Hilbert–Kunz functions and multiplicities for full flag varieties and elliptic curves

We compute the Hilbert–Kunz functions and multiplicities for certain projective embeddings of flag varieties G/B and elliptic curves, over algebraically closed fields of positive characteristics. The group theoretic nature of both these classes of examples is used, albeit in different ways, to explicitly describe the cokernels in each degree of the Frobenius twisted multiplication maps for the corresponding graded rings. This detailed information also enables us to extend our results to arbitrary products of such varieties.