In classification problems, invariants play the most important role: invariants detect non-equivalent objects, characterize specific elements, or detect certain properties of the ambient space. An important example, in a classical setting, and over an algebraically closed field \(k\), considers an algebraic group \(G\) acting on an algebraic variety \(X\). In this setting, the important equivalence is *being in the same orbit* of the action. The classification problem, in this context, consists in the description of the orbit space \(X/G\). To begin with, the natural map \(X\rightarrow X/G\) makes the set of orbits a topological space and continuous functions \(X/G\rightarrow k\) correspond to \(G\)-invariant continuous functions \(X\rightarrow k\). The problem is that \(X/G\) may contain non-closed orbits and thus \(X/G\) may not be an algebraic variety. One important case when the space of orbits \(X/G\) is an algebraic variety is when \(G\) is a reductive affine group and \(X\) is an affine variety (both over the same algebraically closed field \(k\)). Geometric Invariant Theory considers generalizations of this situation, that is, actions of a reductive group on a scheme together with various definitions of what the quotient of this action may be.

In the above-mentioned case when \(G\) is a reductive affine group and \(X\) is an affine variety over \(k\), \(X/G\) is an affine variety and its ring of coordinates \(k[X/G]\) is the subring \(k[X]^G\) of regular functions on \(X\) invariant under the natural action of the group \(G\). By Hilbert’s *Nullstellensatz*, to describe the affine variety \(X/G\) it is enough to describe the ring of invariants \(k[X]^G\). This is, of course, the classical problem that kept Cayley, Sylvester, Gordan, Clebsch, Aronhold, and Cremona quite busy in the second half of the 19th century. They tried, for very specific low dimensional linear groups acting on affine space, to find not only specific generators for the ring of invariants but also generators for the basic relations between these generators, in which case they intuited that the ring of invariants is finitely generated. That this is actually the case is one of the main results of Hilbert’s landmark paper of 1890, a paper which would be the cause of the “first demise” of invariant theory. Besides the plethora of invariants and phenomena that these earlier invariant theorists found and named, one may single out the lovely word *syzygy* that describes the relations between the generators in the ideal(s) that characterize the ring of invariants.

For an algebraically closed field of characteristic zero, Hermann Weyl, in his book *The Classical Groups: Their invariants and representations* (Princeton, 1939, reprinted several times, last one in 1997), describes the ring of invariants in two classical theorems:

- The first fundamental theorem characterizes a finite set of generators for this ring of invariants, and
- The second fundamental theorem gives the relations between the generators provided by the first fundamental theorem.

The generalization of these theorems to fields of arbitrary characteristic is more recent and follows several intertwined paths, from algebraic geometry and representation theory to algebraic combinatorics. One approach (see, for example, Arbarello et al, *Geometry of Algebraic Curves, Vol. 1*, Springer 1985) considers the ring of \(m\times n\) matrices \(\text{Mat}_{m\times n}\) over an algebraically closed field \(k\) and for an integer \(r\leq\min\{m,n\}\), the determinantal variety \(M_r\) given by the \(m\times n\) matrices of rank at most \(r\). There is natural action of the general linear group \(\text{GL}_r\) on the product \(\text{Mat}_{m\times r}\times \text{Mat}_{r\times n}\), and the first fundamental theorem in this case is that the ring of invariants \(k[\text{Mat}_{m\times r}\times \text{Mat}_{r\times n}]^{\text{GL}_r}\) is the ring of coordinates \(k[M_r]\). For the same case, the second fundamental theorem says that the vanishing ideal of \(M_r\) in \(k[\text{Mat}_{m\times n}]\) is generated by the minors of size \(r+1\). Several deep properties of this ring of invariants are codified in terms of the algebraic variety \(M_r\), and their proofs are part of the commutative algebra developed in the last decades of the 20th century.

The authors of the book under review eschew the several algebraic geometry approaches, one of which was sketched in a particular case in the previous paragraph, using instead purely methods of algebraic combinatorics, relaxing the condition on the ground field to being an infinite field or the ring of integers \({\mathbb Z}\). To be able to do so the authors consider just the case of the ring \(M_m\) of square \(m\times m\) matrices and a conjugation action of the general linear group \(\text{GL}_m\) on the product \(S_n\) of \(n\) copies of \(M_n\). The fundamental theorems in this case describe the ring of invariants \(S_n^{\text{GL}_m}\).

As the authors recall, there are corresponding results for other linear groups, but the case that they consider allows them to use only relatively elementary methods and it serves its purpose to give an introduction to classical invariant theory in an almost self-contained manner.

Within this framework the book proceeds systematically. In the first part, chapter one, the authors prove the two fundamental theorems for fields of characteristic \(0\), using some rather elementary results on rational representations of the general linear group due to Frobenius, Schur, Maschke and Young. In the second part, chapters two to six, and to treat the case of finite characteristic, the authors develop the theory of quasi-hereditary algebras, using combinatorial methods when needed, to avoid the original deep methods of the algebraic geometry of line bundles on flag varieties. The first fundamental theorem is proved in chapter four and the second fundamental theorem in chapter five.

The choices made by the authors permit them to highlight the main results and also to keep the material within the reach of an interested reader. At the same time the book remains open-ended, with precise pointers to the literature on other approaches and the cases not treated here.

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Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.