The Rationality Problem in Invariant Theory

ii Contents Introduction v 1 The rationality problem in invariant theory 1 1. iii iv CONTENTS Introduction Invariant theory as a mathematical discipline on its own originated in Eng-land around the middle of the nineteenth century with Cayley's papers on hyperdeterminants and his famous Memoirs on Quantics, followed by Salmon, Sylvester and Boole, and Aronhold, Clebsch and Gordan in Germany. There was also a third school in Italy associated with the names of Brioschi, Cre-mona, Beltrami and Capelli. The techniques employed in this early phase, long before Hilbert transformed the subject with his conceptual ideas, were often computational and symbolic in nature. One of the main questions was, given a linear algebraic group G and finite-dimensional G-representation V over C, to describe the algebra of invariant polynomial functions C[V ] G explicitly; in fact, most attention was given classically to the case where G = SL 2 (C) or G = SL 3 (C) and V is a space of binary or ternary forms of some fixed degree. Suppose now G to be connected and semisimple. Today we know by work of Popov that the algebra of invariants C[V ] G can be arbitrarily complicated: a natural measure for its complexity is the length of its syzygy chain or in other words its homological dimension hd(C[V ] G). Then (see e.g. [Po92], Chapter 3) it is known that if G is nontrivial, then for any n ∈ N, there exists a G-module V with hd(C[V ] G) > n and there exist, up to isomorphism and addition of trivial direct summands, only finitely many G-modules with hd(C[V ] G) ≤ n. Moreover, the complexity of invariant rings increases quite rapidly: classically, a finite generating set and finite set of defining relations for C[Sym d (C 2) ∨ ] SL 2 (C) was only obtained for d ≤ 6 to which the 20th century (Dixmier & Lazard, Shioda) contributed just d = 7, 8. For d > 8 the homological dimension of the algebra of invariants is known to be greater than 10 (cf. [Po-Vi], §8). Thus, algebraically, one is lead to ask: when is the structure of invariants of a G-module V as simple as possible? If we interpret this as asking when v vi INTRODUCTION C[V ] G is free, i.e. has algebraically independent homogeneous generators, then, by Popov's theorem, the classification of such V is a finite problem …