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Representation functors and flag-algebras for the classical groups. II. (English) Zbl 0616.14038
Previously [see J. Towber, J. Algebra 47, 80–104 (1977; Zbl 0358.15033) and part I of this paper, ibid. 59, 16–38 (1979; Zbl 0441.14013)] the authors have constructed functors $$\Lambda^+$$ from $$R$$-modules to $$R$$-algebras ($$R$$ any commutative ring) such that for $$V$$ a finite-dimensional vector space over an algebraically closed field $$R$$ of characteristic 0, $$\Lambda^+(V)$$ is isomorphic to the coordinate ring $$R[G/U]$$ of the base affine space $$G/U$$ ($$U$$ is a unipotent radical of the Borel subgroup of $$G$$) for classical groups $$G=\mathrm{GL}(V)$$, $$G=\mathrm{SO}(V)$$ (with odd $$\dim(V)$$) and $$\mathrm{SP}(V)$$. In the present paper they complete their program for all classical groups by treating similarly the groups $$\mathrm{SO}(V)$$ (with even $$\dim(V)$$) and $$\mathrm{O}(V)$$.
There still exist results in classical 19th century invariant theory which have never been translated into modern terms. The authors express the hope that one possible application of the functors in the paper under review is providing a form in which such a translation is possible.

##### MSC:
 14L35 Classical groups (algebro-geometric aspects) 14L17 Affine algebraic groups, hyperalgebra constructions 20G35 Linear algebraic groups over adèles and other rings and schemes 15A72 Vector and tensor algebra, theory of invariants 14M17 Homogeneous spaces and generalizations
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