The year 1979 saw the publication of Richard Stanley’s celebrated paper on “Invariants of Finite Groups and their Applications to Combinatorics” (Bull. AMS 1 (1979), 475-511) and I. G. Macdonald’s highly influential book Symmetric Functions and Hall Polynomials (Oxford, 1979). Both publications helped to call attention to an important effort to systematize certain aspects of combinatorics by using the tools of commutative algebra, and in the following years there has been a lot of activity at the intersection of these two fields, with many monographs and textbooks trying to keep up with these developments. (See, for example, Combinatorial Commutative Algebra (Springer, 2005) by E. Miller and B. Sturmfels, or Computational Invariant Theory (Springer, 2002) by H. Derksen and G. Kemper.)
Of course, Invariant Theory has a history with roots in both combinatorics and commutative algebra. Almost from its very beginnings it had a computational and combinatorial component that was put to sleep by Hilbert’s landmark paper. He proved that the invariant rings of the classical groups are finitely generated, giving a non-constructive proof that prompted Gordan’s famous claim that that was theology, not mathematics. In addition to the basis theorem, which is behind the mentioned finiteness result, Hilbert’s paper contained many fundamental theorems of commutative algebra, such as the Nullstellensatz, the syzygy theorem and Noether’s normalization theorem. These results are at the foundations of commutative algebra and algebraic geometry, and these subjects gave rise to two rebirths of Invariant Theory, one in the 1960s with Mumford’s geometric approach (for a recent reference, see for example Actions and Invariants of Algebraic Groups by W. Ferrer and A. Rittatore) and a second one in the last three decades with an emphasis on the computational and constructive aspects. With this renewed interest on invariant theory, new connections between classical fields of mathematics such as representation theory, commutative algebra, algebraic geometry and combinatorics have enriched these fields and uncovered deep relations amongst them.
The book under review takes as motivation the study of a two-parameter family of symmetric functions discovered by I. G. Macdonald in 1988. They were obtained by an orthogonalization process for a certain scalar product; when suitably specialized this recovers some well-known symmetric functions, e.g., Schur or Hall-Littlewood functions. By a change of variables and expanding these functions in terms of Schur functions, Macdonald obtained as coefficients some rational functions in two variables indexed by pairs of partitions of a given integer n, and Macdonald conjectured that these rational functions are indeed polynomials with integer positive coefficients.
To explain these constructions and to put them into context, the book starts by recalling some basic combinatorial objects: permutations, partitions, Young diagrams, generalized binomial coefficients and tableaux are discussed in the first two chapters. In the third, fourth and fifth chapters the basics of the algebraic approach to combinatorics is laid out: From the elements of invariant theory in chapter three and an introduction to representation and character theory in chapter five, having as main example the symmetric group to show the relevance of invariant theory for combinatorics, and proving, for example, the hook-length formula that relates the dimension of the irreducible representations of the symmetric group Sn and the number of standard tableaux for a given partition of n. In this chapter it is also shown that there is a linear isomorphism between the space of characters of the symmetric group Sn and the space of symmetric functions of degree n. Moreover, under this correspondence, irreducible representations are associated to symmetric Schur functions, previously studied on chapter four.
Thus, by this point the author has introduced the main theme that will allow him to sketch the approach taken by M. Haiman to solve Macdonald’s conjecture. But in order to explain this approach more algebraic tools are needed, and chapter seven starts by recalling the basic concepts of commutative algebra, from Gröbner bases to the Cohen-Macaulay property.
The fundamental problem of invariant theory is to characterize the ring of invariants RG of a (finite) group G acting on a ring or vector space R. In chapters eight and ten the author introduces the main tools, namely associating to a (finite) group of orthogonal matrices of size n acting on a polynomial ring R on n variables a quotient space RG given by moding out an ideal IG of elements of RG with constant term zero, observing that G acts naturally on RG. The main properties of these coinvariant spaces RG are studied in chapter eight. In chapter ten the author considers a further generalization that can be illustrated by the important case when R=C[x1, …, xn; y1, … , yn] is the polynomial ring in 2n variables with coefficients in the field of complex numbers, and letting the symmetric group Sn act on R by the diagonal action (acting simultaneously on the xi and the yi) and modding out by certain ideals of R indexed by partitions of the integer n. This is a central chapter since it develops the language, concepts and results that are behind the formulation, interpretation in several contexts, and eventually the proof of Macdonald’s conjectures.
As the author points out, in a landmark paper (J. Amer. Math. Soc. 14 (2001), 941-1006) M. Haiman shows that the corresponding quotients (coinvariant spaces) are representations of degree n! of the symmetric group Sn and that the corresponding Frobenius character is the Macdonald symmetric function associated to the given partition of n. The proof by Haiman is beyond the scope of the book, requiring deep properties of the Hilbert scheme of n-tuples of points in the plane, but after having read this book an interested reader would have understood the formulation of the conjecture, its relevance and relation to such diverse fields as algebraic geometry as formulated by Grothendieck, e.g., “Techniques de construction et théorèmes d’existence en géométrie algébrique. IV: Les schémas de Hilbert”. Expose 221, Séminaire Bourbaki Vol. 6, 1960/61, SMF, Paris, 249-276).
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is email@example.com