Projective resolutions of q-Weyl modules

Here projective resolutions for the q-Weyl modules over the q-Schur algebra are given using the quantum multilinear algebra of Hashimoto and Hayashi and the implicit description of projective modules over the q-Schur algebra as tensor products of quantum divided powers. They are given by a mapping cone construction on short exact sequences of generalized Weyl modules attached to shapes and are a specific instance of a general algorithm included in the paper. The proof of the short exactness of these sequences requires a proof of the Pieri rule, giving a filtration for the tensor product of a q-Weyl module with a quantum tensor divided power with sections that are q-Weyl modules, here proven for the quantum situation. The proof further builds on the letterplace techniques developed by Rota and Buchsbaum and realized for the quantum situation as codeterminants by Green.