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Unitary symmetry and combinatorics. (English) Zbl 1190.51002

Hackensack, NJ: World Scientific (ISBN 978-981-281-472-2/hbk). xxi, 619 p. (2008).
This monograph is an outgrowth of the books “Angular momentum in quantum physics” [Reading, Massachusetts: etc.: Addison-Wesley Publishing Company (1981; Zbl 0474.00023); reprint (2009; Zbl 1158.00013)] and ”The Racah-Wigner algebra in quantum theory” [Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1981; Zbl 0474.00024)] by L. C. Biedenharn and J. D. Louck. This book consists of three distinct, but interrelated parts: Chapters 1-4, Chapters 5-9 and Chapters 10-11. The last two chapters are compendiums which define, develop, and summarize concepts used in the first nine chapters.
Chapters 1-4 deal with basic angular momentum theory and the properties of the famous Wigner \(D\)-functions, now extended to polynomial forms over four commuting indeterminates, and with the properties of arbitrary many multiple Kronecker products of these extended \(D\)-polynomials.
Chapters 5-9 deal with the generalization of the solid harmonics to polynomials called \(D^\lambda\)-polynomials, where \(\lambda\) is a partition, and these polynomials are defined over \(n^2\) commuting indeterminates, which when specialized to the elements of a complex matrix of order \(n\) give the integral irreducible representations of the general linear group of complex matrices of order \(n\), and, in particular, all nonequivalent irreducible representations of the general unitary group of matrices of order \(n\).
Chapters 10-11 comprise the third part of this monograph. They consist of two extensive Compendiums A and B of results from algebra, analysis and combinatorics that relate to the first two parts.

MSC:

51-02 Research exposition (monographs, survey articles) pertaining to geometry
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
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