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Mathematics of Computation

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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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Article copyright: © Copyright 1993 American Mathematical Society