Weyl's Polarization in Classical Invariant Theory: A Primer, with Worked Examples

Abstract

Hermann Weyl's The Classical Groups is a landmark work connecting classical invariant theory with modern representation theory. It shows how polynomial invariants of the general linear, orthogonal, and symplectic groups can be systematically understood through linear representations and tensor methods. The current note is primarily based on a personal reading of the book of Weyl and of the more accessible books Classical Invariant Theory by Kraft and Procesi and by Olver. It is neither exhaustive, nor original, nor state of the art. We focus on a few selected aspects, aiming for an elementary and concrete approach. We work over the field of reals R with the classical groups GL(n), SL(n), O(n), and SO(n). Most of our efforts have been devoted to carefully worked examples, introducing just enough of the general theory to handle them effectively.