×

zbMATH — the first resource for mathematics

On Schur algebras and related algebras. I. (English) Zbl 0606.20038
The author defines an \(R\)-algebra \(S_ R(\pi)\) for each finite saturated set \(\pi\) of dominant weights of a semisimple, complex, finite dimensional Lie algebra \(\mathfrak g\). When \(\mathfrak g=\mathfrak{sl}_ n(C)\), then if \(\pi\) is chosen carefully, the construction generalizes that of the Schur algebras. The author obtains results on the representation theory of the symmetric groups by applying the Schur functor.
Reviewer: J.H.Lindsey, II

MSC:
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20G05 Representation theory for linear algebraic groups
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20C30 Representations of finite symmetric groups
20F40 Associated Lie structures for groups
20G10 Cohomology theory for linear algebraic groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akin, K; Buchsbaum, D.A; Weyman, J, Schur functors and Schur complexes, Advan. in math., 44, 207-278, (1982) · Zbl 0497.15020
[2] Akin, K; Buchsbaum, D.A, Characteristic-free representation theory of the general linear group, Advan. in math., 39, 149-200, (1985) · Zbl 0607.20021
[3] {\scK. Akin and D. A. Buchsbaum}, Characteristic-free representation theory of the general linear group. II. Homological considerations, preprint. · Zbl 0681.20028
[4] Borel, A, Properties and linear representations of Chevalley groups, () · Zbl 0197.30501
[5] 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
[6] Chevalley, C, Sur certains schémas en groupes simples, () · Zbl 0066.01503
[7] Cline, E; Parshall, B; Scott, L.L, Cohomology, hyperalgebras and representations, J. algebra, 63, 98-123, (1980) · Zbl 0434.20024
[8] Cline, E; Parshall, B; Scott, L.L; van der Kallen, W, Rational and generic cohomology, Invent. math., 39, 143-163, (1977) · Zbl 0336.20036
[9] Donkin, S, Hopf complements and injective comodules for algebraic groups, (), 298-319 · Zbl 0442.20033
[10] Donkin, S, A filtration for rational modules, Math. Z., 177, 1-8, (1981) · Zbl 0455.20029
[11] Donkin, S, Rational representations of algebraic groups: tensor products and filtrations, () · Zbl 0586.20017
[12] {\scS. Donkin}, On Schur algebras and related algebras, II, J. Algebra, in press. · Zbl 0634.20019
[13] {\scS. Donkin}, Skew modules for reductive groups, preprint. · Zbl 0662.20032
[14] {\scE. Friedlander and B. Parshall}, Cohomology of Lie algebras and algebraic groups, preprint. · Zbl 0601.20042
[15] Grabmeier, J, Unzerlegbare moduln mit trivialer youngquelle und darstellungenstheorie der schuralgebra, Doctoral thesis, (1985), Bayreuth · Zbl 0683.20015
[16] Green, J.A, Locally finite representations, J. algebra, 41, 137-171, (1976) · Zbl 0369.16008
[17] Green, J.A, Polynomial representations of GLn, ()
[18] {\scJ. A. Green}, Functor categories and group representations, preprint. · Zbl 0596.20008
[19] Grothendieck, A, Sur quelques points d’algèbre homologique, Tohoku math. J., 9, 119-221, (1957) · Zbl 0118.26104
[20] Hartshorne, R, Algebraic geometry, () · Zbl 0532.14001
[21] Hochschild, G, Cohomology of algebraic linear groups, Illinois J. math., 5, 492-519, (1961) · Zbl 0103.26502
[22] James, G.D, Trivial source modules for symmetric groups, Arch. math., 41, 294-300, (1983) · Zbl 0506.20004
[23] Jantzen, J.C, Darstellungen halbeinfacher gruppen und kontravariante formen, J. reine angew. math., 290, 117-141, (1977) · Zbl 0342.20022
[24] Jantzen, J.C, Darstellungen halbeinfacher gruppen und ihrer Frobenius-kerne, J. reine angew. math., 317, 157-199, (1980) · Zbl 0451.20040
[25] Koppinen, M, Good bimodule filtrations for coordinate rings, J. London math. soc., 30, 244-250, (1984) · Zbl 0566.20020
[26] Kostant, B, Groups over \(Z\), ()
[27] {\scR. Steinberg}, “Lectures on Chevalley Groups,” mimeographed lecture notes, Department of Mathematics, Yale Univ., New Haven, Conn.
[28] Sweedler, M, Hopf algebras, (1969), Benjamin New York · Zbl 0194.32901
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.