THE DECOMPOSITION NUMBERS OF THE HECKE ALGEBRA OF TYPE-E(6)

Let E6(q) be the Chevalley group of type E6 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 E6(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 E6 (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 E6 over the ring A = 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 Q or values in a finite field. The computations were done by using the computer algebra systems MAPLE and GAP.