Classical Compact Groups Research Articles

Non-Abelian Kuramoto models and synchronization

We describe non-Abelian generalizations of the Kuramoto model for any classical compact Lie group and identify their main properties. These models may be defined on any complex network where the variable at each node is an element of the unitary group U(n), or a subgroup of U(n). The nonlinear evolution equations maintain the unitarity of all variables which therefore evolve on the compact manifold of U(n). Synchronization of trajectories with phase locking occurs as for the Kuramoto model, for values of the coupling constant larger than a critical value, and may be measured by various order and disorder parameters. Limit cycles are characterized by a frequency matrix which is independent of the node and is determined by minimizing a function which is quadratic in the variables. We perform numerical computations for n = 2, for which the SU(2) group manifold is S3, for a range of natural frequencies and all-to-all coupling, in order to confirm synchronization properties. We also describe a second generalization of the Kuramoto model which is formulated in terms of real m-vectors confined to the (m − 1)-sphere for any positive integer m, and investigate trajectories numerically for the S2 model. This model displays a variety of synchronization phenomena in which trajectories generally synchronize spatially but are not necessarily phase-locked, even for large values of the coupling constant.

The Entries of Haar-Invariant Matrices from the Classical Compact Groups

Let Γ n =(γ ij )n×n be a random matrix with the Haar probability measure on the orthogonal group O(n), the unitary group U(n), or the symplectic group Sp(n). Given 1≤m<n, a probability inequality for a distance between (γ ij )n×m and some mn independent F-valued normal random variables is obtained, where F=ℝ, ℂ, or ℍ (the set of real quaternions). The result is universal for the three cases. In particular, the inequality for Sp(n) is new.

Formula omitted]-singular dichotomy for orbital measures of classical compact Lie groups

We prove that for any classical, compact, simple, connected Lie group G, the G-invariant orbital measures supported on non-trivial conjugacy classes satisfy a surprising L 2 -singular dichotomy: Either μ h k ∈ L 2 ( G ) or μ h k is singular to the Haar measure on G. The minimum exponent k for which μ h k ∈ L 2 is specified; it depends on Lie properties of the element h ∈ G . As a corollary, we complete the solution to a classical problem – to determine the minimum exponent k such that μ k ∈ L 1 ( G ) for all central, continuous measures μ on G. Our approach to the singularity problem is geometric and involves studying the size of tangent spaces to the products of the conjugacy classes.

Conditional Haar measures on classical compact groups

We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension $n$. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension $p$. The developed method leads to the following result: for this conditional measure, writing $Z_U^{(p)}$ for the first nonzero derivative of the characteristic polynomial at 1, \[\frac{Z_U^{(p)}}{p!}\stackrel{\mathrm{law}}{=}\prod_{\ell =1}^{n-p}(1-X_{\ell}),\] the $X_{\ell}$'s being explicit independent random variables. This implies a central limit theorem for $\log Z_U^{(p)}$ and asymptotics for the density of $Z_U^{(p)}$ near 0. Similar limit theorems are given for the orthogonal and symplectic groups, relying on results of Killip and Nenciu.

The effect of convolving families of L -functions on the underlying group symmetries

L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N,M --> oo, the statistical behavior (1-level density) of the low-lying zeros of L-functions in F_N (resp., G_M) agrees with that of the eigenvalues near 1 of matrices in G_1 (resp., G_2) as the size of the matrices tend to infinity, where each G_i is one of the classical compact groups (unitary, symplectic or orthogonal). Assuming that the convolved families of L-functions F_N x G_M are automorphic, we study their 1-level density. (We also study convolved families of the form f x G_M for a fixed f.) Under natural assumptions on the families (which hold in many cases) we can associate to each family L of L-functions a symmetry constant c_L equal to 0 (resp., 1 or -1) if the corresponding low-lying zero statistics agree with those of the unitary (resp., symplectic or orthogonal) group. Our main result is that c_{F x G} = c_G * c_G: the symmetry type of the convolved family is the product of the symmetry types of the two families. A similar statement holds for the convolved families f x G_M. We provide examples built from Dirichlet L-functions and holomorphic modular forms and their symmetric powers. An interesting special case is to convolve two families of elliptic curves with rank. In this case the symmetry group of the convolution is independent of the ranks, in accordance with the general principle of multiplicativity of the symmetry constants (but the ranks persist, before taking the limit N,M --> oo, as lower-order terms).

A variance formula related to a quantum conductance problem

Let t be a block of an Haar-invariant orthogonal ( β = 1 ), unitary ( β = 2 ) or symplectic ( β = 4 ) matrix from the classical compact groups O ( n ) , U ( n ) or Sp ( n ) , respectively. We obtain a close form for Var ( tr ( t ∗ t ) ) . The case for β = 2 is related to a quantum conductance problem, and our formula recovers a result obtained by several authors. Moreover, our result shows that the variance has a limit ( 8 β ) −1 for β = 1 , 2 and 4 as the sizes of t go to infinity in a special way. Although t in our formulation comes from a block of an Haar-invariant matrix from the classical compact groups, the above limit is consistent with a formula by Beenakker, where t is a block of a circular ensemble.

Borel theorems for random matrices from the classical compact symmetric spaces

We study random vectors of the form $(\operatorname {Tr}(A^{(1)}V),...,\operatorname {Tr}(A^{(r)}V))$, where $V$ is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the $A^{(\nu)}$ are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et al. for Haar distributed matrices from the classical compact groups. The proof uses integration formulas, due to Collins and \'{S}niady, for polynomial functions on the classical compact groups.

Autocorrelation of ratios of $L$-functions

We give a new heuristic for all of the main terms in the quotient of products of L-functions averaged over a family. These conjectures generalize the recent conjectures for mean values of L-functions. Comparison is made to the analogous quantities for the characteristic polynomials of matrices averaged over a classical compact group.

Harmonic morphisms from the classical compact semisimple Lie groups

In this article, we introduce a new method for manufacturing harmonic morphisms from semi-Riemannian manifolds. This is employed to yield a variety of new examples from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with their standard Riemannian metrics. We develop a duality principle and show how this can be used to construct the first known examples of harmonic morphisms from the non-compact Lie groups $${\bf SL}_n({\mathbb{R}})$$ , SU *(2n), $${\bf Sp}(n, {\mathbb{R}})$$ , SO *(2n), SO(p, q), SU(p, q) and Sp(p, q) equipped with their standard dual semi-Riemannian metrics.

A Characterization of Right Coideals of Quotient Type and its Application to Classification of Poisson Boundaries

Let $G$ be a co-amenable compact quantum group. We show that a right coideal of $G$ is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on $SU_q(N)$ for co-amenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson boundary and the right coideal of quotient type by maximal quantum subgroup of Kac type. In particular, the Poisson boundary and the quantum flag manifold are isomorphic for any q-deformed classical compact Lie group.

Averages over classical Lie groups, twisted by characters

We compute E G ( ∏ i tr ( g λ i ) ) , where g ∈ G = Sp ( 2 n ) or SO ( m ) ( m = 2 n , 2 n + 1 ) with Haar measure. This was first obtained by Diaconis and Shahshahani [Persi Diaconis, Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994) 49–62. Studies in applied probability], but our proof is more self-contained and gives a combinatorial description for the answer. We also consider how averages of general symmetric functions E G Φ n are affected when we introduce a character χ λ G into the integrand. We show that the value of E G χ λ G Φ n / E G Φ n approaches a constant for large n. More surprisingly, the ratio we obtain only changes with Φ n and λ and is independent of the Cartan type of G. Even in the unitary case, Bump and Diaconis [Daniel Bump, Persi Diaconis, Toeplitz minors, J. Combin. Theory Ser. A 97 (2) (2002) 252–271. Erratum for the proof of Theorem 4 available at http://sporadic.stanford.edu/bump/correction.ps and in a third reference in the abstract] have obtained the same ratio. Finally, those ratios can be combined with asymptotics for E G Φ n due to Johansson [Kurt Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (3) (1997) 519–545] and provide asymptotics for E G χ λ G Φ n .

A probabilistic meaning of certain quasinormal subgroups

The role of the cyclic quasinormal subgroups has been recently described in groups both finite and infinite by S.Stonehewer and G.Zacher. This role can be better analyzed in the class of compact groups, obtaining restrictions for the probability that two randomly chosen elements commute. Mathematcs Subject Classification: 20D60, 20P05, 20D08

The low lying zeros of a GL(4) and a GL(6) family of $L$-functions

We investigate the large weight (k --> oo) limiting statistics for the low lying zeros of a GL(4) and a GL(6) family of L-functions, {L(s,phi x f): f in H_k(1)} and {L(s,phi times sym^2 f): f in H_k(1)}; here phi is a fixed even Hecke-Maass cusp form and H_k(1) is a Hecke eigenbasis for the space H_k(1) of holomorphic cusp forms of weight k for the full modular group. Katz and Sarnak conjecture that the behavior of zeros near the central point should be well modeled by the behavior of eigenvalues near 1 of a classical compact group. By studying the 1- and 2-level densities, we find evidence of underlying symplectic and SO(even) symmetry, respectively. This should be contrasted with previous results of Iwaniec-Luo-Sarnak for the families {L(s,f): f in H_k(1)} and {L(s,sym^2f): f in H_k(1)}, where they find evidence of orthogonal and symplectic symmetry, respectively. The present examples suggest a relation between the symmetry type of a family and that of its twistings, which will be further studied in a subsequent paper. Both the GL(4) and the GL(6) families above have all even functional equations, and neither is naturally split from an orthogonal family. A folklore conjecture states that such families must be symplectic, which is true for the first family but false for the second. Thus the theory of low lying zeros is more than just a theory of signs of functional equations. An analysis of these families suggest that it is the second moment of the Satake parameters that determines the symmetry group.

Harmonic morphisms from the compact semisimple Lie groups and their non-compact duals

In this paper we prove the local existence of complex-valued harmonic morphisms from any compact semisimple Lie group and their non-compact duals. These include all Riemannian symmetric spaces of types II and IV. We produce a variety of concrete harmonic morphisms from the classical compact simple Lie groups SO ( n ) , SU ( n ) , Sp ( n ) and globally defined solutions on their non-compact duals SO ( n , C ) / SO ( n ) , SL n ( C ) / SU ( n ) and Sp ( n , C ) / Sp ( n ) .

Quantum Bohr compactification

We introduce a non commutative analog of the Bohr compactification. Starting from a general quantum group G we define a compact quantum group bG which has a universal property such as the universal property of the classical Bohr compactification for topological groups. We study the object bG in special cases when G is a classical locally compact group, dual of a classical group, discrete or compact quantum group as well a quantum group arising from a manageable multiplicative unitary. We also use our construction to give new examples of compact quantum groups.

Isospectral metrics and potentials on classical compact simple Lie groups

Given a compact Riemannian manifold (M, g), the eigenvalues of the Laplace operator � form a discrete sequence known as the spectrum of (M, g). (In the case the M has boundary, we stipulate either Dirichlet or Neumann boundary conditions.) We say that two Riemannian manifolds are isospectral if they have the same spectrum. For a fixed manifold M, an isospectral deformation of a metric g0 on M is a continuous family F of metrics on M containing g0 such that each metric g ∈ F is isospectral to g0. We say that the deformation is nontrivial if none of the other metrics in F are isometric to g0 and that the deformation is multidimensional if F can be parameterized by more that one variable. For two functions φ, ψ ∈ C ∞ (M), we say that φ and ψ are isospectral potentials on (M, g) if the eigenvalue spectra of the Schrodinger operators ~ � + φ and ~ � + ψ are equal for any choice of Planck's constant ~ . In this paper, we prove the existence of multiparameter isospectral deformations of metrics on SO(n) (n = 9 or n ≥ 11), SU(n) (n ≥ 8), and Sp(n) (n ≥ 4). For these exam- ples, we follow a metric construction developed by Schueth who had given one-parameter families of isospectral metrics on orthogonal and unitary groups. Our multiparameter families are obtained by a new proof of nontriviality establishing a generic condition for nonisometry of metrics arising from the construction. We also show the existence of non-congruent pairs of isospectral potentials and nonisometric pairs of isospectral conformally equivalent metrics on Sp(n) for n ≥ 6. The industry of producing isospectral manifolds began in 1964 with Milnor's pair of 16-dimensional isospectral, nonisometric tori (Mi). Several years later, in the early 1980's, new examples began to appear sporadically (e.g. (Vi), (Ik), (GWil1)). These isospectral constructions were ad hoc and did not appear to be related until 1985, when Sunada began developing the first unified approach for producing isospectral manifolds. The method described a program for taking quotients of a given manifold so that the resulting manifolds were isospectral. Sunada's original theorem and subsequent generalizations ((Bd1), (Bd2), (DG), (Pes), (Sut)) explained most of the previously known isospectral examples and led to a wide variety of new examples. See, for example, (BGG), (Bus), and (GWW). In 1993 Gordon produced the first examples of closed isospectral manifolds with dif- ferent local geometry (Gor) and then, in a series of papers, generalized the construction to the following principle based on torus actions.

Averages of ratios of characteristic polynomials for the compact classical groups

Averages of ratios of characteristic polynomials for the compact classical groups are evaluated in terms of determinants whose dimensions are independent of the matrix rank. These formulas are shown to be equivalent to expressions for the same averages obtained in a previous study, which was motivated by applications to analytic number theory. Our approach uses classical methods of random matrix theory, in particular determinants and orthogonal polynomials, and can be considered more elementary than the method of Howe pairs used in the previous study.

Random Matrix Theory and Entanglement in Quantum Spin Chains

We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians - those that are related to quadratic forms of Fermi operators - between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N --> infinity . This is shown to grow logarithmically with N. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlev\'e type. In some cases these solutions can be evaluated to all orders using recurrence relations.

Mock-Gaussian behaviour for linear statistics of classical compact groups

We consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups. We compute their moments and find that the first few moments are Gaussian, whereas the limiting distribution is not. The precise number of Gaussian moments depends upon the particular statistic considered.

Random matrix theory, the exceptional Lie groups andL-functions

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are here extended to non-classical groups. We focus on an explicit example: the exceptional Lie group G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G_2, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the 7-dimensional representation of G_2. The random matrix calculations extend to all exceptional Lie groups