Kostant's convexity theorem and the compact classical groups

The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered.     

High powers of random elements of compact Lie groups

Summary. If a random unitary matrix is raised to a sufficiently high power, its eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the variance of the number of eigenvalues of falling in a given arc, as the dimension of tends to infinity. The independence result, it turns out, extends to arbitrary representations of arbitrary compact Lie groups. We state and prove this more general theorem, paying special attention to the compact classical groups and to wreath products. This paper is excerpted from the author's doctoral thesis, [9]. Received: 15 October 1995 / In revised form: 7 March 1996     

Homogeneous nilmanifolds attached to representations of compact Lie groups

For each compact Lie algebra ? and each real representation V of ? we consider a two-step nilpotent Lie group N(?,V), endowed with a natural left-invariant riemannian metric. The homogeneous nilmanifolds so obtained are precisely those which are naturally reductive. We study some geometric aspects of these manifolds, finding many parallels with H-type groups. We also obtain, within the class of manifolds N(?,V), the first examples of non-weakly symmetric, naturally reductive spaces and new examples of non-commutative naturally reductive spaces. Received: 16 September 1998 / Revised version: 24 February 1999     

A spectral radius formula for the Fourier transform on compact groups and applications to random walks

Let G be a compact group, not necessarily abelian, let ? be its unitary dual, and for f∈L¹(G), let fn?f∗?∗f denote n-fold convolution of f with itself and f? the Fourier transform of f. In this paper, we derive the following spectral radius formula

     

On the first cohomology of arithmetic groups

We study the restriction to smaller subgroups, of cohomology classes on arithmetic groups (possibly after moving the class by Hecke correspondences), especially in the context of first cohomology of arithmetic groups. We obtain vanishing results for the first cohomology of cocompact arithmetic lattices in SU(n,1) which arise from hermitian forms over division algebras D of degree p ², p an odd prime, equipped with an involution of the second kind. We show that it is not possible for a ‘naive’ restriction of cohomology to be injective in general. We also establish that the restriction map is injective at the level of first cohomology for non co-compact lattices, extending a result of Raghunathan and Venkataramana for co-compact lattices. Received: 14 September 2000 / Accepted: 6 June 2001     

Multiple levels of symmetry breaking

In the previous paper in this volume we have studied the p-spin interaction model just below the critical temperature, and we have rigorously proved several aspects of the physicists prediction that this model exhibits “one level of symmetry breaking”. In the present paper we show how to construct systems that exhibit an arbitrarily large, but finite number of “levels of symmetry-breaking”. As the temperature decreases, such systems exhibit many phase transitions, as the structure of the overlaps gains complexity. This phenomenon does not seem to have been described previously, even in the physics literature. Received: 15 January 1998 / Revised version: 10 November 1999 / Published online: 21 June 2000     

On limit theorem for the eigenvalues of product of two random matrices

The existence of limiting spectral distribution (LSD) of the product of two random matrices is proved. One of the random matrices is a sample covariance matrix and the other is an arbitrary Hermitian matrix. Specially, the density function of LSD of S_nW_n is established, where S_n is a sample covariance matrix and W_n is Wigner matrix.     

Rigorous results for the Hopfield model with many patterns

Summary. We perform a thorough investigation of the main aspects of the Hopfield model with many patterns. Advances are made toward the validity of the “replica symmetric” solution. Strong evidence of the validity of this solution is given over the entire domain where this validity is conjectured; complete proof is given in a subregion that contains strictly the ergodic region. Received: 22 May 1996 / In revised form: 20 May 1997     

Unimodality of Stable Gaussian Laws on the Heisenberg Group

 We prove unimodality of all dilatation-stable Gaussian laws on the Heisenberg group. (Received 23 March 1998; in final form 19 May 1999)     

Rigorous low-temperature results for the mean field p-spins interaction model

We prove that, just below the critical temperature, the mean field p-spins interaction model, for p suitably large, spontaneously decomposes into different states. The asymptotic overlaps between any two different states are zero. Under a mild (unproven) hypothesis on the weight distribution of these states, we prove that they are pure states. This situation is called in physics “one level of symmetry breaking”. Received: 15 January 1998 / Revised version: 10 November 1999 / Published online: 21 June 2000     

Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution

Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every measure satisfying a Poincaré inequality shares the same concentration phenomenon. We also discuss exponential integrability under Poincaré inequalities and its consequence to sharp diameter upper bounds on spectral gaps. Received: 10 June 1996 / In revised form: 9 August 1996     

Minimum permanents of doubly stochastic matrices with at least one zero entry †

It is shown that the minimum value of the permanent on the n× ndoubly stochastic matrices which contain at least one zero entry is achieved at those matrices nearest to J_n in Euclidean norm, where J_n is the n× nmatrix each of whose entries is n^-1 . In case n ≠ 3 the minimum permanent is achieved only at those matrices nearest J_n ; for n= 3 it is achieved at other matrices containing one or more zero entries as well.     

Factorisation Theorems for T-decomposable Measures on Groups

 For a real or p-adic unipotent algebraic group G, given a T∈ Hom(G, G) and T-decomposable measure on G which is either ‘full’ or symmetric, we get a decomposition , where μ₀ is T-invariant and , and this decomposition is unique upto a shift. We also show that ν₀ is T-decomposable under some additional sufficient condition and give a counter example to justify this. We generalise the above to power bounded operators on p-adic Banach spaces. We also prove some convergence-of-types theorems on p-adic groups as well as Banach spaces. (Received 21 October 2000; in revised form 21 February 2001)     

The generalized Schur complement in group inverses and (k 1)-potent matrices

In this article, two facts related to the generalized Schur complement are studied. The first one is to find necessary and sufficient conditions to characterize when the group inverse of a partitioned matrix can be expressed in the Schur form. The other one is to develop a formula for any power of the generalized Schur complement of an idempotent partitioned matrix and then to characterize when this generalized Schur complement is a (k+1)-potent matrix. In addition, some spectral theory related to this complement is analyzed.     

Foliations invariant under Lie group transverse actions

In this paper we study (smooth and holomorphic) foliations which are invariant under transverse actions of Lie groups. Authors’ address: Alexandre Behague and Bruno Scárdua, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brazil     

The Sherrington–Kirkpatrick model: a challenge for mathematicians

Summary. The Sherrington–Kirkpatrick (SK) model for spin glasses is deceptively simple to state. Yet its rigorous study represents a considerable challenge. We report here some modest progresses (obtained through elementary methods). Even in the supposedly simple high temperature region, a number of basic questions remain unsolved. Received: 7 December 1995 / In revised form: 6 March 1997     

Long-time existence for signed solutions of the heat equation with a noise term

Summary.   Let ? be the circle [0,J] with the ends identified. We prove long-time existence for the following equation.