  Let $X$ be an irreducible Hermitian symmetric space of compact type (IHSS for
short). In this paper, we give the irreducible decomposition of $Sym^r T_X$. As
a by-product, we give a cohomological characterization of the rank of $X$.
Moreover, we introduce pseudoeffective thresholds to measure the bigness of
tangent bundles of smooth complex projective varieties precisely and calculate
them for irreducible Hermitian symmetric spaces of compact type.
