Pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces

Let X be an irreducible Hermitian symmetric space of compact type (IHSS for short). In this paper, we give the irreducible decomposition of Sym^rT_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.