Approaches to Resolution of Weyl Modules

A characteristic-free example of a Lascoux resolution, and letter-place methods for intertwining numbers

“…The resolutions of two-rowed skew shapes were given in [2] and [6], while those for very special three-rowed shapes were given in [7]. The one given here also appeared in [8], but the explicit connection with the Lascoux resolution was not given there.…”

“…where the subscripts on the X indicate the dimension in which these terms appear. The symbols Z (t) a,b are the formal 'polarization' operators defined in [8], and the letters x, y, z are the separator variables also explained in that paper.…”

“…(of which, in this example, there aren't that many). For these we have to use identities of Capelli type, or some easy variants of them ( [8]). The boundary map on such a term sends it to…”

“…In [8], we had made extensive use of place polarizations, and had made an explicit correspondence between the algebra of such polarizations and the Heisenberg-Weyl algebra. The new idea in attacking the intertwining numbers is to express some of the basic targets of our investigation in terms of place polarizations, and to then simplify our problem by proving some identities in the Heisenberg-Weyl algebra.…”

“…The study of such resolutions was helpful in the Roberts-Weyman [13] presentation of the Hashimoto [10] example of the dependence of the Betti numbers of determinantal ideals on characteristic. Further work on these resolutions with Rota [8] led to the use of letter-place methods and place polarizations in a systematic way in this area.…”