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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The decomposition numbers of the Hecke algebra of type $ E\sp \ast\sb 6$


Author: Meinolf Geck
Journal: Math. Comp. 61 (1993), 889-899
MSC: Primary 20C20
MathSciNet review: 1195429
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Abstract: Let $ {E_6}(q)$ be the Chevalley group of type $ {E_6}$, over a finite field with q elements, l be a prime not dividing q, and $ {H_R}(q)$ be the endomorphism ring of the permutation representation (over a valuation ring R with residue class field of characteristic l) of $ {E_6}(q)$ on the cosets of a standard Borel subgroup $ B(q)$. Then the l-modular decomposition matrix $ {D_l}$ of the algebra $ {H_R}(q)$ is a submatrix of the l-modular decomposition matrix of the finite group $ {E_6}(q)$. In this paper we determine the matrices $ {D_l}$, for all l, q as above. For this purpose, we consider the generic Hecke algebra H associated with the finite Weyl group of type $ {E_6}$ over the ring $ A = \mathbb{Z}[v,{v^{ - 1}}]$ 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 $ \mathbb{Q}$ or values in a finite field. The computations were done by using the computer algebra systems MAPLE and GAP.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1993-1195429-5
PII: S 0025-5718(1993)1195429-5
Article copyright: © Copyright 1993 American Mathematical Society