The decomposition numbers of the Hecke algebra of type
Author:
Meinolf Geck
Journal:
Math. Comp. 61 (1993), 889899
MSC:
Primary 20C20
MathSciNet review:
1195429
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Abstract: Let be the Chevalley group of type , over a finite field with q elements, l be a prime not dividing q, and be the endomorphism ring of the permutation representation (over a valuation ring R with residue class field of characteristic l) of on the cosets of a standard Borel subgroup . Then the lmodular decomposition matrix of the algebra is a submatrix of the lmodular decomposition matrix of the finite group . In this paper we determine the matrices , for all l, q as above. For this purpose, we consider the generic Hecke algebra H associated with the finite Weyl group of type over the ring of Laurent polynomials in an indeterminate v, and calculate the decomposition matrices of H which are associated with specializations of v to roots of unity over or values in a finite field. The computations were done by using the computer algebra systems MAPLE and GAP.
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M. Schönert (Editor), GAP 3.0 manual, Lehrstuhl D für Mathematik, RWTH Aachen, 1991.
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 R. W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, New York, 1985. MR 794307 (87d:20060)
 [2]
 B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt, MAPLE Reference manual, 5th ed., University of Waterloo, 1988.
 [3]
 R. Dipper, Polynomial representations of finite general linear groups in nondescribing characteristic, Progr. Math., vol. 95, Birkhäuser Verlag, Basel, 1991, pp. 343370. MR 1112168 (92h:20018)
 [4]
 M. Geck, On the decomposition numbers of the finite unitary groups in nondefining characteristic, Math. Z. 207 (1991), 8389. MR 1106814 (92f:20013)
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 , On the classification of lblocks of finite groups of Lie type, J. Algebra 151 (1992), 180191. MR 1182021 (93g:20029)
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 , Brauer trees of Hecke algebras, Comm. Algebra 20 (1992), 29372973. MR 1179271 (94a:20019)
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 M. Geck and K. Lux, The decomposition numbers of the Hecke algebra of type , Manuscripta Math. 70 (1991), 285306. MR 1089065 (92a:20043)
 [8]
 G. Hiss, Decomposition numbers of finite groups of Lie type in nondefining characteristic, Progr. Math., vol. 95, Birkhäuser Verlag, Basel, 1991, pp. 405418. MR 1112171 (93a:20019)
 [9]
 R. A. Parker, The computer calculation of modular characters (the MeatAxe), Computational Group Theory (M. D. Atkinson, ed.), Academic Press, London, 1984. MR 760660 (85k:20041)
 [10]
 M. Schönert (Editor), GAP 3.0 manual, Lehrstuhl D für Mathematik, RWTH Aachen, 1991.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199311954295
PII:
S 00255718(1993)11954295
Article copyright:
© Copyright 1993
American Mathematical Society
