The work of Gian-Carlo rota on invariant theory

The purpose of this paper is twofold: first, to explain Gian-Carlo Rotas workon invariant theory; second, to place this work in a broad historical and mathematicalcontext. Rotas work falls under three specific cases: vector invariants, the invariants ofbinary forms, and the invariants of skew-symmetric tensors. We discuss each of these casesand show how determinants and straightening play central roles. In fact, determinantsconstitute all invariants in the vector case; for binary forms and skew-symmetric tensors,they constitute all invariants when invariants are represented symbolically. Consequently,we explain the symbolic method both for binary forms and for skew-symmetric tensors,where Rota developed generalizations of the usual notion of a determinant. We also discussthe Grassmann algebra, with its two operations of meet and join, which was a theme whichran through Rotas work on invariant theory almost from the very beginning.To the memory of Gian-Carlo Rota