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
http://www.math.niu.edu/~rusin/known-math/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