modules using Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Gelfand-Tsetlin modules for finite W-algebras.     

Graded Modules Over the q-Analog Virasoro-Like Algebra

In this paper, we deal with the classification of the irreducible Z-graded and Z ²-graded modules with finite dimensional homogeneous subspaces for the q analog Virasoro-like algebra L. We first prove that a Z-graded L-module must be a uniformly bounded module or a generalized highest weight module. Then we show that an irreducible generalized highest weight Z-graded module with finite dimensional homogeneous subspaces must be a highest (or lowest) weight module and give a necessary and sufficient condition for such a module with finite dimensional homogeneous subspaces. We use the Z-graded modules to construct a class of Z ²-graded irreducible generalized highest weight modules with finite dimensional homogeneous subspaces. Finally, we classify the Z ²-graded L-modules. We first prove that a Z ²-graded module must be either a uniformly bounded module or a generalized highest weight module. Then we prove that an irreducible nontrivial Z ²-graded module with finite dimensional homogeneous subspaces must be isomorphic to a module constructed as above. As a consequence, we also classify the irreducible Z-graded modules and the irreducible Z ²-graded modules with finite dimensional homogeneous subspaces and center acting nontrivial. Supported by the National Science Foundation of China (No 10671160), the China Postdoctoral Science Foundation (No. 20060390693), the Specialized Research fund for the Doctoral Program of Higher Education (No.20060384002), and the New Century Talents Supported Program from the Education Department of Fujian Province.     

Galois Theory for Braided Tensor Categories and the Modular Closure

Given a braided tensor *-category with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory we define a crossed product . This construction yields a tensor *-category with conjugates and an irreducible unit. (A *-category is a category enriched over Vect with positive *-operation.) A Galois correspondence is established between intermediate categories sitting between and and closed subgroups of the Galois group Gal( / )=Aut ( ) of , the latter being isomorphic to the compact group associated with by the duality theorem of Doplicher and Roberts. Denoting by the full subcategory of degenerate objects, i.e., objects which have trivial monodromy with all objects of , the braiding of extends to a braiding of iff . Under this condition, has no non-trivial degenerate objects iff = . If the original category is rational (i.e., has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category ≡ is called the modular closure of since in the rational case it is modular, i.e., gives rise to a unitary representation of the modular group SL(2,  ). If all simple objects of have dimension one the structure of the category can be clarified quite explicitly in terms of group cohomology.     

How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert”

Navier?CCauchy format for Continuum Mechanics is based on the concept of contact interaction between sub-bodies of a given continuous body. In this paper, it is shown how??by means of the Principle of Virtual Powers??it is possible to generalize Cauchy representation formulas for contact interactions to the case of Nth gradient continua, that is, continua in which the deformation energy depends on the deformation Green?CSaint-Venant tensor and all its N ? 1 order gradients. In particular, in this paper, the explicit representation formulas to be used in Nth gradient continua to determine contact interactions as functions of the shape of Cauchy cuts are derived. It is therefore shown that (i) these interactions must include edge (i.e., concentrated on curves) and wedge (i.e., concentrated on points) interactions, and (ii) these interactions cannot reduce simply to forces: indeed, the concept of K-forces (generalizing similar concepts introduced by Rivlin, Mindlin, Green, and Germain) is fundamental and unavoidable in the theory of Nth gradient continua.     

Realizing irreducible semigroups and real algebras of compact operators

Let B(X) be the algebra of bounded operators on a complex Banach space X. Viewing B(X) as an algebra over R, we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions, answering the question: is an irreducible semigroup of compact operators with real spectra realizable, i.e., simultaneously similar to a semigroup whose matrices are real?