Structure theorems for vector valued Siegel modular forms of degree 2 and weight det(k) circle times Sym(10)

We prove the explicit structure theorems of modules circle plus(k) M-detk circle times Sym(10)(Sp(2)(Z(2))) of vector valued Siegel modular forms of degree 2, where k runs over the set of even integers or odd integers. We also check the conjecture given by Ibukiyama [Vector valued Siegel modular forms of symmetric tensor weight of small degrees, Comment. Math. Univ. St. Pauli 61 (2012) 51-75.] for modules of vector valued Siegel modular forms of degree 2 of weights det* circle times Sym(8) and det* circle times Sym(10)