The work of Gian-Carlo rota on invariant theory

Abstract.

The purpose of this paper is twofold: first, to explain Gian-Carlo Rota’s work on invariant theory; second, to place this work in a broad historical and mathematical context. Rota’s work falls under three specific cases: vector invariants, the invariants of binary forms, and the invariants of skew-symmetric tensors. We discuss each of these cases and show how determinants and straightening play central roles. In fact, determinants constitute 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 discuss the Grassmann algebra, with its two operations of meet and join, which was a theme which ran through Rota’s work on invariant theory almost from the very beginning.