The Cartan Invariant Matrix for the Finite Group G₂(13) of Type G₂

DOI: 10.4236/alamt.2014.41003, PP. 40-63

Keywords: Cartan Invariant Matrix, Weyl Module, Projective Indecomposable Module, Simple Module

Full-Text   Cite this paper   Add to My Lib

Abstract:

The Cartan invariant matrixfor the finite group G₂(13) of type G₂ over the finite field with 13 elements is determined in this paper.

References

[1]	Alperlin, J.L. (1975) On Modules for the Linear Fractional Groups. International Symposium on Theory of Finite Groups. RIMS, Kyoto.
[2]	Alperlin, J.L. (1979) Projective Modules for SL(2, 2n). Journal of Pure and Applied Algebra, 15, 219-234. http://dx.doi.org/10.1016/0022-4049(79)90017-3
[3]	Upadhyaya, B.S. (1978) Composition Factors of the Principal Indecomposable Modules for the Speciallinear Group SL(2, q). Journal of London Mathematical Society, 17, 437-445.
[4]	Benson, D.J. and Martin, S. (1991) Mod 2 Cohomology of the Unitary Groups U3(2 n). Communications in Algebra, 19, 3125-3144. http://dx.doi.org/10.1080/00927879108824311
[5]	Humphreys, J.E. (1976) Ordinary and Modular Representations of Chevalley Groups. Lecture Notes in Mathematics 528, Springer, Berlin.
[6]	Humphreys, J.E. (1973) Some Computations of Cartan Invariants for Finite Groups of Lie Type. Communications on Pure and Applied Mathematics, 26, 745-755. http://dx.doi.org/10.1002/cpa.3160260516
[7]	Ye, J.C. (1982) The Cartan Invariants of SL(3, pn). Journal of Mathematical Research with Applications, 2, 9-19. (In Chinese)
[8]	Zaslawsky, E. (1974) Computational Methods Applied to Ordinary and Modular Characters of Some Finite Simple Groups. Ph.D. Thesis, University of California, Santa Cruz.
[9]	Benson, D.J. (1983) The Loewy Structure of the Projective Indecomposable Modules for a8 in Characteristic 2. Communications in Algebra, 11, 1395-1432. http://dx.doi.org/10.1080/00927878308822912
[10]	Du, J. (1986) The Cartan Invariants of SL(4, 2n). Journal of East China Normal University (Natural Sciences Ed.), 4, 17-25. (In Chinese)
[11]	Ye, J.C. (1997) The Cartan Invariant Matrix for the Finite Group SL(5, 2). Algebra Colloquium, 4, 203-211.
[12]	Thackray, J.G. (1980) Modular Representations of Some Finite Groups. Ph.D. Thesis, Cambridge University, Cambridge.
[13]	Humphreys, J.E. (1980) Modular Representations of Finite Groups of Lie Type. Finite Simple Groups II, Academic Press, London, 259-290.
[14]	Ye, J.C. (1985) The Cartan Invariants of Sp(4, pn). Journal of East China Normal University (Natural Sciences Ed.), 2, 18-25. (In Chinese)
[15]	Ye, J.C. and Zhou, Z.G. (2001) Irreducible Characters of Algebraic Groups in Characteristic Three. Communications in Algebra, 29, 201-223. http://dx.doi.org/10.1081/AGB-100000795
[16]	Liu, A.L. and Ye, J.C. (2003) Cartan Matrix for the Finite Symplectic Group Sp(6, 3). Algebra Colloquium, 10, 33-39. http://dx.doi.org/10.1007/s100110300005
[17]	Hu, Y.W. and Xu, L. (2011) The Cartan Invariant Matrix for Finite Symplectic Group Sp(4, 11). Advances in Mathematics (China), 40, 483-491.
[18]	Mertens, D. (1985) Zur Darstellungstheorie der endlichen Chevalley-Gruppen vom Typ G2. Diplomarbeit, University of Bonn, Bonn.
[19]	Hu, Y.W. and Ye, J.C. (2001) Cartan Invariant Matrix for Finite Group of Lie Type G2(5). Journal of Tongji University (Natural Sciences Ed.), 29, 1205-1208. (In Chinese)
[20]	Lusztig, G. (1980) Hecke Algebras and Jantzen’s Generic Decomposition Patterns. Advances in Mathematics, 37, 121-164. http://dx.doi.org/10.1016/0001-8708(80)90031-6
[21]	Jantzen, J.C. (1977) Uber das Dekompositionsverhalten Gewisser Modularer Darstellungen halbeinfacher Gruppen und ihrer Lie-Algebren. Journal of Algebra, 49, 441-469. http://dx.doi.org/10.1016/0021-8693(77)90252-6
[22]	Humphreys, J.E. (1987) Cohomology of Link Bundles on G/B for the Ecceptional Group G2. Journal of Pure and Applied Algebra, 44, 160-168.
[23]	Andersen, H.H. (1981) On the Structure of the Conhomology of Line Bundles on G/B. Journal of Algebra, 71, 245-258. http://dx.doi.org/10.1016/0021-8693(81)90119-8
[24]	Jantzen, J.C. (1980) Darstellungen Halbeinfacher Gruppen and Ihrer Frobenius-Kerne. Journal für die Reine und Angewandte Mathematik, 317, 157-199.
[25]	Chastkofsky, L. (1981) Projective Characters for Finite Chevalley Groups. Journal of Algebra, 69, 347-357. http://dx.doi.org/10.1016/0021-8693(81)90209-X
[26]	Jantzen, J.C. (1981) Zur Reduktion modulo p der Character von Deligne und Lusztig. Journal of Algebra, 70, 452-474. http://dx.doi.org/10.1016/0021-8693(81)90229-5

Full-Text