Modular invariant theory concerns the representations of finite groups over vector spaces of positive characteristic. Thus it is ab initio quite distinct from, and ostensibly more expressly algebraic in flavor than, say, the unitary representation theory of Lie groups, to go the opposite end of the spectrum. Perhaps it is not wrong, accordingly, to characterize modular representation theory as substantially a bit narrower than its fellows: the tools that come into the game here are taken almost exclusively from commutative algebra and algebraic geometry.
The book under review is a contribution to the “Encyclopaedia of Mathematical Sciences” series published by Springer-Verlag: it’s number VIII in the subseries titled, “Invariant Theory and Algebraic Transformation Groups,” and this attests to the fact that it is not intended as a course text as such. Indeed, we are told on the back-cover that “[l]argely self-contained, the book develops the theory from its origins up to modern results … and is aimed at both graduate students and researchers — an introduction to many important topics in modern algebra … for the former, an exploration of a fascinating subfield of algebraic geometry for the latter.” To these ends, Modular Invariant Theory, is equipped with many examples, in order to establish something of a “concrete setting” for the indicated material, but there are no exercises to be had. To boot, the presentation is a bit on the terse side and is pitched at a level presupposing an experienced and motivated reader.
All this having been said, the authors take the reader from first principles or “First Steps” to much more serious material, at a pretty reasonable but non-trivial pace. The first two chapters set the stage, building up to commutative algebra at the level of Hilbert’s syzygy theorem; algebraic geometry is touched on a handful of pages earlier, very briefly (i.e., to the extent needed, which is modest at least for the moment). With the third chapter it’s off to the races: invariant theory properly so-called is now in full flower: Molien’s Theorem, Hilbert series, etc.
The rest of the book goes on to present ever more sophisticated material, with the focus being modular invariant theory of rather a special sort: the prevailing hypothesis entails that the characteristic of the base field of the representation space divides the order of the group being represented. The back-cover once again: “[Modular Invariant Theory] explains a theory that is more complicated than the study of the classical non-modular case, and it describes many open questions.”
Modular Invariant Theory is a fitting entry into the “Encyclopaedia of Mathematical Sciences” series: it deals with important living mathematics in a way suited to researchers both at the rookie and more advanced levels.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.