Standard monomial theory. Invariant theoretic approach.

*(English)*Zbl 1137.14036
Encyclopaedia of Mathematical Sciences 137. Invariant Theory and Algebraic Transformation Groups 8. Berlin: Springer (ISBN 978-3-540-76756-5/hbk). xiv, 265 p. (2008).

The book aims to describe the beautiful connection between Schubert varieties and their Standard Monomial Theory (SMT) on the one hand and Classical Invariant Theory (CIT) on the other. The roots of SMT are to be found in the work of Hodge, who described nice bases for the homogeneous coordinate ring of Schubert varieties of the Grassmannian in the Plücker embedding (over a field of characteristic zero). Grassmannians being precisely the homogeneous spaces that arise as quotients of special linear groups by maximal parabolic subgroups, it is natural to try to generalize Hodge’s approach to projective embeddings of other spaces \(G/Q\), where \(G\) is a semisimple algebraic group and \(Q\) a parabolic subgroup. In the early ’70s Seshadri initiated this generalization and called it SMT.

CIT concerns diagonal actions of classical groups on direct sums of the tautological representations and their duals. A description of the algebra of invariants for these actions comprises of two theorems. The First Fundamental Theorem specifies a finite set of generators for the algebra of invariants, and the Second Fundamental Theorem provides a finite set of generators of the ideal of relations among the algebra generators. A central role here plays the article [C. De Concini and C. Procesi, “A characteristic free approach to Invariant Theory”, Adv. Math. 21, 330–354 (1976; Zbl 0347.20025)], where the First Fundamental Theorem for all classical groups and the Second Fundamental Theorem for general, orthogonal and symplectic linear groups were obtained (the case when the characteristic of the ground field is zero goes back to H. Weyl). The idea to use SMT in the proof of the First and the Second Fundamental Theorems appeared in [V. Lakshmibai and C. S. Seshadri, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1–54 (1978; Zbl 0447.14011)] and turned out to be very fruitful. More precisely, one should realize the subalgebra of the algebra of invariants generated by “basic” invariants (which will in fact coincide with the algebra of invariants) as the algebra of regular functions on an affine variety related to a Schubert variety. Then there is a morphism from the spectrum of the algebra of invariants to this affine variety. Using Zariski’s Main Theorem, one shows that this is an isomorphism. A difficult part of this program is to prove that our affine variety is normal. Normality follows from normality of Schubert varieties, and that is a consequence of SMT. Nowadays this approach is realized in complete generality, and the book under review provides an excellent account of there results.

The authors tried to make the presentation self-contained keeping in mind the needs of prospective graduate students and young researchers. After a detailed introduction, generalities on algebraic varieties and algebraic groups are given. Next chapters are devoted to classical, symplectic and orthogonal Grassmannians, determinantal varieties, Geometric Invariant Theory (GIT), basic results of SMT and their interrelations with CIT. The proof of the main theorem of SMT is given in an appendix. The authors also included some important applications of SMT: to the determination of singular loci of Schubert varieties, to the study of some affine varieties related to Schubert varieties — ladder determinantal varieties, quiver varieties, varieties of complexes, etc. — and to toric degenerations of Schubert varieties.

The book may be recommended as a nice introduction to SMT and related active research areas. It may be used for a year long course on Invariant Theory and Schubert varieties.

CIT concerns diagonal actions of classical groups on direct sums of the tautological representations and their duals. A description of the algebra of invariants for these actions comprises of two theorems. The First Fundamental Theorem specifies a finite set of generators for the algebra of invariants, and the Second Fundamental Theorem provides a finite set of generators of the ideal of relations among the algebra generators. A central role here plays the article [C. De Concini and C. Procesi, “A characteristic free approach to Invariant Theory”, Adv. Math. 21, 330–354 (1976; Zbl 0347.20025)], where the First Fundamental Theorem for all classical groups and the Second Fundamental Theorem for general, orthogonal and symplectic linear groups were obtained (the case when the characteristic of the ground field is zero goes back to H. Weyl). The idea to use SMT in the proof of the First and the Second Fundamental Theorems appeared in [V. Lakshmibai and C. S. Seshadri, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1–54 (1978; Zbl 0447.14011)] and turned out to be very fruitful. More precisely, one should realize the subalgebra of the algebra of invariants generated by “basic” invariants (which will in fact coincide with the algebra of invariants) as the algebra of regular functions on an affine variety related to a Schubert variety. Then there is a morphism from the spectrum of the algebra of invariants to this affine variety. Using Zariski’s Main Theorem, one shows that this is an isomorphism. A difficult part of this program is to prove that our affine variety is normal. Normality follows from normality of Schubert varieties, and that is a consequence of SMT. Nowadays this approach is realized in complete generality, and the book under review provides an excellent account of there results.

The authors tried to make the presentation self-contained keeping in mind the needs of prospective graduate students and young researchers. After a detailed introduction, generalities on algebraic varieties and algebraic groups are given. Next chapters are devoted to classical, symplectic and orthogonal Grassmannians, determinantal varieties, Geometric Invariant Theory (GIT), basic results of SMT and their interrelations with CIT. The proof of the main theorem of SMT is given in an appendix. The authors also included some important applications of SMT: to the determination of singular loci of Schubert varieties, to the study of some affine varieties related to Schubert varieties — ladder determinantal varieties, quiver varieties, varieties of complexes, etc. — and to toric degenerations of Schubert varieties.

The book may be recommended as a nice introduction to SMT and related active research areas. It may be used for a year long course on Invariant Theory and Schubert varieties.

Reviewer: Ivan V. Arzhantsev (Moskva)