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Book Information:
Authors:
Roe Goodman and
Nolan R. Wallach
Title:
Representations and invariants of the classical groups
Additional book information:
Cambridge Univ. Press,
Cambridge,
1998,
xvi + 685 pp.,
ISBN 0-521-58273-3,
$100.00$,
hardcover;
ISBN 0-521-66348-2,
$39.95$,
paperback
L. C. Biedenharn and J. D. Louck, Angular momentum in quantum physics, Encyclopedia of Mathematics and its Applications, vol. 8, Addison-Wesley Publishing Co., Reading, Mass., 1981. Theory and application; With a foreword by Peter A. Carruthers. MR 635121
Hermann Boerner, Representations of groups. With special consideration for the needs of modern physics, Second English edition, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1970. Translated from the German by P. G. Murphy in cooperation with J. Mayer-Kalkschmidt and P. Carr. MR 0272911
Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344, DOI 10.1007/978-3-662-12918-0
5. A. Cayley, A Second Memoir upon Quantics, CLXVI (1856) Philosophical Transactions of the Royal Society of London [Collected Works, Vol. II (1889), Cambridge at the University Press (141), pp. 533-538].
6. A. Clebsch and F. Lindemann, Vorlesungen über Geometrie, (1876-1891) Leipzig.
Jacques Dixmier, Quelques aspects de la théorie des invariants, Gaz. Math. 43 (1990), 39–64 (French). Translated by J.-R. Billuard. MR 1035388
J. Dixmier and D. Lazard, Le nombre minimum d’invariants fondamentaux pour les formes binaires de degré $7$, Portugal. Math. 43 (1985/86), no. 3, 377–392 (French, with English summary). MR 886462
9. E.B. Elliott, An Introduction to the Algebra of Quantics, 1895, Oxford at the Clarendon Press.
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Hans Freudenthal and H. de Vries, Linear Lie groups, Pure and Applied Mathematics, Vol. 35, Academic Press, New York-London, 1969. MR 0260926
12. F. von Gall, Das vollständige Formensystem der binären Formen 7ten Ordnung, 31 (1888) Math. Ann., 318-336.
13. O.E. Glenn, A Treatise on the Theory of Invariants, (1915) Ginn and Company, Boston.
Paul Gordan, Vorlesungen über Invariantentheorie, 2nd ed., Chelsea Publishing Co., New York, 1987 (German). Erster Band: Determinanten. [Vol. I: Determinants]; Zweiter Band: Binäre Formen. [Vol. II: Binary forms]; Edited by Georg Kerschensteiner. MR 917266
15. J.H. Grace and A. Young, The Algebra of Invariants, (1903) Cambridge University Press, (Reprinted Chelsea Publishing Company, Bronx, New York).
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR 1321638
Roger E. Howe, The first fundamental theorem of invariant theory and spherical subgroups, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 333–346. MR 1278717, DOI 10.4310/pamq.2005.v1.n1.a11
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
Nathan Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR 559927
David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. MR 719371, DOI 10.1007/978-3-642-96676-7
Peter J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995. MR 1337276, DOI 10.1017/CBO9780511609565
23. G. Salmon, A Treatise on Higher Plane Curves, 3rd Edition (1889), Hodges, Foster and Figgis, Dublin.
24. G. Salmon, Lessons Introductory to the Modern Higher Algebra, 3rd Edition, (1885), Hodges, Foster and Figgis, Dublin.
Issai Schur, Vorlesungen über Invariantentheorie, Die Grundlehren der mathematischen Wissenschaften, Band 143, Springer-Verlag, Berlin-New York, 1968 (German). Bearbeitet und herausgegeben von Helmut Grunsky. MR 0229674
Tetsuji Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89 (1967), 1022–1046. MR 220738, DOI 10.2307/2373415
Jacques Tits, Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Springer-Verlag, Berlin-New York, 1967 (German). MR 0218489
28. H.W. Turnbull, The Theory of Determinants, Matrices and Invariants, (1929) Blackie (London, Glasgow).
V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0376938
Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Alfred Young, The collected papers of Alfred Young (1873–1940), Mathematical Expositions, No. 21, University of Toronto Press, Toronto, Ont.-Buffalo, N.Y., 1977. With a foreword by G. de B. Robinson and a biography by H. W. Turnbull. MR 0439548
D. P. Zhelobenko, Kompaktnye gruppy Li i ikh predstavleniya, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0473097
- 1.
- L.C. Biedenharn and J.D. Louck, Angular Momentum in Quantum Physics, Theory and Application, (1981) Encyclopedia of Mathematics and Its Applications, G.-C. Rota, editor, vol. 8, Addison-Wesley Publishing Company, Reading, Mass. MR 83a:81001
- 2.
- H. Boerner, Representations of Groups with Special Consideration for the Needs of Modern Physics, (1970) North Holland, New York. MR 42:7792
- 3.
- A. Borel, Linear Algebraic Groups, (1969) W.A. Benjamin, New York. MR 40:4273
- 4.
- T. Bröcker and T. tomDieck, Representations of Compact Lie Groups, (1985) Graduate Texts in Mathematics 98, Springer-Verlag (New York, Berlin, Heidelberg, Tokyo). MR 86i:22023
- 5.
- A. Cayley, A Second Memoir upon Quantics, CLXVI (1856) Philosophical Transactions of the Royal Society of London [Collected Works, Vol. II (1889), Cambridge at the University Press (141), pp. 533-538].
- 6.
- A. Clebsch and F. Lindemann, Vorlesungen über Geometrie, (1876-1891) Leipzig.
- 7.
- J. Dixmier, Quelques aspects de la théorie des invariants 43 (1990), Gazette des Math., 39-64. MR 90m:15047
- 8.
- J. Dixmier and D. Lazard, Le Nombre Minimum d'Invariants Fondamentaux Pour Les Formes Binaires de Degré 7, 43 (1985-1986) Portugaliae Mathematica. MR 88f:15045
- 9.
- E.B. Elliott, An Introduction to the Algebra of Quantics, 1895, Oxford at the Clarendon Press.
- 10.
- W. Fulton and J. Harris, Representation Theory: A First Course, (1991) Graduate Texts in Mathematics 129, Springer-Verlag, (New York, Berlin, Heidelberg). MR 93a:20069
- 11.
- H. Freudenthal and H. de Vries, Linear Lie Groups, (1969) Academic Press, New York. MR 41:5546
- 12.
- F. von Gall, Das vollständige Formensystem der binären Formen 7ten Ordnung, 31 (1888) Math. Ann., 318-336.
- 13.
- O.E. Glenn, A Treatise on the Theory of Invariants, (1915) Ginn and Company, Boston.
- 14.
- P. Gordan, Vorlesungen über Invariantentheorie, (ed. G. Kerschensteiner) Vol. 1 (1885), Vol. 2 (1887) Leipzig, 2nd Edition (1987) Chelsea Publishing Company, New York. MR 89g:01034
- 15.
- J.H. Grace and A. Young, The Algebra of Invariants, (1903) Cambridge University Press, (Reprinted Chelsea Publishing Company, Bronx, New York).
- 16.
- S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, (1978) Pure and Applied Math. 80, Academic Press, NY. MR 80k:53081
- 17.
- R. Howe, Perspectives on Invariant Theory, The Schur Lectures (1992) (ed., I. Piatetski-Shapiro and S. Gelbart), Israel Mathematical Conference Proceedings 8. MR 96e:13006
- 18.
- R. Howe, The first fundamental theorem of invariant theory and spherical subgroups, (333-346), Vol. 56 Part One, Proc. of Summer Research Institute on Algebraic Groups and Their Generalizations, (1991) (Eds. Haboush and Parshall), Proceedings of Symposia in Pure Mathematics, AMS, Providence, Rhode Island. MR 95f:13008
- 19.
- J.E. Humphreys, Introduction to Lie Algebras and Representation Theory (1972) Springer-Verlag. MR 48:2197
- 20.
- N. Jacobson, Lie Algebras, (1962) Wiley-Interscience, New York (Reprinted Dover 1979). MR 80k:17001
- 21.
- D. Mumford and J. Fogarty, Geometric Invariant Theory, 2nd edition (1982) Ergebnisse der Math. und Ihrer Grenzgebiete 34, Springer-Verlag (Berlin, Heidelberg, New York). MR 86a:14006
- 22.
- P.J. Olver, Equivalence, Invariants and Symmetry, (1995) Cambridge University Press. MR 96i:58005
- 23.
- G. Salmon, A Treatise on Higher Plane Curves, 3rd Edition (1889), Hodges, Foster and Figgis, Dublin.
- 24.
- G. Salmon, Lessons Introductory to the Modern Higher Algebra, 3rd Edition, (1885), Hodges, Foster and Figgis, Dublin.
- 25.
- I. Schur (H. Grunsky), Vorlesungen über Invariantentheorie, (1968) Springer-Verlag. MR 37:5248
- 26.
- T. Shioda, On the graded ring of invariants of binary octavics, 89 (1967) Amer. J. Math., 1022-1046. MR 36:3790
- 27.
- J. Tits, Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, (1967) Lecture Notes in Math. 40, Springer-Verlag, Berlin. MR 36:1575
- 28.
- H.W. Turnbull, The Theory of Determinants, Matrices and Invariants, (1929) Blackie (London, Glasgow).
- 29.
- V.S. Varadarajan, Lie Groups, Lie algebras and Their Representations, (1974; 1984) Graduate Texts in Math. 102, Springer-Verlag, New York. MR 51:13113; MR 85e:22001
- 30.
- N.R. Wallach, Harmonic Analysis on Homogeneous Spaces, (1973) Marcel Dekker, Inc., New York. MR 58:16978
- 31.
- H. Weyl, The Classical Groups, Their Invariants and Representations (First Edition, 1939, Second Edition with supplements, 1946) Princeton University Press. MR 1:42c; MR 98k:01049
- 32.
- A. Young, Collected Papers, Univ. of Toronto Press Math. Expositions, no. 21 (1977), Univ. of Toronto Press. MR 55:12438
- 33.
- D.P. \v{Z}elobenko, Compact Lie Groups and Their Representations, (1973) Translations of Mathematical Monographs 40, Amer. Math. Soc., Providence, RI. MR 57:12776b
Review Information:
Reviewer:
Jacob Towber
Affiliation:
De Paul University
Email:
jtowber@math.uchicago.edu
Journal:
Bull. Amer. Math. Soc.
36 (1999), 533-538
DOI:
https://doi.org/10.1090/S0273-0979-99-00795-8
Published electronically:
July 28, 1999
Review copyright:
© Copyright 1999
American Mathematical Society
