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

Although Alain Lascoux and I never collaborated on a mathematical paper, his ideas have been a constant influence on my students’, my collaborators’ and my work ever since the appearance of the first draft of his doctoral thesis [12] His systematic use of characteristic-zero representation theory to find the resolutions of ideals generated by the minors, of any order, of generic matrices, impelled me and my students in that period to develop a characteristicfree theory of Schur and Weyl modules [4]. The attempt by Akin, Weyman and myself to use these characteristic-free methods to reproduce Lascoux’ resolutions led to the realization that there were many mysteries hidden in Zforms of rational representations that had to be uncovered in order to move ahead with this project [3]. Akin and I soon discovered that the study of Zforms was intimately bound up with the resolutions of Weyl modules [1, 2], the characteristic-zero version of which Lascoux had already presented. 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. In sections 2 and 3, we will give a few examples of the way in which Lascoux’ work has been incoporated into a number of the above-mentioned areas of investigation. Where possible, we will point out the similarities and differences between the classical and ‘neoclassical’ results. In section 4, we give a very brief indication of how place polarization methods, Capelli identities, and resolutions come into play in the study of intertwining numbers.