Noncommutative classical invariant theory

Abstract

In this thesis, we consider some aspects of noncommutative classical invariant theory, i.e., noncommutative invariants of the classical group SL(2, k). We develop a symbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspace Ĩ ${}_{d}^{m}$ of the noncommutative invariant algebra Ĩ_d consisting of homogeneous elements of degree m has the structure of a module over the symmetric group S_m. We find the explicit decomposition into irreducible modules. As a consequence, we obtain the Hilbert series of the commutative classical invariant algebras. The Cayley—Sylvester theorem and the Hermite reciprocity law are studied in some detail. We consider a new power series H(Ĩ_d, t) whose coefficients are the number of irreducible S_m-modules in the decomposition of Ĩ ${}_{d}^{m}$ , and show that it is rational. Finally, we develop some analogues of all this for covariants.