The decomposition numbers of the Hecke algebra of type

Author:
Meinolf Geck

Journal:
Math. Comp. **61** (1993), 889-899

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 *l*-modular decomposition matrix of the algebra is a submatrix of the *l*-modular 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.

**[1]**Roger W. Carter,*Finite groups of Lie type*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR**794307****[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]**Richard Dipper,*Polynomial representations of finite general linear groups in nondescribing characteristic*, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 343–370. MR**1112168****[4]**Meinolf Geck,*On the decomposition numbers of the finite unitary groups in nondefining characteristic*, Math. Z.**207**(1991), no. 1, 83–89. MR**1106814**, 10.1007/BF02571376**[5]**Meinolf Geck,*On the classification of 𝑙-blocks of finite groups of Lie type*, J. Algebra**151**(1992), no. 1, 180–191. MR**1182021**, 10.1016/0021-8693(92)90138-C**[6]**Meinolf Geck,*Brauer trees of Hecke algebras*, Comm. Algebra**20**(1992), no. 10, 2937–2973. MR**1179271**, 10.1080/00927879208824499**[7]**Meinolf Geck and Klaus Lux,*The decomposition numbers of the Hecke algebra of type 𝐹₄*, Manuscripta Math.**70**(1991), no. 3, 285–306. MR**1089065**, 10.1007/BF02568379**[8]**Gerhard Hiss,*Decomposition numbers of finite groups of Lie type in nondefining characteristic*, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 405–418. MR**1112171**, 10.1007/978-3-0348-8658-1_17**[9]**R. A. Parker,*The computer calculation of modular characters (the meat-axe)*, Computational group theory (Durham, 1982) Academic Press, London, 1984, pp. 267–274. MR**760660****[10]**M. Schönert (Editor), GAP 3.0*manual*, Lehrstuhl D für Mathematik, RWTH Aachen, 1991.

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DOI:
https://doi.org/10.1090/S0025-5718-1993-1195429-5

Article copyright:
© Copyright 1993
American Mathematical Society