Joe Repka

Parentage analysis with incomplete sampling of candidate parents and offspring

Many breeding systems include ‘multiple mating’ in which males or females mate with multiple partners. We identify two forms of multiple mating: ‘single‐sex’, where the next‐generation individuals (NGIs) are the product of multiple mating by one sex; and ‘two‐sex’, where the NGIs are the product of multiple mating by both sexes. For both mating systems we develop models that estimate the proportion of NGIs that is fathered (paternity) or mothered (maternity) by the putative parents. The models only require genetic data from the parent or parents in question and the sample of NGIs, as well as an estimate of population allele frequencies. The models provide unbiased estimates, can accommodate loci with many alleles and are robust to violations of their assumptions. They allow researchers to address intractable problems such as the parentage of seeds found on the ground, juvenile fish in a stream, and nestlings in a communal breeding bird. We demonstrate the models using genetic data from a nest of the bluegill sunfish Lepomis macrochirus, where the NGIs may be from multiple females that have spawned with multiple males from different life histories (cuckolder and parental).

Statistical confidence in parentage analysis with incomplete sampling: how many loci and offspring are needed?

We have recently presented models to estimate parentage in breeding systems with multiple mating and incomplete sampling of the candidate parents. Here we provide formulas to calculate the statistical confidence and the optimal trade‐off between the number of loci and offspring. These calculations allow an understanding of the statistical significance of the parentage estimates as well as the appropriate sampling regime required to obtain a desired level of confidence. We show that the trade‐off generally depends on the parentage of the putative parents. When parentage is low, sampling effort should concentrate on increasing the number of loci. Otherwise, there are similar benefits from increasing the number of loci or offspring. We demonstrate these methods using genetic data from a nest of the bluegill sunfish (Lepomis macrochirus).

Vector‐coherent‐state theory as a theory of induced representations

A general definition is given of vector‐coherent state (VCS) representation theory. It is shown that the theory is more general than suggested by previous applications and that it incorporates the standard theories of induced representations as special cases. The associated K‐matrix theory is also given a fuller treatment than hitherto and shown to provide a rather general algorithm both for projecting VCS representations from larger representations in which they are embedded and for determining the Hermitian form, with respect to which an isometric‐equivalent representation is, in fact, isometric.

Spherical harmonics and basic coupling coefficients for the group SO(5) in an SO(3) basis

An easily programmable algorithm is given for the computation of SO(5) spherical harmonics needed to complement the radial (beta) wave functions to form an orthonormal basis of wave functions for the five-dimensional harmonic oscillator. It is shown how these functions can be used to compute the (Clebsch–Gordan a.k.a. Wigner) coupling coefficients for combining pairs of irreps in this space to other irreps. This is of particular value for the construction of the matrices of Hamiltonians and transition operators that arise in applications of nuclear collective models. Tables of the most useful coupling coefficients are given in the Appendix.

Sums of adjoint orbits

We investigate a natural generalization of the problem of the description of the eigenvalues of the sum of two Hermitian matrices both of whose eigenvalues are known. We describe more generally the convolution of the invariant probability measures supported on any two adjoint orbits of a compact Lie group. Our techniques utilize the convexity results of Guillemin and Sternberg and Kirwan on the one hand, and the character formulae of Weyl and Kirillov on the other. Applications to representation theory are discussed.

Dynamic structure and embedded representation in physics: The group theory of the adiabatic approximation

The group theoretical concepts of embedded representations and dynamical structure groups, distinct from dynamical symmetry groups, are introduced in order to describe the common physical situation in which collective bands of states of a many‐body system are well described by an algebraic collective model even though the states may not span an invariant subspace of the many‐body Hilbert space.

A rotor expansion of the su(3) Lie algebra

Vector coherent state (VCS) theory is used to give a rotor expansion of the su(3) Lie algebra in a way that parallels the boson expansions that have been made for other Lie algebras. The construction provides a systematic procedure for calculating Hermitian matrix elements of su(3) in an SO(3)-coupled basis and represents a new development in VCS theory and in the theory of induced representations.

On the subalgebra of H∗((RP∞)n; F2) annihilated by Steenrod operations☆

We define a homomorphism θ on H∗((RP∞)n; F2) having the property that it is zero on elements hit by the positive degree elements of the Steenrod algebra. We describe the subalgebra (Im θ)∗ of Steenrod-annihilated elements of H∗((RP∞)n; F2) and in particular we show that it is nilpotent of order n + 1. We make some conjectures as to properties of H∗((RP∞)n; F2) including a nilpotency conjecture that is a strengthening of the conjecture of Peterson, proved by Wood, concerning the degrees containing elements not hit by positive degree Steenrod operations.

Embeddings of the Euclidean algebra e(3) into sl(4,C) and restrictions of irreducible representations of sl(4,C)

The Euclidean group E(3) is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. It is the noncompact, semidirect product group E(3)≅SO(3)⋉R3. The Euclidean algebra e(3) is the complexification of the Lie algebra of E(3). We classify the embeddings of the Euclidean algebra e(3) into the simple Lie algebra sl(4,C) and, as an application of this classification, discuss the restriction to various embeddings of e(3) of certain irreducible representations of sl(4,C). In particular, we consider which of these restrictions are e(3)-indecomposable.

Vector coherent state representations, induced representations and geometric quantization: I. Scalar coherent state representations

Coherent state theory is shown to reproduce three categories of representations of the spectrum generating algebra for an algebraic model: (i) classical realizations which are the starting point for geometric quantization, (ii) induced unitary representations corresponding to prequantization and (iii) irreducible unitary representations obtained in geometric quantization by choice of a polarization. These representations establish an intimate relation between coherent state theory and geometric quantization in the context of induced representations.