Noncommutative classical invariant theory
In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspaceĨdm of the noncommutative invariant algebraĨd consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group Sm. We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(Ĩd,t) whose coefficients are the number of irreducibleSm-modules in the decomposition ofĨdm, and show that it is rational. Finally, we develop some analogues of all this for covariants.