A Glimpse of Vector Invariant Theory. The Points of View of Weyl, Rota, De Concini and Procesi, and Grosshans

We provide a description of some of the founding ideas of the characteristic-zero theory as well as of the characteristic-free theory of vector invariants for the classical groups GL(d), SL(d), Sp2m, O(d). The characteristic zero approach to vector invariants is traditionally based on the so-called Weyl's Theorem which essentially says that a system of generators of the algebra of (absolute) vector invariants for any subgroup G of GL(d) can be constructed by polarizing a system of generators of the invariants in the first d vector variables (or even, in a more refined form, a system of generators of the invariants in the first in d - 1 vector variables, in case by adding the determinant of the first d variables). We discuss the way to derive from Weyl's Theorem the basic invariant theory for the groups SL(d), Sp2m, SO(d), O(d). Weyl's Theorem is not valid in positive characteristic. In their fundamental 1976's paper, De Concini and Procesi wrote: "The classical proofs of invariant theoretic results follow essentially two equivalent paths. Polarization and the Gordan-Capelli expansion (via the theorem of E. Pascal) or double centralizer theorem, owing to the linear reductivity of the group in consideration. There is, on the other hand, another line of approach based on the standard Young tableaux which should be traced at least to Hodge (1943), to Igusa (1954) (who proves the first fundamental theorem of vector invariants in a characteristic free approach) and finally to Doubilet, Rota and Stein (1974), which gives the main technical tool: the straightening formula." The straightening formula for bideterminants of Doubilet, Rota and Stein is to be regarded as the characteristic-free version of the Gordan-Capelli-Deruyts polar expansion formula. This result allowed Rota and his collaborators to provide characteristic-free combinatorial proofs of the First and Second Fundamental Theorems for relative vector invariants of GL(d), or, equivalently, absolute invariants of SL(d). A quite relevant step forward was made by De Concini and Procesi in 1976.They proved two variants of Rota's Straightening Formula which hold for Pfaffians and Gramians. Furthermore, they realized (in the extremely general language of schemes and formalinvariants), that a common strategy in the proofs of the First Fundamental Theorems for vector invariants of classical groups could be that of proving versions of these results on the localized rings (with respect to suitable Zariski open sets) of naturally chosen a±ne varieties, and then to get the global result by using the cancellation laws, which follow, in turn, from the Straightening Formulae. In 2007, Grosshans proved a quite general result which provides a bridge between Weyl's approach and the approaches of Rota, De Concini and Procesi. Grosshans's main result is a far-reaching generalization of Weyl's Theorem. Informally speaking, Grosshans's main result says that, given an algebraically closed field K of arbitrary characteristic, for any subgroup H of GL(d;K), the H-vector invariants in n variables (n > d) can still be obtained from the H-vector invariants in d variables; more precisely, the algebra of H-vector invariants in n variables is the p-root closure of the algebra of the "polarized" H-invariants in d variables.