Cartan Invariant Matrix For Finite Symplectic Group Sp(4,13) Of Type C₂

Let G=Sp(2n, K) be a simply-connected semisimple algebraic group of type Cn over an al-gebraically closed field K of Fpr, Fr the standard Frobenius map of G relative to pr, and G(r)= Sp(2n,pr) the finite subgroup of G which consist of fixed points in G under Fr, so called finite sym-plectic group of type Cn. The case of type C2 is considered in this paper. Firstly, we determine decomposition patterns of simple G-module for Weyl G-module V(λ) and Weyl G-module filtra-tions of principal indecomposable G1T-module Qi(λ). Secondly, we give decomposition pattens of simple G(1)-module for tensor product L(λ)(?)L(μ) of simple G-modules and decomposition pattens of Q1(λ) into principal indecomposable projective G(1)-modules U1(μ). Finally, we compute Cartan invariant matrix C=(cλμ(1))λ,μ∈X1(T) for finie symplectic group G(1)=Sp(4,13) of type C2, and it turns out that determinant of C is 1343.