Resolutions, homological dimensions, and extension of hook representations

Abstract

We describe explicit universal projective resolutions of the Weyl modules for the general linear groups that correspond to (skew) hook diagrams. Using these resolutions the homological dimensions of the hook Weyl modules are determined in many cases. Our results are applied to obtain explicit projective resolutions and to find the homological dimensions of certain irreducible representations over fields of positive characteristics. Finally, we determine various Ext groups involving hook Wey μ modules. For example, let X and μ be two hooks, where IJL is obtained by raising the bottom box of λ to the top row. Then for i ¦ 1 while for i=1 is cyclic of order the weight of λ. Here A is the appropriate integral Schur algebra.