Alan Huckleberry

Symmetry Classes of Disordered Fermions

Building upon Dyson’s fundamental 1962 article known in random-matrix theory as the threefold way, we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important physical examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds.The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V, a space of endomorphisms in End(), a bilinear form on which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V*. Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way.This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.

Symplectic Geometry of Entanglement

We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using the Kostant-Sternberg theorem, we show that separable states form a unique symplectic orbit, whereas orbits of entangled states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement. The above statements remain true for systems with an arbitrary number of components, moreover the presented method is general and can be applied also under different additional symmetry conditions stemming, e.g., from the indistinguishability of particles. We show how to calculate the degeneracy for various multiparticle systems providing also simple criteria of separability.

A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients

We present an algorithm for the explicit numerical calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients, based on the Gelfand-Tsetlin pattern calculus. Our algorithm is well-suited for numerical implementation; we include a computer code in an appendix. Our exposition presumes only familiarity with the representation theory of SU(2).

Actions of Groups of Holomorphic Transformations

This paper primarily deals with three topics: Classification results for homogeneous and almost homogeneous spaces, complex analytic questions on homogeneous spaces, and certain types of actions, e.g. of compact Lie groups on complex spaces. Our goal here is to indicate our own current view of these areas, as opposed to presenting a comprehensive survey.

Schubert varieties and cycle spaces

Let G0 be a real semisimple Lie group. It acts naturally on every complex flag manifold Z = G/Q of its complexification. Given an Iwasawa decomposition G0 = K0A0N0 , a G0- orbitZ, and the dual K-orbit � � Z, Schubert varieties are studied and a theory of Schubert slices for arbitrary G0-orbits is developed. For this, certain geometric properties of dual pairs (,�) are underlined. Canonical complex analytic slices contained in a given G0-orbit which are transversal to the dual K0-orbit \ � are constructed and analyzed. Associated algebraic incidence divisors are used to study complex analytic properties of certain cycle domains. In particular, it is shown that the linear cycle space W(D) is a Stein domain that contains the universally defined Iwasawa domain I. This is one of the main ingredients in the proof that W(D) = AG for all but a few hermitian exceptions. In the hermitian case, W(D) is concretely described in terms of the associated bounded symmetric domain.

Group actions on S6 and complex structures on ℙ3

It is proved that if S 6 possesses an integrable complex structure, then there exists a 1-dimensional family of pairwise different exotic complex structures on P3(C). This follows immediately from the main result of the paper: S 6 is not the underlying differentiable manifold of an almost homoge- neous complex manifold X. Via elementary Lie theoretic techniques this is reduced to ruling out the possibility of a C � -action on a certain non-normal surface EX. A contradiction is reached by analyzing combinatorial aspects of the non-normal locus N of E and its preimage ˆ N in the normalization ˆ E.

Homogeneous Spaces from a Complex Analytic Viewpoint

Manifolds having many automorphisms play a fundamental role in geometry. If X is a compact complex manifold, then the group Aut(X) of holomorphic automorphisms of X is, when equipped with the compact-open topology, a complex Lie group. If G = Aut(X), then an orbit G(p) , p∈X, may be holomorphical ly identified with the quotient manifold G/H, where H := {g ∈ G|g(p) = p} is the isotropy group of the G-action at p. Thus, studying quotients G/H of a complex Lie group G by a closed subgroup H becomes relevant. A natural first step is to analyze the structure of compact homogeneous spaces X = G/H. In the early 1950’s, with development of Lie theory along with the theory of algebraic groups, a number of very sharp results were obtained by algebraic methods. The works of Borel (e.g., [10]), Goto [22], Tits [64] and Wang [67] are typical of this direction. Later, but still in an algebraic geometry spirit, many general methods were developed (e.g., see the papers of Hochschild, Mostow, Rosenlicht et al.).

Haar expectations of ratios of random characteristic polynomials

AbstractWe compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups

Introduction to Group Actions in Symplectic and Complex Geometry

K = \mathrm {O}_N\,