×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Boerner, H., Darstellungen von gruppen, (1963), Springer-Verlag Berlin · Zbl 0166.29301
[2] Carter, R.W.; Lusztig, G., On the modular representations of the general linear and symmetric groups, Math. Z., 136, 193-242, (1974) · Zbl 0298.20009
[3] DeConcini, C., Symplectic standard tableaux, Adv. in math., 34, 1-27, (1979)
[4] Flanders, H., On free exterior powers, Trans. amer. math. soc., 14, 357-367, (1969) · Zbl 0202.03701
[5] Lakshimbai, V.; Musili, C.; Seshadri, C.S., Geometry ofG/P, Bull. amer. math soc. (N.S.), 1, 432-435, (1979) · Zbl 0466.14020
[6] Lancaster, G.; Towber, J., Representation-functors and flag-algebras for the classical groups, I, J. algebra, 59, 16-37, (1979) · Zbl 0441.14013
[7] Samelson, H., Notes on Lie algebras, (1969), Von Nostrand New York · Zbl 0209.06601
[8] Towber, J., Two new functors from modules to algebras, J. algebra, 47, 80-104, (1977) · Zbl 0358.15033
[9] Towber, J., Young symmetry, the flag manifold, and representations ofG/L(n), J. algebra, 61, 414-462, (1979) · Zbl 0437.14030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.