Multiplicity one theorem for ({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))

Let F be either or . Consider the standard embedding and the action of GL_n(F) on GL_n+1(F) by conjugation. We show that any GL_n(F)-invariant distribution on GL_n+1(F) is invariant with respect to transposition. We prove that this implies that for any irreducible admissible smooth Fréchet representations π of GL_n+1(F) and of GL_n(F),

Un théorème de finitude (A Theorem of Finitude)

We prove that, for an inner form of GL _n , the number of classes of cuspidal automorphic representations with fixed central character and fixed factor at almost every place is finite. We also prove, in the local situation, relations between the level and the -factor of an irreducible smooth representation.     

A character relationship on GLn(ℂ)

In this paper we consider the character of an irreducible finite-dimensional algebraic representation of GL_mn(?) restricted to a particular disconnected component of the normalizer of the Levi subgroup GL_m(?)ⁿ of GL_mn(?), generalizing a theorem of Kostant on the character values at the Coxeter element.     

On completely irreducible operators

Let T be a bounded linear operator on Hilbert space H, M an invariant subspace of T. If there exists another invariant subspace N of T such that H = M + N and M ∩ N = 0, then M is said to be a completely reduced subspace of T. If T has a nontrivial completely reduced subspace, then T is said to be completely reducible; otherwise T is said to be completely irreducible. In the present paper we briefly sum up works on completely irreducible operators that have been done by the Functional Analysis Seminar of Jilin University in the past ten years and more. The paper contains four sections. In section 1 the background of completely irreducible operators is given in detail. Section 2 shows which operator in some well-known classes of operators, for example, weighted shifts, Toeplitz operators, etc., is completely irreducible. In section 3 it is proved that every bounded linear operator on the Hilbert space can be approximated by the finite direct sum of completely irreducible operators. It is clear that a completely irreducible operator is a rather suitable analogue of Jordan blocks in L(H), the set of all bounded linear operators on Hilbert space H. In section 4 several questions concerning completely irreducible operators are discussed and it is shown that some properties of completely irreducible operators are different from properties of unicellular operators. __________ Translated from Acta Sci. Nat. Univ. Jilin, 1992, (4): 20–29     

Hyperinvariant Subspace Problem for Quasinilpotent Operators

In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. Let denote the class of quasinilpotent quasiaffinities Q in such that Q ^* Q has an infinite dimensional reducing subspace M with Q ^* Q| _M compact. It was known that if every quasinilpotent operator in has a nontrivial hyperinvariant subspace, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. Thus it suffices to solve the hyperinvariant subspace problem for elements in . The purpose of this paper is to provide sufficient conditions for elements in to have nontrivial hyperinvariant subspaces. We also introduce the notion of “stability” of extremal vectors to give partial solutions to the hyperinvariant subspace problem.      

Transient analysis of queues with heterogeneous arrivals

In this paper we consider a discrete time queueing model where the time axis is divided into time slots of unit length. The model satisfies the following assumptions: (i) an event is either an arrival of typei of batch sizeb _i, i=1,...,r with probability _i or is a depature of a single customer with probability or zero depending on whether the queue is busy or empty; (ii) no more than one event can occur in a slot, therefore the probability that neither an arrival nor a departure occurs in a slot is 1––_{i i} or 1–_{i i} according as the queue is busy or empty; (iii) events in different slots are independent. Using a lattice path representation in higher dimensional space we will derive the time dependent joint distribution of the number of arrivals of various types and the number of completed services. The distribution for the corresponding continuous time model is found by using weak convergence.     

On a lower bound for the perron eigenvalue

A lower boundn ^–1 _i,k a_ik for the Perron eigenvalue of a symmetric non-negative irreducible matrixA=(a _ik) is studied and compared with certain other lower bounds.     

Some properties of rational approximations of degree (k, 1) in the hardy spaceH 2(D)

We prove that the well-known interpolation conditions for rational approximations with free poles are not sufficient for finding a rational function of the least deviation. For rational approximations of degree (k, 1), we establish that these interpolation conditions are equivalent to the assertion that the interpolation pointc is a stationary point of the functionk(c) defined as the squared deviation off from the subspace of rational functions with numerator of degree k and with a given pole 1/¯c. For any positive integersk ands, we construct a functiong H₂(D) such thatR _k ,1(g)=R _k +s,1(g) > 0. whereR _k ,1(g) is the least deviation ofg from the class of rational function of degree (k, 1).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 251–259, August, 1998.The author is keenly grateful to N. S. Vyacheslavov, E. P. Dolzhenko, and V. G. Zinov for useful discussions.     

Parameter-dependent renewal theorems with applications

Let ₁, ₂, ... be a sequence of i.i.d. random variables with positive mean and finite variance and letr(b), b0, be real numbers tending to 0 asb . Definings _n=₁+...+_n andS _n=S_n(b)=s_n+r(b)n, the stopping time =(b)=inf {n>/1:S_n >b} whereb=b(b) , will be considered with special regard to the excess over the boundaryR _b=s+r(b)–b. It turns out that the limiting distribution ofR _b is the same as in the caser(b)0 for allb. Proving this, Blackwell's renewal theorem and its integral version have to be established first in the above stated situation. Finally, an expansion ofE to vanishing terms asb will be provided and applied to some examples arising in economics.

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Zusammenfassung Seien ₁, ₂, ... unabhängige identisch verteilte Zufallsgrößen mit positivem Erwartungswert und endlicher Varianz sowier(b), b0, reelle Zahlen mitr(b)0 für b. Sei ferners ₁, s₂, ... der zugehörige Summenprozeß,S _n= S_n(b)=s_n+r(b)n fürn1 und =(b)=inf {n1: S_n>b, wobeib=b(b) fürb . Es wird gezeigt, daß die asymptotische Verteilung des ExzessesR _b=s +r(b) –b mit der im Fallr(·)0 übereinstimmt. Dazu werden sowohl das Blackwellsche Erneuerungstheorem als auch seine Integralversion in der vorher beschriebenen parameterabhängigen Situation geeignet formuliert und bewiesen. Als Folgerung ergibt sich dann eine asymptotische Entwicklung vonE(b) fürb bis zu Termen o(1). Anh- and einiger Beispiele aus dem ökonomischen Bereich wird schließlich noch aufgezeigt, wo Approximationen fürE(b) von Interesse sein können.

     

On functorial decompositions of self-smash products

We give a decomposition formula for n-fold self smash of a two-cell suspension X localized at 2. The mod 2 homology of each factor in the decomposition is explicitly given as a module over the Steenrod algebra and in the case where X is formed by suspending one of P²,P²,P² or P², this is a complete decomposition into indecomposable pieces. The method has consequences in the modular representation theory of the symmetric group where it leads to a computation of the submatrix for the decomposition matrix of the group algebra /2[S_n] which correspond to partitions of length 2. In particular this yields a derivation of the explicit formula due to Erdmann which gives the multiplicities in the decomposition of /2[S_n] of the indecomposable projective modules which correspond to those partitions.     

Tests for a given linear structure of the mean direction of the langevin distribution

This paper deals with Watson statistic T _w and likelihood ratio (LR) statistic T _l for testing hypothesis H _0s: V (a given s-dimensional subspace) based on a sample of size n from a p-variate Langevin distribution M _p(, ). Asymptotic expansions of the null and non-null distributions of T _w and T _l are obtained when n is large. Asymptotic expressions of those powers are also obtained. It is shown that the powers of them are coincident up to the order n ^-1 when is unknown.     

Multiplication and Compact-friendly Operators

During the last few years the authors have studied extensively the invariant subspace problem of positive operators; see [6] for a survey of this investigation. In [4] the authors introduced the class of compact-friendly operators and proved for them a general theorem on the existence of invariant subspaces. It was then asked if every positive operator is compact-friendly. In this note, we present an example of a positive operator which is not compact-friendly but which, nevertheless, has a non-trivial closed invariant subspace.In the process of presenting this example, we also characterize the multiplication operators that commute with non-zero finite-rank operators. We show, among other things, that a multiplication operator M commutes with a non-zero finite-rank operator if and only the multiplier function is constant on some non-empty open set.     

Representations of Extended Affine Lie Algebras Coordinatized by Certain Quantum Tori

An irreducible representation of the extended affine Lie algebra of type A _n-1 coordinatized by a quantum torus of variables is constructed by using the Fock space for the principal vertex operator realization of the affine Lie algebra .     

Sur quelques représentations modulaires et p-adiques de GL2(Q p ): I

Let p be a prime number and F a complete local field with residue field of characteristic p. In 1993, Barthel and Livné proved the existence of a new kind of -representations of GL₂(F) that they called 'supersingular' and on which one knows almost nothing. In this article, we determine all the supersingular representations of GL₂(Q _p ) with their intertwinings. This classification shows a natural bijection between the set of isomorphism classes of supersingular representations of GL₂(Q _p ) and the set of isomorphism classes of two-dimensional irreducible -representations of .     

Birth times,death times and time substitutions in Markov chains

Summary Given a Markov chain (X _n ) _n0, random times are studied which are birth times or death times in the sense that the post- and pre- processes are independent given the present (X _–1, X ) at time and the conditional post- process (birth times) or the conditional pre- process (death times) is again Markovian. The main result for birth times characterizes all time substitutions through homogeneous random sets with the property that all points in the set are birth times. The main result for death times is the dual of this and appears as the birth time theorem with the direction of time reversed.Part of this work was done while the author was visiting the Department of Mathematics, University of California at San DiegoThe support of The Danish Natural Science Research Council is gratefully acknowledged     

Representations of twisted Yangians associated with skew Young diagrams

Let $G_M$ be either the orthogonal group $O_M$ or the symplectic group $Sp_M$ over the complex field; in the latter case the non-negative integer $M$ has to be even. Classically, the irreducible polynomial representations of the group $G_M$ are labeled by partitions $\mu=(\mu_{1},\mu_{2},\,\ldots)$ such that $\mu^{\prime}_1+\mu^{\prime}_2\le M$ in the case $G_M=O_M$, or $2\mu^{\prime}_1\le M$ in the case $G_M=Sp_M$. Here $\mu^{\prime}=(\mu^{\prime}_{1},\mu^{\prime}_{2},\,\ldots)$ is the partition conjugate to $\mu$. Let $W_\mu$ be the irreducible polynomial representation of the group $G_M$ corresponding to $\mu$. Regard $G_N\times G_M$ as a subgroup of $G_{N+M}$. Then take any irreducible polynomial representation $W_\lambda$ of the group $G_{N+M}$. The vector space $W_{\lambda}(\mu)={\rm Hom}_{\,G_M}( W_\mu, W_\lambda)$ comes with a natural action of the group $G_N$. Put $n=\lambda_1-\mu_1+\lambda_2-\mu_2+\ldots\,$. In this article, for any standard Young tableau $\varOmega$ of skew shape $\lm$ we give a realization of $W_{\lambda}(\mu)$ as a subspace in the $n$-fold tensor product $(\mathbb{C}^N)^{\bigotimes n}$, compatible with the action of the group $G_N$. This subspace is determined as the image of a certain linear operator $F_\varOmega (M)$ on $(\mathbb{C}^N)^{\bigotimes n}$, given by an explicit formula. When $M=0$ and $W_{\lambda}(\mu)=W_\lambda$ is an irreducible representation of the group $G_N$, we recover the classical realization of $W_\lambda$ as a subspace in the space of all traceless tensors in $(\mathbb{C}^N)^{\bigotimes n}$. Then the operator $F_\varOmega\(0)$ may be regarded as the analogue for $G_N$ of the Young symmetrizer, corresponding to the standard tableau $\varOmega$ of shape $\lambda$. This symmetrizer is a certain linear operator on $\CNn$$(\mathbb{C}^N)^{\bigotimes n} $ with the image equivalent to the irreducible polynomial representation of the complex general linear group $GL_N$, corresponding to the partition $\lambda$. Even in the case $M=0$, our formula for the operator $F_\varOmega(M)$ is new. Our results are applications of the representation theory of the twisted Yangian, corresponding to the subgroup $G_N$ of $GL_N$. This twisted Yangian is a certain one-sided coideal subalgebra of the Yangian corresponding to $GL_N$. In particular, $F_\varOmega(M)$ is an intertwining operator between certain representations of the twisted Yangian in $(\mathbb{C}^N)^{\bigotimes n}$.     

Actions of the dipper-donkin quantization GL 2 on the clifford algebra C(l,3)

Following the method already developed for studying the actions of GL_q (2,C) on the Clifford algebra C(l,3) and its quantum invariants [1], we study the action on C(l, 3) of the quantum GL ₂ constructed by Dipper and Donkin [2]. We are able of proving that there exits only two non-equivalent cases of actions with nontrivial “perturbation” [1]. The spaces of invariants are trivial in both cases.

We also prove that each irreducible finite dimensional algebra representation of the quantum GL ₂ q^m ≠1, is one dimensional.

By studying the cases with zero “perturbation” we find that the cases with nonzero “perturbation” are the only ones with maximal possible dimension for the operator algebra ?.     

Hilbert Series of Group Representations and Gröbner Bases for Generic Modules

Each matrix representation :G GL_n() of a finite Group G over a field induces an action of G on the module ⁿ over the polynomial algebra The graded -submodule M() of _n generated by the orbit of is studied. A decomposition of M() into generic modules is given. Relations between the numerical invariants of and those of M(), the latter being efficiently computable by Gröbner bases methods, are examined. It is shown that if is multiplicity-free, then the dimensions of the irreducible constituents of can be read off from the Hilbert series of M(Pi;). It is proved that determinantal relations form Gröbner bases for the syzygies on generic matrices with respect to any lexicographic order. Gröbner bases for generic modules are also constructed, and their Hilbert series are derived. Consequently, the Hilbert series of M(Pi;) is obtained for an arbitrary representation.