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Multiplicative invariant theory, as a research area in its own right within the wider spectrum of invariant theory, is of relatively recent vintage. The present text offers a coherent account of the basic results achieved thus far..
Multiplicative invariant theory is intimately tied to integral representations of finite groups. Therefore, the field has a predominantly discrete, algebraic flavor. Geometry, specifically the theory of algebraic groups, enters through Weyl groups and their root lattices as well as via character lattices of algebraic tori.
Throughout the text, numerous explicit examples of multiplicative invariant algebras and fields are presented, including the complete list of all multiplicative invariant algebras for lattices of rank 2.
The book is intended for graduate and postgraduate students as well as researchers in integral representation theory, commutative algebra and, mostly, invariant theory.
Multiplicative invariant theory is intimately tied to integral representations of finite groups. Therefore, the field has a predominantly discrete, algebraic flavor. Geometry, specifically the theory of algebraic groups, enters through Weyl groups and their root lattices as well as via character lattices of algebraic tori.
Throughout the text, numerous explicit examples of multiplicative invariant algebras and fields are presented, including the complete list of all multiplicative invariant algebras for lattices of rank 2.
The book is intended for graduate and postgraduate students as well as researchers in integral representation theory, commutative algebra and, mostly, invariant theory.
From the reviews:
"[...] Multiplicative Invariant Theory by Martin Lorenz is a beautiful book on an exciting new subject, written by an expert and major contributor to the field. Indeed, Chapter 4 on class groups is substantially due to the author [...], as is much of the progress discussed in Chapter 8 on understanding when the fixed ring R^G inherits the Cohen-Macaulay property [...]. Chapter 5 on Picard groups benefits greatly from his insight [...]. The book includes all of the above discussed material and a good deal more. Most of the proofs have been completely reworked, and many of the results appear to be new. The author is especially careful to explain where each chapter is going, why it matters, and what background material is required. The last chapter on open problems, with a good deal of annotation, is certainly welcome, since there is much yet to be done. Be aware, this is definitely a research monograph. The subject matter is broad and deep, and the prerequisites on the reader can sometimes be daunting. Still, it is wonderful stuff and well worth the effort. [...]"
D.S.Passman, Bulletin of the American Mathematical Society, Vol. 44, Number 1, Jan. 2007
"... Martin Lorenz has written an excellent book treating the theory of invariants of groups acting on lattices. ... The choice of topics and the order in which they are presented is very good. The proofs are easy to follow, the references are many and thorough. The author brings many diverse topics together in one place. ..."
Robert M. Fossum, SIAM Review, Vol. 48 (2), 2006
"The book under review is the first systematic treatment of multiplicative invariant theory in the form of a textbook written by an author who has contributed several research articles ... . The book is recommended for graduate and postgraduate students as well as researchers in representation theory, commutative algebra, and invariant theory. It opens a freshview to research problems on these fields related to multiplicative invariants."
Peter Schenzel, Zentralblatt MATH, Vol. 1078, 2006
"... So it is not surprising that the body of the book, which gives for the first time a full account of the algebraic side of the theory, is concerned with classical themes: class group, Picard group, regularity and Cohen-Macaulay property of multiplicative invariant algebras. As it turns out the results differ sometimes strongly from the linear case and the proofs are much more involved. A chapter on ordered and twisted invariant fields, which are connected intimately with Noether's rationality problem, and one on open problems complete the book."
G.Kowohl, Monatshefte für Mathematik 148:4, p. 352-353, 2006
Aus den Rezensionen: "... Das vorliegende Buch stellt den aktuellen Stand der multiplikativen Invariantentheorie dar. Nach einführenden Kapiteln folgen Kapitel über die Klassengruppe, die Picard-Gruppe, Invarianten von Spiegelungsgruppen, Regularität und die Cohen-Macaulay-Eigenschaft. ... Das Buch ist übersichtlich und gut lesbar geschrieben, ich kann es allen an der Invariantentheorie Interessierten empfehlen." (Franz Pauer, in: IMN Internationale Mathematische Nachrichten, August/2009, Issue 211, S. 76)
"[...] Multiplicative Invariant Theory by Martin Lorenz is a beautiful book on an exciting new subject, written by an expert and major contributor to the field. Indeed, Chapter 4 on class groups is substantially due to the author [...], as is much of the progress discussed in Chapter 8 on understanding when the fixed ring R^G inherits the Cohen-Macaulay property [...]. Chapter 5 on Picard groups benefits greatly from his insight [...]. The book includes all of the above discussed material and a good deal more. Most of the proofs have been completely reworked, and many of the results appear to be new. The author is especially careful to explain where each chapter is going, why it matters, and what background material is required. The last chapter on open problems, with a good deal of annotation, is certainly welcome, since there is much yet to be done. Be aware, this is definitely a research monograph. The subject matter is broad and deep, and the prerequisites on the reader can sometimes be daunting. Still, it is wonderful stuff and well worth the effort. [...]"
D.S.Passman, Bulletin of the American Mathematical Society, Vol. 44, Number 1, Jan. 2007
"... Martin Lorenz has written an excellent book treating the theory of invariants of groups acting on lattices. ... The choice of topics and the order in which they are presented is very good. The proofs are easy to follow, the references are many and thorough. The author brings many diverse topics together in one place. ..."
Robert M. Fossum, SIAM Review, Vol. 48 (2), 2006
"The book under review is the first systematic treatment of multiplicative invariant theory in the form of a textbook written by an author who has contributed several research articles ... . The book is recommended for graduate and postgraduate students as well as researchers in representation theory, commutative algebra, and invariant theory. It opens a freshview to research problems on these fields related to multiplicative invariants."
Peter Schenzel, Zentralblatt MATH, Vol. 1078, 2006
"... So it is not surprising that the body of the book, which gives for the first time a full account of the algebraic side of the theory, is concerned with classical themes: class group, Picard group, regularity and Cohen-Macaulay property of multiplicative invariant algebras. As it turns out the results differ sometimes strongly from the linear case and the proofs are much more involved. A chapter on ordered and twisted invariant fields, which are connected intimately with Noether's rationality problem, and one on open problems complete the book."
G.Kowohl, Monatshefte für Mathematik 148:4, p. 352-353, 2006
Aus den Rezensionen: "... Das vorliegende Buch stellt den aktuellen Stand der multiplikativen Invariantentheorie dar. Nach einführenden Kapiteln folgen Kapitel über die Klassengruppe, die Picard-Gruppe, Invarianten von Spiegelungsgruppen, Regularität und die Cohen-Macaulay-Eigenschaft. ... Das Buch ist übersichtlich und gut lesbar geschrieben, ich kann es allen an der Invariantentheorie Interessierten empfehlen." (Franz Pauer, in: IMN Internationale Mathematische Nachrichten, August/2009, Issue 211, S. 76)