Polynomial Invariant Theory of the Classical Groups

The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of basic invariants for the classical groups GL$(V)$, O$(n)$, and Sp$(n)$ for $n$ even. In the first half of the paper we set up relevant definitions and theorems for our search for the set of basic invariants, starting with linear algebraic groups and then discussing associative algebras. We then state and prove a monumental theorem that will allow us to proceed with hope: it says that the set of basic invariants is finite if $G$ is reductive. Finally we state without proof the First Fundamental Theorems, which aim to list explicitly the relevant sets of basic invariants, for the classical groups above. We end by commenting on some applications of invariant theory, on the history of its development, and stating a useful theorem in the appendix whose proof lies beyond the scope of this work.

💡 Research Summary

The paper “Polynomial Invariant Theory of the Classical Groups” provides a concise yet thorough overview of invariant theory as it applies to the three classical groups: the general linear group GL(V), the orthogonal group O(n), and the symplectic group Sp(n) (with n even). It begins by establishing the necessary algebraic background, defining linear algebraic groups as subgroups of GL(n, ℂ) cut out by polynomial equations, and introducing regular functions on such groups as elements of the coordinate ring Aff(G) = ℂ