On The First Cartan Invariant For The Finite Symplectic Group Sp(4,11~r)

Let G=Sp(2n, K) be a simply-connected semisimple algebraic group of type Cn over an alge-braically closed field K of Fp, Fr the standard Frobenius map of G relative to pr and G(r)=Sp(2n, pr) the finite subgroup consisting of fixed points under Fr in G. The case of type C2=B2 is considered in this paper. Firstly, we determine G-composition factors L(μ) s of Weyl G- modules V(λ) with some dominant weights, give decomposition patterns of G-composition factors L(μ)'s and principal indecomposable projective G(r)-modules U1(μ)'s- summand of the principal indecomposable G1T-module Q1(λ) with the restricted dominant weights. Then we deduce the formulas of computing the first Cartan invariant C00(r) of G(r) and the dimension of Ur(0). At last, we calculate exactly C00(r) and dimUr(0) for the finite symplectic group G(r)=Sp(4,11r) of type C2=B2.