Noncommutative classical invariant theory

Torbjörn Tambour Department of Mathematics, Lunds Universitet

TBD mathscidoc:1701.332746

Arkiv for Matematik, 29, (1), 127-182, 1987.4
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.
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  title={Noncommutative classical invariant theory},
  author={Torbjörn Tambour},
  booktitle={Arkiv for Matematik},
Torbjörn Tambour. Noncommutative classical invariant theory. 1987. Vol. 29. In Arkiv for Matematik. pp.127-182.
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