Degree bounds — An invitation to postmodern invariant theory

Denote by V = Fn the n-dimensional vector space over F and by F[V ] the graded algebra of homogeneous polynomial functions on V . If :G ! GL(n,F) is a representation of a finite group G over the field F, the fixed points of the action of G on F[V ], induced by the action of GL(n,F), form a ring F[V ]G, the ring of invariants of G. Invariant theory is concerned with the study of group actions and their fixed points. Its connection with other areas of research has produced a fruitful exchange of information. In this paper the author gives a survey of known results concerning one of the problems in the field of postmodern invariant theory: find an upper bound for the maximal degree of the algebra generators in a minimal generating set of F[V ]G. The author presents the results differentiating between modular representations and nonmodular representations. Permutation representations and particular families of rings of invariants that occur in the modular case are treated separately.