The First Cartan Invariant Of The Finite Chevalley Group SL(3,11~r)

Let G=SL(n+1,K) be a simply-connected semisimple algebraic group of type An over an algebraically closed field K of Fp, Fr the standard Frobenius map of G relative to pr, Gr=KerFr the kernel of Fr,πan automorphism of G coming from that of the Dynkin diagram of G, and Gπ(r) the finite subgroup consisting of fixed points underπFr in G. It is assumed that An is of type A2 in this paper. We firstly determine decompositions of simple G-modules of V(λ) with some dominant weights, and give decomposition patterns of simple G-modules and principal indecomposable projective Gπ(r)-modules U1(μ) of principal indecomposable G1T- modules Q1(λ) with the restricted dominant weights. Secondly, we conclude the formulas for computing c00(r) and dimUr (0) for finite group of Lie type. At the end, we calculate precisely c00(r) of Chevalley group SL(3, 11r) and dimension of principal indecomposable projective SL(3, 11r)-module Ur(0).