The aim of this book is to develop the combinatorics of Young tableaux, and to see them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties. There are three parts: Part I develops the basic combinatorics of Young tableaux; Part II applies this to representation theory; and Part III applies the first two parts to geometry.

Part I is a combinatorial study of some remarkable constructions one can make on Young tableaux, each of which can be used to make the set of tableaux into a monoid: the Schensted “bumping” algorithm and the Schützenberger “sliding” algorithm; the relations with words developed by Knuth and Lascoux and Schützenberger, and the Robinson–Schensted–Knuth correspondence between matrices with nonnegative integer entries and pairs of tableaux on the same shape. These constructions are used for the combinatorial version of the Littlewood–Richardson rule, and for counting the numbers of tableaux of various types.

One theme of Parts II and III is the ubiquity of certain basic quadratic equations that appear in constructions of representations of Sn and GLmℂ, as well as defining equations for Grassmannians and flag varieties. The basic linear algebra behind this, which is valid over any commutative ring, is explained in Chapter 8. Part III contains, in addition to the basic Schubert calculus on a Grassmannian, a last chapter working out the Schubert calculus on a flag manifold; here the geometry of flag varieties is used to construct the Schubert polynomials of Lascoux and Schützenberger.