construction of conformally invariant higher spin operators using transvector algebras this paper deals with a systematic construction of higher spin operators defined as conformally invariant differential operators acting on functions on flat space rm with values in an arbitrary half integer irreducible representation for the spin group to be more precise the higher spin version of the dirac operator and associated twistor operators will be constructed as generators of a transvector algebra hereby generalising the well known fact that the classical dirac operator on rm and its symbol generate the orthosymplectic lie superalgebra osp 1 2 to do so we will use the extremal projection operator and its relation to transvector algebras in the second part of the article the conformal invariance of the constructed higher spin operators will be proven explicitly

young tableaux gelfand patterns and branching rules for classical groups abstract young tableaux and gelfand patterns which combinatorially describe characters of irreducible finite dimensional representations of complex classical lie groups are presented some branching rules and or character recursions are given explicit combinatorial descriptions these recursions are derived from manipulations of determinantal character formulas

on the decomposition of tensor products of the representations of the classical groups by means of the universal characters on definit les caracteres universels d une representation rationnelle de gl n c et on donne les formules de decomposition des produits tensoriels entre eux en termes de coefficients de littlewood richardson on demontre les formules de decomposition du produit tensoriel de deux representations irreductibles de sp 2n c et de so n c

outlier tests for logistic regression a conditional approach summary we consider exact conditional methods for identifying outliers in logistic regression data tests for a single outlier and multiple outliers are developed assuming a logistic slippage model the p values for these tests are determined using an explicit enumeration of all possible responses consistent with the observed value of the sufficient statistic justifications are given for preferring this computationally intensive approach to standard methods based on asymptotic approximations the techniques are applied to two examples

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

holomorphic discrete series and harmonic series unitary irreducible representations of non compact lie groups sp 2n r u p q and so 2n generalised characters of the infinite dimensional holomorphic discrete series unitary irreducible representations of the non compact groups u p q sp 2n r and so 2n are explicitly expressed in terms of characters of finite dimensional unitary group representations these formulae are remarkably succinct despite involving certain infinite series of schur functions similar formulae are derived for harmonic series unitary representation of both u p q and sp 2n r consideration of the branching rules from u p q to u q p and from sp 2n r to u n enables holomorphic representations to be identified as a subset of the harmonic representations the branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations in the case of the former a known result is recast in terms of closed formulae involving schur functions and for the latter various generalisations of these formulae are given a conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of sp 2n r and u p q illustrative examples are presented

generalized hua operators and parabolic subgroups let g be a connected noncompact semisimple group with finite center fix g kan an iwasawa decomposition of g that is k is a maximal compact subgroup of g a a maximal vector subgroup of g with ad a consisting of semisimple elements normalizing n a maximal simply connected nilpotent subgroup of g the space 52 g k is a riemannian symmetric space and if m is the centralizer of a in k the group b man is a minimal parabolic subgroup of g and the space g b k m is called the maximal or furstenberg boundary of u2 in general a boundary of 52 will be a space of the form g p where p is some parabolic subgroup of g which we may assume contains b fixing a parabolic subgroup p of g there is a poisson kernel

theta lifting of unitary lowest weight modules and their associated cycles we consider a reductive dual pair g g in the stable range with g the smaller member and of hermitian symmetric type we study theta lifting of holomorphic nilpotent k c orbits in relation to theta lifting of unitary lowest weight modules of g we determine the associated cycles of all such representations in particular we prove that the multiplicity in the associated cycle is preserved under theta lifting

special functions of matrix argument i algebraic induction zonal polynomials and hypergeometric functions hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis multivariate statistics quantum physics molecular chemistry and number theory this paper presents a general theory of such functions for real division algebras these functions which generalize the classical hypergeometric functions are defined by infinite series on the space s s n f of all n x n hermitian matrices over the division algebra f the theory depends intrinsically upon the representation theory of the general linear group g gl n f of invertible n x n matrices over f and the theme of this work is the full exploitation of the inherent group theory the main technique is the use of the method of algebraic induction to realize explicitly the appropriate representations of g to decompose the space of polynomial functions on s and to describe the zonal polynomials from which the hypergeometric functions are constructed detailed descriptions of the convergence properties of the series expansions are given and integral representations are provided future papers in this series will develop the fine structure of these functions 0 introduction we begin a series of articles in which we develop the fine structure of generalized hypergeometric functions of matrix argument by fine structure we allude to the analogues of such classical results as series expansions integral formulas asymptotics differential equations summation formulas addition theorems composition formulas and recurrence relations this first paper in which we simultaneously treat real complex and quaternionic analysis is the result of our desire to present a complete theory of hypergeometric functions of matrix argument over real division algebras not only as a framework for the body of detailed results to follow in later papers but also to clarify the representationtheoretic foundation for such a theory although these hypergeometric functions are of interest on purely analytic grounds they arise in a wide range of applications indeed various classes of received by the editors may 19 1986 1980 malhema1zics s ject claszficain 1985 ren primary 22e30 22e45 33a75 43a85 43a90 62h10 secondary 20g20 32a07 32m15 44a10 62e15 key w and phrases generalized hypergeometric functions zonal polynomials representation theory algebraic induction multivariate statistics general linear group generalized gamma functions pochhammer symbols laplace transforms maximal compact subgroup invariant polynomials positive cones symmetric spaces schur functions special functions of matrix argument the first author is on leave from the university of wyoming the second author was partially supported by the national science foundation under grant mcs 8403381 and by the research council of the university of north carolina b1987 american mathematical society 0002 9947 87 1 00 25 per page

brill gordan loci transvectants and an analogue of the foulkes conjecture abstract the hypersurfaces of degree d in the projective space p n correspond to points of p n where n n d d 1 now assume d 2 e is even and let x n d p n denote the subvariety of two e fold hyperplanes we exhibit an upper bound on the castelnuovo regularity of the ideal of x n d and show that this variety is r normal for r 2 the latter result is representation theoretic and says that a certain gl n 1 equivariant morphism s r s 2 e c n 1 s 2 s r e c n 1 is surjective for r 2 a statement which is reminiscent of the foulkes howe conjecture for its proof we reduce the statement to the case n 1 and then show that certain transvectants of binary forms are nonzero the latter part uses explicit calculations with feynman diagrams and hypergeometric series for ternary quartics and binary d ics we give explicit generators for the defining ideal of x n d expressed in the language of classical invariant theory

erratum to remarks on classical invariant theory although this paper circulated as a preprint for 12 years an error in the argument for the capelli identity went unnoticed until professor m wakayama read the paper in order to review it i thank him for his care and for bringing this error to my attention the error occurs on page 566 the result at issue that det ii is central in gyn is true and classical and the general line of argument is sufficient to prove it but the details of the computation are described incorrectly the identity on line 3 page 566 is correct but the description of the calculation drawn from it is not instead of terms of the form

assessing the adequacy of the logistic regression model for matched case control studies we apply results for the detection of outliers in logistic regression to the analysis of matched pairs data and illustrate the techniques with data from a matched pairs study of the relationship between iron binding proteins and mortality in the solomon islands although the study includes 90 matched pairs the conclusions change substantially with deletion of one or two pairs thus even with reasonably large data sets single observations may have a great impact on the results of an analysis the identification of influential observations should constitute part of the analysis of matched case control data

graph theory and classical invariant theory abstract this paper presents a simple graphical method closely related to the algebrochemical method of clifford and sylvester for computations in the classical invariant theory of binary forms applications to syzygies and transvectants of covariants and the determination of a hilbert basis of covariants using gordan s method are presented

operator valued bessel functions holomorphic discrete series and harmonic representations of u p q abstract this paper presents representation theoretic applications of the theory of operator valued bessel functions which was developed in the first paper of this series using the reduced bessel function k we study the harmonic representation and holomorphic discrete series for the group u p q of biholomorphic automorphisms of the siegel domains of type ii

unitary representations branching rules and matrix elements for the non compact symplectic groups the complementarity of the symplectic and orthogonal groups is used to infer properties of the infinite dimensional unirreps of the former from the character theory of the latter the complete set of d series metaplectic unirreps of sp n r is identified and branching rules are given for their restrictions to the maximal compact subgroup u n developed in terms of the properties of schur functions a known algorithm for the evaluation of matrix elements of the sp 3 r lie algebra is extended to any sp n r and analytic expressions are given for important classes of unirreps and multiplicity free states

hilbert series howe duality and branching rules abstract let be a partition with l parts and let f be the irreducible finite dimensional representation of gl m associated to when l m the littlewood restriction rule describes how f decomposes when restricted to the orthogonal group o m or to the symplectic group sp m 2 under the condition that l m 2 in this paper this result is extended to all partitions our method combines resolutions of unitary highest weight modules by generalized verma modules with reciprocity laws from the theory of dual pairs in classical invariant theory corollaries include determination of the gelfand kirillov dimension of any unitary highest weight representation occurring in a dual pair setting and the determination of their hilbert series as a graded module for p let l be a unitary highest weight representation of sp n r so 2n or u p q when the highest weight of l plus satisfies a partial dominance condition called quasi dominance we associate to l a reductive lie algebra gl and a graded finite dimensional representation bl of gl the representation bl will have a hilbert series p q that is a polynomial in q with positive integer coefficients let l be the gelfand kirillov dimension of l and set cl equal to the ratio of the dimensions of the zeroth levels in the gradings of l and bl then the hilbert series of l may be expressed in the form in the easiest example of the correspondence l bl the two components of the weil representation of the symplectic group correspond to the two spin representations of an orthogonal group

sampling for conditional inference on case control data summary the problem of exact conditional inference for discrete multivariate case control data has two forms the first is grouped case control data where monte carlo computations can be done using the importance sampling method of booth and butler 1999 biometrika86 321 332 or a proposed alternative sequential importance sampling method the second form is matched case control data for this analysis we propose a new exact sampling method based on the conditional poisson distribution for conditional testing with one binary and one integral ordered covariate this method makes computations on data sets with large numbers of matched sets fast and accurate we provide detailed derivation of the constraints and conditional distributions for conditional inference on grouped and matched data the methods are illustrated on several new and old data sets

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

weighted laplace transforms and bessel functions on hermitian symmetric spaces this paper defines ir weighted laplace transforms on the spaces of ir covariant functions by the inverse laplace transform we define operator valued bessel functions we also study the holomorphic discrete series of the automorphism group of a siegel domain of type ii

a strong generalization of helgason s theorem let c be a simple lie group with ka n an iwasawa decomposition of c and let m be the centralizer of a in k suppose k1 is a fixed closed normal analytic subgroup of k and set pf kl equal to the set of all parabolic subgroups p of c which contain man such that k1p c and k1 n p is normal in the reductive part of p suppose qr c clf v is an irreducible representation of c then if p kl 0 we obtain necessary and sufficient conditions for vkl the space of kl fixed vectors to be 0 moreover reciprocity formulas are obtained which determine dim vkt

data analytic methods for matched case control studies the recent introduction of complex multivariate statistical models in matched case control studies is a mixed blessing their use can lead to a better understanding of the way in which many variables contribute to the risk of disease on the other hand these powerful methods can obscure salient features in the data that might have been detected by other less sophisticated methods this shortcoming is due to a lack of support methodology for the routine use of these models satisfactory computation of estimated relative risks and their standard errors is not sufficient justification for the fitted model goodness of fit must be examined if inferences are to be trusted this paper is concerned with the analysis of matched case control studies with logistic models analogies of these models to linear regression models are emphasized in particular basic concepts such as analysis of variance multiple correlation coefficient one degree of freedom tests and residual analysis are discussed the fairly new field of regression diagnostics is also introduced all procedures are illustrated on a study of bladder cancer in males

classical invariant theory and the equivalence problem for particle lagrangians on montre que le probleme de l equivalence des formes binaires sous le groupe lineaire general est un cas particulier du probleme de l equivalence des lagrangiens de particule sous le pseudo groupe de transformations des variables dependantes et independantes

statistical inference for familial disease clusters in many epidemiologic studies the first indication of an environmental or genetic contribution to the disease is the way in which the diseased cases cluster within the same family units the concept of clustering is contrasted with incidence we assume that all individuals are exchangeable except for their disease status this assumption is used to provide an exact test of the initial hypothesis of no familial link with the disease conditional on the number of diseased cases and the distribution of the sizes of the various family units new parametric generalizations of binomial sampling models are described to provide measures of the effect size of the disease clustering we consider models and an example that takes covariates into account ascertainment bias is described and the appropriate sampling distribution is demonstrated four numerical examples with real data illustrate these methods

the equations for the moduli space of n points on the line a central question in invariant theory is that of determining the relations among invariants geometric invariant theory git quotients come with a natural ample line bundle and hence often a natural projective embedding this question translates to determining the equations of the moduli space under this embedding this article deals with one of the most classical quotients the space of ordered points on the projective line we show that under any weighting of the points this quotient is cut out scheme theoretically by a particularly simple set of quadric relations with the single exception of the segre cubic threefold the space of six points with equal weight we also show that the ideal of relations is generated in degree at most four and we give an explicit description of the generators if all the weights are even e g in the case of equal weight for odd n then we show that the ideal of relations is generated by quadrics

stable branching rules for classical symmetric pairs we approach the problem of obtaining branching rules from the point of view of dual reductive pairs specifically we obtain a stable branching rule for each of 10 classical families of symmetric pairs in each case the branching multiplicities are expressed in terms of littlewood richardson coefficients some of the formulas are classical and include for example littlewood s restriction rule as a special case

spherical representations of lie supergroups abstract the classical cartan helgason theorem characterises finite dimensional spherical representations of reductive lie groups in terms of their highest weights we generalise the theorem to the case of a reductive symmetric supergroup pair g k of even type along the way we compute the harish chandra c function of the symmetric superspace g k by way of an application we show that in type a iii a iii all spherical representations are self dual

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

the conditional distribution of goodness of fit statistics for discrete data abstract i consider the distribution of pearson s statistic and of the likelihood ratio goodness of fit statistic for discrete data in the important case where the data are extensive but sparse it is argued that the appropriate reference distribution is conditional on the sufficient statistic for the unknown regression parameters the first three conditional asymptotic cumulants are derived by edgeworth expansion and these are used for the computation of tail probabilities the principal advantage of the limit considered here as opposed to the more usual x 2 limit is that the cell counts need not be large

logistic regression diagnostics a maximum likelihood fit of a logistic regression model and other similar models is extremely sensitive to outlying responses and extreme points in the design space we develop diagnostic measures to aid the analyst in detecting such observations and in quantifying their effect on various aspects of the maximum likelihood fit the elements of the fitting process which constitute the usual output parameter estimates standard errors residuals etc will be used for this purpose with a properly designed computing package for fitting the usual maximum likelihood model the diagnostics are essentially free for the asking in particular good data analysis for logistic regression models need not be expensive or time consuming

function spaces and reproducing kernels on bounded symmetric domains there are several natural hilbert spaces of holomorphic functions on a bounded symmetric domain such are the bergman type spaces on which the holomorphic discrete series operates and the hardy type spaces which are related to the analytic continuation of the holomorphic discrete series also closely related is the bergmann space of entire functions on the ambient which arises as the closure of the polynomials with respect to a natural inner product the space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group k of the domain the main facts about this decomposition were proved by schmid 21 for another proof see 22 each irreducible subspace contains a unique normalized l invariant spherical polynomial where l is the isotropy group of the shilov boundary in k our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the hilbert spaces considered for the domains of classical type they were considered by hua 9 and for some of the hilbert spaces this was done before by upmeier 24 using different methods for certain others there are partial results in 22 we are able to do this in a fairly simple unified way by making strong use of gindikin s generalized gamma function s next we obtain a description of the reproducing kernels of the k irreducible subspaces in each of our hilbert spaces and an expansion in terms of these for

hilbert series howe duality and branching for classical groups an extension of the littlewood restriction rule is given that covers all pertinent parameters and simplifies to the original under littlewood s hypotheses two formulas are derived for the gelfand kirillov dimension of any unitary highest weight representation occurring in a dual pair setting one in terms of the dual pair index and the other in terms of the highest weight for a fixed dual pair setting all the irreducible highest weight representations which occur have the same gelfand kirillov dimension we define a class of unitary highest weight representations and show that each of these representations l has a hilbert series hl q of the form

products and plethysms for the fundamental harmonic series representations of u p q we give the decomposition of the kronecker products and the symmetrized kronecker squares of all the fundamental representations of the harmonic series of unitary irreducible representations of u p q the results for u 2 2 are relevant to two electron hydrogenic like atoms

assessing the fit of the logistic regression model to individual matched sets of case control data we develop exact conditional methods for checking the fit of a logistic regression model to individual matched sets in case control studies justifications are given for preferring this approach to conventional maximum likelihood based methods data from two studies are used to illustrate the techniques

operator valued bessel functions on schrodinger fock spaces and siegel domains of type ii in this paper a general theory of operator valued bessel functions on mixed schrodinger fock spaces and siegel domains of type ii is presented in general bessel functions j are associated to the partial fourier transform on a mixed schrodinger fock space j can be realized as a reduced bessel function k which is a generalized inverse laplace transform on a siegel domain of type ii the application of this theory to representation theory will be studied in the next paper

geometric interpretations of two branching theorems of d e littlewood abstract d e littlewood proved two branching theorems for decomposing the restriction of an irreducible finite dimensional representation of a unitary group to a symmetric subgroup one is for restriction of a representation of u n to the rotation group so n when the given representation of u n has nonnegative highest weight of depth n 2 it says that the multiplicity in so n of an irreducible representation of so n of highest weight is the sum over of the multiplicities of in the u n tensor product the allowable s being all even nonnegative highest weights for u n littlewood s proof is character theoretic the present paper gives a geometric interpretation of this theorem involving the tensor products explicitly the geometric interpretation has an application to the construction of small infinite dimensional unitary representations of indefinite orthogonal groups and for each of these representations to the determination of its restriction to a maximal compact subgroup the other littlewood branching theorem is for restriction from u 2 r to the rank r quaternion unitary group sp r it concerns nonnegative highest weights for u 2 r of depth r and its statement is of the same general kind the present paper finds an analogous geometric interpretation for this theorem also

products and symmetrized powers of irreducible representations of the calculation of branching rules tensor products and plethysms of the infinite dimensional harmonic series unitary irreducible representations of the non compact group is considered and the duality between and sp 2k exploited the branching rule for the restriction of an arbitrary harmonic series irreducible representation of to u n is derived and the decomposition is given explicitly for each of the infinite number of fundamental harmonic series irreducible representations of whose direct sum constitutes the metaplectic representation h of a concise expression for the decomposition of tensor products is derived and a complete analysis of the terms in both and is given a general formula for plethysms of arbitrary irreducible representations of is derived and its implementation illustrated both by means of a detailed generic example and by a complete determination of the symmetric and antisymmetric terms of finally relationships that arise from the embedding of the product groups and in the metaplectic group mp 4nk are discussed