Block decomposition via the geometric Satake equivalence

We give a new proof for the description of the blocks in the category of representations of a reductive algebraic group $\mathbf{G}$ over a field of positive characteristic $\mathrm{\ell }$ (originally due to Donkin), by working in the Satake category of the Langlands dual group and applying Smith-Treumann theory as developed by Riche and Williamson. On the representation theoretic side, our methods enable us to give a bound for the length of a minimum chain linking two weights in the same block, and to give a new proof for the block decomposition of a quantum group at an $\mathrm{\ell }$-th root of unity.