The Grassmannian Variety

Grassmannian varieties are a class of well-understood examples of algebraic projective varieties that play an essential role in the classical approach to the representation theory of algebraic groups. As is usually the case with fundamental examples, the starting point is just plain linear algebra: the Grassmanian \(G(m,n)\) is defined by fixing a vector space of finite dimension \(n\) over an arbitrary (algebraically closed) field and considering the family of vector subspaces of the given vector space of a fixed dimension \(m\). Using a bit more of linear algebra one embeds the set \(G(m,n)\) in an appropriate projective space and shows that the Grassmannian is cut-out by a set of quadratic polynomials.

What is not obvious at all is that a Grassmannian is a homogeneous space of the general linear group \(GL(n)\) and consequently the homogeneous coordinate ring of the Grassmannian comes equipped with a \(GL(n)\)-action; this is one point of entrance for the representation theory of algebraic groups, in particular for the general linear group.

Classical invariant theory deals with the finite generation of the ring of polynomial invariants of the general linear group, or more generally of a semisimple linear group, and the relations or syzygies, between these generators. Actually, classical invariant theory is in characteristic zero, where the fundamental theorems were obtained at the beginning of the 20^th century, and can be found in Weyl’s The Classical Groups: Their Invariants and Representations (Princeton, 1987).

These fundamental theorems where generalized to all characteristics by de Concini and Procesi in the mid-1970s, and standard monomial theory was developed to obtain explicit standard bases for the rings of invariants of actions of classical algebraic groups in terms of the geometry of Schubert subvarieties of the Grassmannian. The first instance where these standard bases appeared is a classical paper by Hodge, where he obtains a set of generators for its homogeneous coordinate ring of the Grassmannian variety in terms of monomials indexed by certain standard Young tableaux. These results were vastly extended to all characteristics and all homogeneous spaces of classical linear groups modulo maximal parabolic subgroups by Musili, Seshadri and the senior author of the book under review. An exposition of most of these results can be found in the monograph Standard Monomial Theory: Invariant-Theoretic Approach, Lakshmibai and Raghavan (Springer, 2008) which is devoted to the standard monomial approach to classical invariant theory in a characteristic free environment.

The book under review is more elementary; it is exclusively devoted to Grassmannians and their Schubert subvarieties. The book is divided into three parts. Part I is a short summary of some facts and terminology from commutative algebra and algebraic geometry that are used throughout the text. This is mostly standard, except perhaps that the authors still call a prescheme what is now called a scheme, so that a scheme for the authors is a separated scheme. This summary runs from the very elementary, e.g. the Hilbert basis theorem and the Zariski topology in affine or projective spaces, to sheaf and local cohomology. The reader is assumed to be acquainted with Hartshorne’s Algebraic Geometry (Springer, 1977) and Matsumura’s Commutative Ring Theory (Cambridge, 1989).

Part II introduces Grassmannian varieties and their Schubert subvarieties and proves their main geometric properties, from the calculation of their dimension to normality and the Cohen-Macaulay property. Something that is new here is the study of flat degenerations of the cone on a Schubert variety with applications to obtaining the degree of a Schubert variety and the characterization of which Schubert varieties are Gorenstein. Save for some calls to the literature in the case of toric degenerations, all proofs are given in detail.

Part III deals with the representation-theory aspects. It includes a detailed treatment of determinantal varieties and a sketchy overview of quotients of algebraic group actions. The main objective is to formulate the two fundamental theorems of classical invariant theory and discuss the relevance of Grassmannians, or more generally flag varieties, in this context. Here some proofs are given and some are omitted, due to limitations in the size and audience for the book.

This is a nicely written book, one that may appeal to students and researchers in related areas. It complements, and overlaps, some previous books by the senior author, such as the one mentioned before or the more comprehensive treatment of flag varieties in the monograph Flag Varieties of the same authors, published by the Hindustan Book Agency in 2009.

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