Invariant theory, old and new.

*(English)*Zbl 0258.14011
New York-London: Academic Press. VIII, 85 p. $ 4.75 (1971).

This book is reprinted from an article of the authors [Adv. Math. 4, 1–80 (1970; Zbl 0196.05802)] and has been reviewed by the senior author. Because it has now appeared in bock form making it more readily available, because of the renewed interest in the subject and because it is a well written, exciting introduction to invariant theary, it is worth another review. The preface gives a brief history of invariant theory which serves also as an outline of the book itself.

Chapter 1 introduces the terminology and ends with the important restitution process and the reduction from the multilinear to the linear case. Stated briefly, we have a group \(G\) acting an a set \(E\) and we wish to describe the set of invariant elements \(E^G=\{e\in E: s\cdot e=e\) for all \(s\in G\}\). Often the space \(E\) is a vector space and the action of \(G\) is required to be linear. Real geometric problems arise when \(G\) is an algebraic group defined over a field \(k\), the set \(E\) is a \(k\) scheme and the action is a morphism of \(k\)-schemes. The question then is whether the space of orbits is again a \(k\)-scheme.

In Chapter 2, entitled: “Rational representations of the general linear group”, the synthetic method of the 19th century is developed and used firstly to show that the general linear group \(\text{GL}(n,k)\) defined over an algebraically closed field \(k\) of characteristic zero is completely reducible and then secondly to find all rational eharacters. Finally Aronhold’s rule is developed so that all irreducible rational representations ran be obtained. [Compare this method with the more modern method found, for example, in the article of R. Hartshorne, Publ. Math., Inst. Hautes Étud. Sci. 29, 63–94 (1966; Zbl 0173.49003) where Borel’s fixed point theorem is used in an essential manner. See also J. Dieudonné, Sem. Bourbaki 1970/71, Lect. Notes Math. 244, 257–274 (1971; Zbl 0231.01001) for a description of the f irst two chapters of this bock under review.]

Chapter 3, “Post Hilbert invariant theory”, the problem of the finite generation of the invariants of a finitely generated commutative \(k\)-algebra is discussed. A complete proof is given of the Hilbert-Nagata theorem: Suppose \(G\) is a group of automorphisms of the finitely generated \(k\)-algebra \(R\) subject to the two conditions; (i) the orbit of each element of \(R\) under \(G\) is contained in a finite dimensional vertor space over \(k\), and (ii) each linear representation of \(G\) an a finite dimensional vector space over \(k\) which leaves invariant a hyperplane is equivalent to a representation of the same dimension fixing pointwise a line complementary to an invariant hyperplane. Then the ring of invariants \(R^G\) is finitely generated over \(k\). This is related to Hilbert’s 14th problem: Suppose \(\text{GL}(n,k)\) actc in the usual manner on the ring of polynomials \(k[X_1,\dots,X_n]\). Let \(G\) be a subgroup of \(\text{GL}(n,k)\). Is the ring \(k[X_1,\dots,X_n]^G\) a finitely generated \(k\)-algebra? It is easily seen that the action of \(G\) satisfies condition (i) above. If \(G\) is reductive, then condition (ii) is satisfied. However M. Nagata [Notes by M. Pavaman Murthy. (Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Mathematics. Vol. 31) Bombay: Tata Institute of Fundamental Research. 78 + III p. (1965; Zbl 0182.54101)] has found a group \(G\) for which the fixed subring is not finitely generated. This counterexample to Hilbert’s 14th problem is reported in the last section of this chapter with most of the details verifying its existence given.

The final Chapter 4, “Introduction to Hilbert-Mumford theory”, is exactly as the title states. Whereas Chapter 3 is concerned with construction of quotients of affine schemes, the main concern of Chapter 4 is the construction of orbit spaces (or quotients) of projective varieties. This is a more difficult, not to say insoluble, problem since it must be solved first locally and then the local orbit spaces must be glued together coherently. This is perhaps the most interesting chapter because it leads the reader naturally to Mumford’s recent revival of geometric invariant theory [D. Mumford, Geometric invariant theory, Berlin-Heidelberg-New York (1965; Zbl 0147.39304)]. The chapter ends with several illustrative examples.

Finally there is an appendix, “A short digest of non-commutative algebra”, which contains the theory of semi-simple algebras, including Maschke’s theorem and Schur’s commutation theorem, necessary for showing the complete reducibility of \(\text{GL}(n,k)\). This material is used only in Chapter 2.

This book is an excellent introduction to the subject giving not only the important contributions of the last century, but also the most important contributions of the last decades, as well as an introduction to the recent developments due to Mumford. Recent successes of invariant theory have been made by Hochster, Eagon and Roberts. M. Hochster and J. A. Eagon have shown, using invariant theory, that the homogeneous coordinate ring for the Plücker embedding of the Grassmannian is Gorenstein [Am. J. Math. 93, 1020–1058 (1971; Zbl 0244.13012)], while M. Hochster has shown that the ring of invariants of a torus is Cohen-Macaulay [Ann. Math. (2) 96, 318–337 (1972; Zbl 0233.14010)]. Most recently M. Hochster and Roberts have shown that a linearly reductive affine linear group defined over a field \(k\) (of arbitrary characteristic) acting on a regular \(k\) algebra has a Cohen-Macaulay ring of invariants [Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay. Mimeographed. Univ. of Minnesota, Minneapolis, Minnesota (1973)].

Chapter 1 introduces the terminology and ends with the important restitution process and the reduction from the multilinear to the linear case. Stated briefly, we have a group \(G\) acting an a set \(E\) and we wish to describe the set of invariant elements \(E^G=\{e\in E: s\cdot e=e\) for all \(s\in G\}\). Often the space \(E\) is a vector space and the action of \(G\) is required to be linear. Real geometric problems arise when \(G\) is an algebraic group defined over a field \(k\), the set \(E\) is a \(k\) scheme and the action is a morphism of \(k\)-schemes. The question then is whether the space of orbits is again a \(k\)-scheme.

In Chapter 2, entitled: “Rational representations of the general linear group”, the synthetic method of the 19th century is developed and used firstly to show that the general linear group \(\text{GL}(n,k)\) defined over an algebraically closed field \(k\) of characteristic zero is completely reducible and then secondly to find all rational eharacters. Finally Aronhold’s rule is developed so that all irreducible rational representations ran be obtained. [Compare this method with the more modern method found, for example, in the article of R. Hartshorne, Publ. Math., Inst. Hautes Étud. Sci. 29, 63–94 (1966; Zbl 0173.49003) where Borel’s fixed point theorem is used in an essential manner. See also J. Dieudonné, Sem. Bourbaki 1970/71, Lect. Notes Math. 244, 257–274 (1971; Zbl 0231.01001) for a description of the f irst two chapters of this bock under review.]

Chapter 3, “Post Hilbert invariant theory”, the problem of the finite generation of the invariants of a finitely generated commutative \(k\)-algebra is discussed. A complete proof is given of the Hilbert-Nagata theorem: Suppose \(G\) is a group of automorphisms of the finitely generated \(k\)-algebra \(R\) subject to the two conditions; (i) the orbit of each element of \(R\) under \(G\) is contained in a finite dimensional vertor space over \(k\), and (ii) each linear representation of \(G\) an a finite dimensional vector space over \(k\) which leaves invariant a hyperplane is equivalent to a representation of the same dimension fixing pointwise a line complementary to an invariant hyperplane. Then the ring of invariants \(R^G\) is finitely generated over \(k\). This is related to Hilbert’s 14th problem: Suppose \(\text{GL}(n,k)\) actc in the usual manner on the ring of polynomials \(k[X_1,\dots,X_n]\). Let \(G\) be a subgroup of \(\text{GL}(n,k)\). Is the ring \(k[X_1,\dots,X_n]^G\) a finitely generated \(k\)-algebra? It is easily seen that the action of \(G\) satisfies condition (i) above. If \(G\) is reductive, then condition (ii) is satisfied. However M. Nagata [Notes by M. Pavaman Murthy. (Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Mathematics. Vol. 31) Bombay: Tata Institute of Fundamental Research. 78 + III p. (1965; Zbl 0182.54101)] has found a group \(G\) for which the fixed subring is not finitely generated. This counterexample to Hilbert’s 14th problem is reported in the last section of this chapter with most of the details verifying its existence given.

The final Chapter 4, “Introduction to Hilbert-Mumford theory”, is exactly as the title states. Whereas Chapter 3 is concerned with construction of quotients of affine schemes, the main concern of Chapter 4 is the construction of orbit spaces (or quotients) of projective varieties. This is a more difficult, not to say insoluble, problem since it must be solved first locally and then the local orbit spaces must be glued together coherently. This is perhaps the most interesting chapter because it leads the reader naturally to Mumford’s recent revival of geometric invariant theory [D. Mumford, Geometric invariant theory, Berlin-Heidelberg-New York (1965; Zbl 0147.39304)]. The chapter ends with several illustrative examples.

Finally there is an appendix, “A short digest of non-commutative algebra”, which contains the theory of semi-simple algebras, including Maschke’s theorem and Schur’s commutation theorem, necessary for showing the complete reducibility of \(\text{GL}(n,k)\). This material is used only in Chapter 2.

This book is an excellent introduction to the subject giving not only the important contributions of the last century, but also the most important contributions of the last decades, as well as an introduction to the recent developments due to Mumford. Recent successes of invariant theory have been made by Hochster, Eagon and Roberts. M. Hochster and J. A. Eagon have shown, using invariant theory, that the homogeneous coordinate ring for the Plücker embedding of the Grassmannian is Gorenstein [Am. J. Math. 93, 1020–1058 (1971; Zbl 0244.13012)], while M. Hochster has shown that the ring of invariants of a torus is Cohen-Macaulay [Ann. Math. (2) 96, 318–337 (1972; Zbl 0233.14010)]. Most recently M. Hochster and Roberts have shown that a linearly reductive affine linear group defined over a field \(k\) (of arbitrary characteristic) acting on a regular \(k\) algebra has a Cohen-Macaulay ring of invariants [Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay. Mimeographed. Univ. of Minnesota, Minneapolis, Minnesota (1973)].

Reviewer: Robert M. Fossum (Champaign)

##### MSC:

14L24 | Geometric invariant theory |

13A50 | Actions of groups on commutative rings; invariant theory |

20G05 | Representation theory for linear algebraic groups |

57S25 | Groups acting on specific manifolds |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |