Computation of 2-modular sheaves and representations for \(L_ 4(2)\), \(A_ 7\), \(3S_ 6\), and \(M_{24}\). (English) Zbl 0676.20007
The authors determine the zero-homology for the panel irreducible presheaves for the projective space for \(L_ 4(2)\), the sporadic \(C_ 3\) geometry for \(A_ 7\), the generalized quadrangle for \(Sp_ 4(2)\) and its triple cover (with group \(3S_ 6)\), and the 2-local geometry for \(M_{24}\) in the sense of the authors [Proc. Symp. Pure Math. 37, 283- 289 (1980; Zbl 0478.20015)]. In the latter they just give a rough description of the zero-homology. It is a nice byresult that all irreducible \(F_ 2\)-modules for the corresponding groups \((L_ 4(2)\), \(A_ 7\), \(S_ 6\), \(3S_ 6\), \(M_{24})\) support a panel irreducible sheaf. This yields the authors to suggest to extend the high-weight theory to simple-group geometries other than buildings.
The proofs in this paper are very elementary. They rely on what the authors call “geometric spanning”, which means that \(H_ 0\) is spanned by the terms at the vertices of any one particular type.
The proofs in this paper are very elementary. They rely on what the authors call “geometric spanning”, which means that \(H_ 0\) is spanned by the terms at the vertices of any one particular type.
Reviewer: G.Stroth
MSC:
| 20C20 | Modular representations and characters |
| 20G10 | Cohomology theory for linear algebraic groups |
| 20D05 | Finite simple groups and their classification |
| 20J05 | Homological methods in group theory |
| 51E25 | Other finite nonlinear geometries |
| 51D20 | Combinatorial geometries and geometric closure systems |
Keywords:
zero-homology; panel irreducible presheaves; projective space; generalized quadrangle; triple cover; 2-local geometry; irreducible \(F_ 2\)-modules; simple-group geometriesCitations:
Zbl 0478.20015References:
| [1] | Aschbacher M., Geom.Dedicate 16 pp 195– (1984) |
| [2] | DOI: 10.1080/00927878308822912 · Zbl 0528.20007 · doi:10.1080/00927878308822912 |
| [3] | DOI: 10.1080/00927877708822186 · Zbl 0355.20011 · doi:10.1080/00927877708822186 |
| [4] | DOI: 10.1016/0021-8693(73)90165-8 · Zbl 0268.20008 · doi:10.1016/0021-8693(73)90165-8 |
| [5] | Lusztig G., Annals of Math Studies 81 (1974) |
| [6] | DOI: 10.1007/BF01198134 · Zbl 0509.05026 · doi:10.1007/BF01198134 |
| [7] | Parker, R.A. The computer calculation of modular characters. Pages 267-278. Proceedings of LMS Durham Symposium. 1982, New York. Edited by: Atkinson, M. Academic Press. |
| [8] | DOI: 10.1007/BF01214721 · Zbl 0478.05020 · doi:10.1007/BF01214721 |
| [9] | Ronan, M.A. and Smith, S.D. 2-Local geometries for some sporadic groups. Proc.Symp.Pure Math.37. 1979, Santa Cruz. Providence, R.I: Amer. Math Soc. Proceedings, ed. Cooperstein-Mason · Zbl 0478.20015 |
| [10] | DOI: 10.1016/0021-8693(85)90013-4 · Zbl 0604.20043 · doi:10.1016/0021-8693(85)90013-4 |
| [11] | DOI: 10.1016/0021-8693(86)90132-8 · Zbl 0602.20014 · doi:10.1016/0021-8693(86)90132-8 |
| [12] | DOI: 10.1016/0021-8693(86)90132-8 · Zbl 0602.20014 · doi:10.1016/0021-8693(86)90132-8 |
| [13] | Ronan M.A., Europ.J.Combinatorics 5 pp 59– (1984) |
| [14] | DOI: 10.1016/0021-8693(85)90088-2 · Zbl 0588.51012 · doi:10.1016/0021-8693(85)90088-2 |
| [15] | Stroth G., Part I:the symplectic case |
| [16] | Tlts J., Lecture Notes in Math 386 (1974) |
| [17] | DOI: 10.1007/BF02564442 · Zbl 0616.20022 · doi:10.1007/BF02564442 |
| [18] | DOI: 10.1016/0021-8693(72)90061-0 · Zbl 0226.20009 · doi:10.1016/0021-8693(72)90061-0 |
| [19] | Ronan M.A., Europ.J.Combinatorics 8 pp 179– (1987) |
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