Resolutions of three-rowed skew- and almost skew-shapes in characteristic zero

In [2], a connection was made between the characteristic-free resolution of the three-rowed partition, (2, 2, 2), and its characteristic-zero resolution described by A. Lascoux. The method used there was to take the known general resolution, modify the boundary map exploiting the fact that we can divide when we’re over the rationals, and then reduce the large general resolution to the much slimmer one of Lascoux ([4]). A similar program was carried out by [3] in his thesis, for the partition (3, 3, 3), but it was clear that the method had reached its limit there, since the explicit description of the boundary maps in the characteristic-free case is not yet available. 1 Since it was still very tempting to have an explicit description of the boundary maps in the characteristic-zero situation, we decided to borrow another tool from the characteristic-free construction, namely the mapping cone, and see if we couldn’t arrive at the Lascoux resolutions in general, that is not just for partitions, but for all the three-rowed shapes that come up in the representation category: the skew-shapes and almost skew-shapes ([1]). The combinatorial complexity of a three-rowed skewor almost skewshape is to a large extent tied up with the number of triple overlaps that occur in the shape. If the shape is a skew-shape: