The ubiquity of Lie algebras in various seemingly distant fields of mathematics, from real and complex analysis, differential geometry and Lie theory to algebraic geometry and representation theory, makes the publication of a book on some aspects of it a welcome opportunity to look at the subject from a fresh perspective.
The author has chosen as his main focus the so-called minuscule representations of Lie algebras. Minuscule representations are usually more manageable than general representations. One example of this is in the interpretation and generalization of Hodge’s description of the ideal of the Grassmannian in terms monomials indexed by certain Young diagrams. Musili and Seshradi first accomplished this generalization for quotients by minuscule parabolic subgroups, where the geometry of the homogeneous space G/P is similar to that of a Grassmannian. As is well known, this was the starting point of standard monomial theory and its relation to classical invariant theory, encompassing now quotients G/P that are not necessarily minuscule. There are other contexts where minuscule representations are also important, mainly because they are more manageable and thus provide a starting point to understand more general representations.
The approach chosen by the author emphasizes the rich combinatorics of Weyl and Coxeter groups, Bruhat decompositions and Dynkin diagrams underlying the study of the representations of Lie algebras. Moreover, by focusing on the combinatorial aspects of minuscule representations, their weights and the action of the Weyl group, the author presents a panorama of this area of representation theory using certain partially ordered sets which allows him to construct most simple Lie algebras and their associated Weyl groups.
The book is not self-contained and assumes from the reader a working knowledge of Lie algebras, as in Humphrey’s Introduction to Lie Algebras and Representation Theory, or any standard book on the subject. However, in sections scattered around the chapters, the author summarizes the basic properties of Lie algebras (chapter one), Weyl groups (chapter one and three), Lie theory (chapter four) or Chevalley bases (chapter seven). The remaining chapters develop, from scratch, the combinatorics of heaps over graphs, with special emphasis on heaps over Dynkin diagrams (chapter two), and how they are used to construct minuscule representations of simple Lie algebras (chapters five and six) and to understand the combinatorics of Weyl groups (chapter eight). There are two additional chapters, nine and ten, that give some applications of the combinatorics associated to certain minuscule representations, to some configurations of lines in cubics, del Pezzo surfaces and some other classical algebraic varieties. The last chapter surveys some recent work on the combinatorics of minuscule representations.
Although not written as a textbook, this monograph could be read with profit not only by the specialist, but also by an interested graduate student with some background on Lie algebras.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.