" Multiplicative Invariant Theory " by Martin Lorenz

Let K be a commutative integral domain and let S = K[x1, x2, . . . , xn] denote the polynomial ring over K in the n variables x1, x2, . . . , xn. If H is a group of automorphisms of the free K-module V = Kx1 +Kx2 + · · ·+Kxn, that is, if H is a subgroup of GL(V ) ∼= GLn(K), then the linear action of H on V extends uniquely to a K-algebra action on S. The relationship between S, H, the H-stable prime ideals of S, and the subring of H-invariants S = {s ∈ S | s = s for all h ∈ H} is the realm of (additive) invariant theory. The adjective “additive” is by no means a standard part of the name, but it is useful in the context of this review. Note that the full group of K-automorphisms of S is appreciably larger than GLn(K), since for example it contains maps of the form xi 7→ xi for all i 6= 1 and x1 7→ x1 + f(x2, . . . , xn), but, for the most part, additive invariant theory is concerned with GLn(K) and its subgroups. Now suppose we modify S slightly by adjoining the inverses x−1 1 , x −1 2 , . . . , x −1 n . Then the new ring R is usually written as R = K[x1, x−1 1 , x2, x −1 2 , . . . , xn, x −1 n ], and clearly few subgroups of GLn(K) will actually determine K-automorphism groups of R. Indeed, if X = 〈x1, x2, . . . , xn〉 ∼= 〈x1〉× 〈x2〉× · · ·× 〈xn〉 is the multiplicative subgroup of the unit group of R, then R is isomorphic to K[X], the group ring of X over K, and the unit group R• of R is easily seen to be given by R• = K• ×X. In particular, any K-automorphism of R must stabilize R• and act faithfully as automorphisms on this group. Note that X is isomorphic to the additive lattice Z, so Aut(X) ∼= GLn(Z), and if G is any subgroup of Aut(X), then the action of G on X extends uniquely to a K-algebra action on R. The relationship between R, G, the G-stable prime ideals of R, and the fixed ring R is the stuff of multiplicative invariant theory. The two invariant theories are obviously similar, and yet they are different in many ways. Additive invariant theory is classical, well over a hundred years old, while the multiplicative version has a much shorter pedigree. Admittedly there are earlier results to be found in the study of algebraic groups, field theory, Lie algebras and group algebras, as we will see below, but multiplicative invariant theory itself was literally born and named in the three fundamental papers [Fa1, Fa2, Fa3] by D. R. Farkas in the mid 1980’s. Thus, the book under review discusses the state of this subject at a mature, but still young, twenty years of age. An interesting example in classical invariant theory is as follows. Suppose we are given n = m indeterminates xij with 1 ≤ i, j ≤ m. We can think of these elements as the entries of anm×m generic matrix [xij ], and we can letH = SLm(K)