## The decomposition numbers of the Hecke algebra of type $E^ \ast _ 6$

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**61**(1993), 889-899 Request permission

## 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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## Additional Information

- © Copyright 1993 American Mathematical Society
- Journal: Math. Comp.
**61**(1993), 889-899 - MSC: Primary 20C20
- DOI: https://doi.org/10.1090/S0025-5718-1993-1195429-5
- MathSciNet review: 1195429