From: Subject: Re: invariant measures on SO(3) and SE(3)? Date: Wed, 5 May 1999 23:25:13 -0500 (CDT) Newsgroups: [missing] To: bruyninc@leland.Stanford.EDU Keywords: Haar measure >Where van I find (references to) invariant measures (``Haar'' measures) >on SO(3) and SE(3)? Gee, how much detail do you want? I've taken the liberty of enclosing the reviews of a few texts in a few of the areas of mathematics where these topics would arise. I think if you don't make your request more specific, we will have every group theorist, measure theorist, and harmonic analyst telling you everything they know... Until you respond with a more specific question I'd rather not approve your post, OK? dave (moderator) PS - one or two more references at index/43-XX.html 98e:28001 28-01 (60-01) Simonnet, Michel(SNG-DAK) Measures and probabilities. (English. English summary) With a foreword by Charles-Michel Marle. Universitext. Springer-Verlag, New York, 1996. xiv+510 pp. $44.00. ISBN 0-387-94644-6 This book seems to be intended both as an advanced textbook and as a reference book on measure theory. It deals with the three usual faces of integration theory (countably additive measures on abstract sets, Daniell integrals on abstract sets, and Radon measures on locally compact Hausdorff spaces), plus some applications to probability theory and to analysis on locally compact groups. The prerequisites for reading the book are basic point-set topology and functional analysis. The book begins with an introduction to ordered groups and vector spaces and then introduces the Daniell construction of the integral (the functions being integrated are allowed to have values in a Banach space). As special cases Simonnet treats integration with respect to abstract measures (including their extension from semirings of sets to $\sigma$-rings of sets) and the theory of Radon measures. The usual material for a course on measure theory is presented in detail. The second half of the book begins with applications to probability theory. This includes the existence of measures on infinite product spaces, Birkhoff's ergodic theorem, an introduction to the central limit theorem (including Fourier transforms on ${R}\sp n$), the strong law of large numbers, and conditional expectations and probabilities. There is no treatment, for example, of martingales, of Brownian motion, or of weak convergence on more general metric spaces. The final part of the book returns to Radon measures, following (as the author points out) Bourbaki rather closely. After some more generalities on Radon measures (including the Radon-Nikodym theorem for Radon measures, products of Radon measures, etc.), the book contains a chapter devoted to Haar measures, and then closes with a chapter on convolutions of measures and functions. Although the treatment is rather abstract and general, it is also very concrete. For example, a thorough treatment of change of variables in multiple integrals is given, followed later in the book with details on the calculation of Haar measures for some concrete groups. There are numerous exercises of all sorts---a bit more than 75 pages of them. On the other hand, the book contains no historical notes and no bibliography. All in all, the book contains a large amount of information, presented in a careful manner. However, its level of generalityq (in particular, the use of the elements of functional analysis throughout the book), plus the fact that abstract measures, Daniell integrals, and Radon measures are simultaneously studied, may make this book more useful as a reference for advanced students than as a textbook for a basic real analysis course. Reviewed by Donald L. Cohn _________________________________________________________________ 98c:43001 43-01 (22-01 46-01) Folland, Gerald B.(1-WA) A course in abstract harmonic analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. $61.95. ISBN 0-8493-8490-7 This deligthful book fills a long-standing gap in the literature on abstract harmonic analysis. For the author the term "harmonic analysis" means those parts of analysis in which the action of a locally compact group plays an essential role: more specifically, the theory of unitary representations of locally compact groups, and the analysis of functions on such groups and their homogeneous spaces. The book contains a careful treatment of certain key results in the subject that were developed from about 1927 (the date of the Peter-Weyl theorem) up to 1970. The focus is on fundamental ideas and theorems in harmonic analysis that are used over and over again, and which can be developed with minimal assumptions on the nature of the underlying group. Its purpose is not to compete in any way with the many existing excellent monographs and treatises on the subject, but to provide a unified picture of the general abstract theory in an introductory book of moderate length. To the reviewer's knowledge no one existing book contains all of the topics that are treated in this one. To be sure, various bits and pieces of what the author covers can be found in one reference or another, and certain aspects of the theory are treated much more extensively in a few lengthy treatises [see, e.g., J. Dixmier, $C\sp*$-algebras, Translated from the French by Francis Jellett, North-Holland, Amsterdam, 1977; MR 56 #16388; J. M. G. Fell and R. S. Doran, Representations of $\sp *$-algebras, locally compact groups, and Banach $\sp *$-algebraic bundles. Vol. 1, Academic Press, Boston, MA, 1988; MR 90c:46001; Vol. 2; MR 90c:46002]. Assuming only a knowledge of real analysis and elementary functional analysis, the author carefully introduces and proves (with a few exceptions), in the first six chapters, classical facts in representation theory. Chapter 1, titled "Banach algebras and spectral theory", contains background material on $C\sp *$-algebras and spectral theory of *-representations that is needed in the remainder of the book. As the author points out, Chapters 2--6 form the core of the book. Chapter 2, titled "Locally compact groups", develops the basic tools for doing analysis on groups and homogeneous spaces. Here the reader will find a nice introductory treatment of topological groups, Haar measure, convolutions, homogeneous spaces and quasi-invariant measures. Chapter 3, titled "Basic representation theory", presents the rudiments of unitary representation theory up through the Gelfand-Raikov existence theorem for irreducible unitary representations. The connections between functions of positive type and representations are also described. Chapters 4 and 5 are respectively entitled "Analysis on locally compact abelian groups" and "Analysis on compact groups". Here the Fourier transform takes center stage, first as a straightforward generalization of the classical Fourier transform ${\scr F}f(\xi)=\int\sb {-\infty}\sp \infty e\sp {-2\pi ix\xi}f(x)dx$ from the real line to locally compact abelian groups, and then to the more representation-theoretic form that is appropriate for the non-abelian, compact case. Chapter 6 presents the theory of induced representations. This is a way of constructing a unitary representation of a locally compact group $G$ out of a unitary representation of a closed subgroup $H$. Geometrically speaking, these induced representations are the unitary representations of $G$ arising from the action of $G$ on functions or sections of homogeneous vector bundles on the homogeneous space $G/H$. After describing the construction of induced representations for locally compact groups, the author proves the Frobenius reciprocity theorem for compact groups. This provides a powerful tool for finding the irreducible decomposition of an induced representation of a compact group. He then develops the notion of pseudomeasures of positive type (a generalization of functions of positive type) and uses it to prove the theorem on induction in stages and the imprimitivity theorem, which is the deepest result of the chapter. It forms the basis for the so-called "Mackey machine", a body of techniques for analyzing representations of a group $G$ in terms of the representations of a normal subgroup $N$ and the representations of various subgroups of $G/N$. It is important to mention that the author includes specific examples throughout the book to illustrate the general theory. In Chapters 2--4 these examples are interwoven with the rest of the text, while in Chapters 5 and 6 they are, for the most part, collected in separate sections at the end of the chapter. Now, a few words need to be said about Chapter 7, which is entitled "Further topics in representation theory". Focusing on the theory of noncompact, nonabelian, locally compact groups, it is more like a survey article than a chapter of the book. The principal object of study is the dual space $\hat{G}$ of a locally compact group $G$, i.e., the set of equivalence classes of irreducible unitary representations of $G$ furnished with a natural topology (which in this book is called the Fell topology). Topics discussed include the group $C\sp *$-algebra of a locally compact group, the dual space, tensor products, direct integrals, and the Plancherel theorem. As the author observes, giving a complete treatment of this material would require a lengthy digression into the theory of von Neumann algebras, representations of $C\sp *$-algebras, and direct integral decompositions that would substantially increase the size of the book. As a result, the author is content with providing definitions and statements of the theoerems, together with a discussion of some concrete cases. References to sources where a detailed treatment of all of these topics can be found are provided throughout the chapter. To help make the book self-contained, three brief appendices are provided, respectively entitled "A Hilbert space miscellany", "Trace-class and Hilbert-Schmidt operators", and "Vector-valued integrals". The bibliography consists of 134 carefully selected references and makes no pretence at completeness. Finally, a few general concluding remarks. This book is aimed at a broad mathematical audience. One of the reasons the author wrote it (see the Preface) is that he believes the material is "beautiful". His respect for the subject shows on every hand. This is apparent through his careful writing style, which is concise, yet simple and elegant. The reviewer would encourage anyone with an interest in harmonic analysis to have this book in his or her personal library. The author is to be congratulated on writing a fine book that the reviewer would have been proud to write. Reviewed by Robert S. Doran _________________________________________________________________ 97c:22001 22-01 (20C05 20C15 22E15) Simon, Barry Representations of finite and compact groups. (English. English summary) Graduate Studies in Mathematics, 10. American Mathematical Society, Providence, RI, 1996. xii+266 pp. $34.00. ISBN 0-8218-0453-7 [AMS Book Store] Although not divided explicitly, the book consists of two parts which should be considered separately. The first one is concerned with the theory of representations of finite groups; it contains 6 chapters and 120 pages. The second part, devoted to the theory of compact Lie groups, contains 3 chapters and 135 pages. Taking into account that the subject of this latter part is much more extensive and complicated, it is obvious that the author has had to apply a different approach in attempting to cover it in almost the same space. In a concise form one can say that, while the first part can be considered as a complete and self-contained introduction to finite group representations, the second one presents selected topics of the theory of compact groups and their representations. Chapter I is devoted to basic information about finite groups, homogeneous spaces, and constructions of the direct and semi-direct products of groups. An exhaustive list of examples is presented, including $Z\sb n$, the permutation group $S\sb n$, finite groups of rotations, Platonic groups, and $p$-groups including Sylow theorems. Chapter II describes the fundamental concepts and results about representations of finite groups: irreducible representations, regular representation, group algebra, matrix elements, Schur's lemma. Special attention is paid to the classification of the irreducible representations as real, complex or quaternionic. Chapter III is devoted to the central components of representation theory, such as the theory of characters and of class functions, and Fourier analysis. The dimension theorem is also proved. Chapter IV is concerned with representations of abelian finite groups, dual groups and Clifford groups. Chapter V is of a more general character. It presents the Frobenius theory of irreducible representations of semidirect products, the general induced representations of finite groups, the Frobenius character formula and the reciprocity theorem, and Mackey's criterion of irreducibility. Chapter VI is totally devoted to the representations of symmetric groups with application of Young frames and Young tableaux. The Frobenius character formula for $S\sb n$ and its applications close the first part of the book, which can be recommended as a very good text about finite groups and their representations. The approach is elementary, and the presentation is clear and well organized, in the form of a course. In the unique case when auxiliary material is necessary (the theory of algebraic integers), the exposition is concise, complete and elegant. Passing to the second part, devoted to compact groups and their representations, we must emphasize that it treats almost exclusively finite-dimensional representations of the compact Lie groups and the approach is much more algebraic than we could expect after reading the introduction. Chapter VII is mostly introductory and contains generalities (without proofs) about $C\sp \infty$-manifolds, homotopy theory and multilinear algebra interspersed with the elements of representation theory. Then Lie groups and their Lie algebras, the exponential mapping and the adjoint representation are introduced. The construction of the Haar measure is carried over for general Lie groups. The classical matrix groups are presented as examples of Lie groups. The detailed description of their structure presented along the whole text is a great advantage of the book. The final 9 pages of this chapter are devoted to the representations of groups. The author, anxious to avoid general concepts, speaks only of compact Lie groups acting on finite-dimensional spaces. The orthogonality relations for matrix elements and the Peter-Weyl theorem are proved only in this context. Obviously it is impossible to avoid infinite-dimensional representations completely; hence the author is sometimes forced to speak of (undefined) "infinite-dimensional representations" or, as in Theorem VII.10.8, of a "strongly continuous map of $G$ to unitary operators on ${\scr H}$" (forgetting at this moment that the map should be a group homomorphism). At the beginning of Chapter VIII, which in fact is an original contribution to the theory of maximal tori in compact Lie groups, the exposition is strangely complicated. First, the existence of the maximal tori and the fact that all of them are conjugate to each other is announced in Theorem VIII.1.1 for compact and semisimple Lie groups. In order to prove that the compactness is critical, the author gives an example of a group without a maximal torus which is neither compact nor semisimple; hence the example fails. Next, he proves the equivalence of Theorem VIII.1.1 to Theorem VII.1.1$'$, where the semisimplicity is not assumed. The above-mentioned counterexample put after Theorem VIII.1.1$'$ would work perfectly. The version VII.1.1$'$ is proved finally but the proof of the existence of the maximal tori appears as a remark outside this proof. This part of the book is interesting but needs polishing. The final sections of the chapter are algebraic and devoted to the concepts of roots, root spaces, to the classification of the fundamental systems of roots, Dynkin diagrams, Weyl groups and Cartan-Stiefel diagrams. Again, the classical groups are presented from this point of view. The last and the most extensive Chapter IX begins with the study of the geometry of the Cartan-Stiefel diagrams and of the integral forms. After proving the Weyl integration formula, the maximal weights are introduced and the Weyl character formula is proved. As applications of the latter, the Weyl dimension formula, and the multiplicity formulas of Kostant and Freudenthal, and the formulas of Racah and of Steinberg for Clebsch-Gordan integers are given. The last sections contain the description of irreducible representations of compact classical groups and their tensor products. The real and quaternionic representations are distinguished. The alternative proof of the Frobenius character formula appears in relation with the tensor products of irreducible representations of the group ${\rm U}(n)$. It must be mentioned that the description of the irreducible representations, although made for groups, not their Lie algebras, is algebraic, being based on the concept of the highest weight. The analytic realizations do not appear even in the examples. The induced representations are not introduced in this part of the book; hence Frobenius reciprocity is also absent. The decomposition theory of representations is practically omitted. The author's promise to give more analytic flavour to the theory is kept only in the part concerning the structure of the compact Lie groups. Surprisingly, the algebraic parts of the book seem to be more complete and better organized. The theory of representations of groups is nowadays a very extensive area. Textbooks presenting particular topics of the theory are very desirable. In particular this book can be recommended as a base for courses about representations of finite groups and finite-dimensional representations of Lie groups. It is a pity that the bibliography is definitely incomplete. The absence of the classical monographs of C. W. Curtis and I. Reiner, H. Weyl, I. M. Gelfand and M. A. Naimark, S. Helgason, and D. P. Zhelobenko is difficult to explain. Reviewed by Antoni Wawrzynczyk _________________________________________________________________ 96b:00001 00A05 (28-01 30-01 46-01) Gelbaum, Bernard R.(1-SUNYB) Modern real and complex analysis. (English. English summary) A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1995. xiv+489 pp. $64.95. ISBN 0-471-10715-8 This book is ambitious in scope and aims to achieve in one volume what older French and German cours d'analyse did in several. Besides the standard topics one would expect from the title, there is much more. The basics of point-set topology are reviewed, uniform structures and simplices discussed, and Tikhonov's product theorem, Brouwer's fixed-point theorem, and the Tietze-Urysohn extension theorem proved. {Here a pedagogical opportunity is missed: although the open map theorem in Banach spaces is later proved, the commonality of proof with the extension theorem [S. Grabiner, Amer. Math. Monthly 93 (1986), no. 3, 190--191; MR 88a:54034] is neither exploited nor mentioned.}There is an admirably concise treatment of the complex exponential and circular functions. Integration is from the Daniell-functional as well as the Caratheodory-outer-measure point of view, and the Haar measure is constructed. Considerable functional analysis (weak topologies, Banach algebras, Hilbert space, the $C\sp *$-algebra version of the spectral theorem) is developed, and we are only up to page 140 in this 490-page book. In the complex analysis half, we see Pompeiu's generalization of the Cauchy integral formula (via Stokes), Riemann surfaces developed in some detail, the uniformization theorem presented as a sequence of exercises, a short introduction to several complex variables, and a very nice short chapter entitled "convexity and complex analysis" (centering around the Riesz-Thorin convexity theorem). For this wealth of topics and length the book's $$65$ price must be considered reasonable nowadays. All this notwithstanding, the book has serious defects. Some are technical/stylistic. For example, the author's laudable concision is often at the expense of readability: symbols are preferred to words, but even the 6-page symbol index is unable to chronicle all of them, and the appearance of the printed page (not to mention the reading of it) is, in the argot, not very user-friendly. Parentheses are rampant where not needed (e.g., we see, for propositions $A$ and $B$, $A\wedge B$, but $\{A\}\Rightarrow\{B\}$) and sometimes absent where needed (e.g., in $\int f+g$). It is often hard to know where hypotheses end and conclusions begin because of the author's sparing use of "then" and "that", and statements of theorems are often convoluted. This is sure to impede foreign readers. (Native speakers of the international scientific language are too often not conscious of their special obligations to handle it meticulously.) These deficiencies are regrettable but bearable. More serious are the logical deficiencies, and unfortunately they are legion (as are routine typographical errors). The reviewer read in detail the first $50%$ of the complex variables part and generated over ten pages of errata. Some things are repairable, but probably not by neophytes, while others are unsalvageable. The attempt to derive Weierstrass' theorem on specifying zeros from Mittag-Leffler's theorem on specifying principal parts is an example of the latter: in the course of it the function $(z-a)/(b-a)$ is exhibited as the exponential of a holomorphic function in a deleted neighborhood of $a$. And the proof of Mittag-Leffler's theorem itself involves a confused misuse of Runge's theorem. Exercise 6.2.26 asks the reader to prove that if $u\sb n$ are uniformly bounded and harmonic in an open disk $D$ and converge on a set with a cluster point in $D$, then the sequence converges throughout $D$. Another exercise (with a hint!) claims that if $u$, $v$ are harmonic in region $\Omega$ and $\limsup u\leq\liminf v$ at each boundary point, then $u\leq v$. But perhaps a student can be expected to observe that $u\sb n(z)=(-1)\sp n{\rm Im}\,z$ and $\Omega=\bold C\sb \infty\sbs\{0,\infty\}$, $v(z)=\log \vert z\vert $, $u=2v$ provide counterexamples. The book is probably valuable to cognoscenti for its breadth of topics, overview and organization, but cannot be recommended as a text---except as a challenge to the more mature student. Any automobile marketed with as many defects would surely be recalled. Does the publisher deserve some blame for not having had this critically read by a mathematician (and paid him/her adequately to do so)? Shouldn't the author himself feel such an obligation to the mathematical public? Ours is, after all, a self-policing profession. Reviewed by R. B. Burckel _________________________________________________________________ 95i:20001 20-01 (20C30 20C35 22E46) Sternberg, S.(1-HRV) Group theory and physics. (English. English summary) Cambridge University Press, Cambridge, 1994. xiv+429 pp. ISBN 0-521-24870-1 There are hundreds of books written on group theory and perhaps a hundred about physical applications, and it seems already impossible to write something very outstanding. Nevertheless it can be done, as we are witnessing here with a new book on applications of group theory in physics where modern mathematics is nicely intertwined with physics, from classical crystallography to fullerenes and from symmetry properties of atoms and molecules to quarks. The book contains a fresh approach to many topics and shows the highest degree of mathematical competency. The text seems to be very friendly to physicists though written in terms of modern mathematics (morphisms, orbits, vector bundles, etc.). In addition there are interesting and valuable excursions into the history of groups and spectroscopy and citations of classical works which make the reading of the book a real pleasure. Perhaps the best introduction of the book would be to reproduce the contents (the headings of sections being somewhat abridged). Chapter 1. Basic definitions and examples (definition of group, examples, homomorphisms, action on a set, conjugation, topology of groups SU(2) and SO(3), morphisms, finite subgroups of SO(3) and O(3); applications to crystallography, icosahedral group and fullerenes). Chapter 2. Representation theory of finite groups (definitions, examples, irreducibility, complete reducibility, Schur lemma, characters, regular representation, acting on function spaces, representations of the symmetric group). Chapter 3. Molecular vibrations and homogeneous vector bundles (small oscillations, molecular displacements and vector bundles, induced representations, principal bundles, tensor products, operators and selection rules, semiclassical theory of radiation, semidirect products and their representations, Wigner's classification of irreps of the Poincare group, parity, Mackey theorems on induced representations with applications to the symmetric group, exchange forces and induced representations). Chapter 4. Compact groups and Lie groups (Haar measure, Peter-Weyl theorem, irreps of SU(2), irreps of SO(3) and spherical harmonics; hydrogen atom, periodic table, shell model of the nucleus, CG-coefficients and isospin, relativistic wave equations; Lie algebras, representations of su(2)). Chapter 5. The irreducible representations of ${\rm SU}(n)$ (tensor representations of ${\rm GL}(V)$, restrictions to some subgroups, decompositions, computational rules, weight vectors, finite-dimensional irreps of ${\rm Sl}(d,\bold C)$; strangeness, the Eightfold Way, quarks, color and beyond. Where do we stand?). Appendices. A. The Bravais lattices and the arithmetical crystal classes. B. Tensor product. C. Integral geometry and the representations of the symmetric group. D. Wigner's theorem on quantum mechanical symmetries. E. Compact groups, Haar measure, and the Peter-Weyl theorem. F. A history of 19th-century spectroscopy. G. Characters and fixed point formulas for Lie groups. This book will certainly become a landmark among the books on group theory in physics as are the classical books by B. L. van der Waerden, H. Weyl, E. P. Wigner, M. Hamermesh, etc. Reviewed by J. Lohmus © Copyright American Mathematical Society 1999