Universal filtration of Schur complexes

Abstract:

The Schur complex ${{\mathbf {L}}_{\lambda /\mu }}\phi$ has proved useful in studying resolutions of determinantal ideals, both in characteristic zero and in a characteristic-free setting. We show here that in every characteristic, ${{\mathbf {L}}_{\lambda /\mu }}\phi$ is isomorphic, up to a filtration, to a sum of Schur complexes $\sum \nolimits _\nu {\gamma (\lambda /\mu ;\nu ){{\mathbf {L}}_\nu }\phi }$, where $\gamma (\lambda /\mu ;\nu )$ is the usual Littlewood-Richardson coefficient. This generalizes a well-known direct sum decomposition of ${{\mathbf {L}}_{\lambda /\mu }}\phi$ in characteristic zero.