Multiplicative Invariant Theory [Paperback]

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.

Notations and Conventions.- Groups Acting on Lattices.- Permutation Lattices and Flasque Equivalence.- Multiplicative Actions.- Class Group.- Picard Group.- Multiplicative Invariants of Reflection Groups.- Regularity.- The Cohen-Macaulay Property.- Multiplicative Invariant Fields.- Problems.

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