Selections from the Letter-place Panoply

There is a fairly extensive literature on letter-place algebras, but mostly for the edification of those working in algebraic combinatorics (see, for instance, [15, 16, 18, 20]). I don’t think that letter-place algebras have had much play yet in commutative and homological algebra, so I thought I’d talk about them here and perhaps arouse a bit of interest in that subject. I was introduced to the letter-place notion by Gian-Carlo Rota, with whom I had the pleasure of collaborating for almost a decade. He attributed it to a physicist, I believe to Feynman. The fact that letter-place techniques, including place polarizations, could simplify a good deal of the work Akin and I had been doing earlier on resolving Weyl modules ([1], [3]), appealed to both of us, and we decided to use them to push ahead to find the projective resolutions of those representations. Much of this is spelled out in great detail in [7], and in a fairly long article written with Rota (posthumously) [13], we focused on the resolutions themselves. In Section 2, we will see some examples to show how representation theory first reared its head for Eisenbud and me in our joint work on generalized Koszul complexes ([8], [10]). From there we discuss Lascoux’s use of classical representation theory to describe the terms of the resolutions of determinantal ideals and of Schur modules ([21]). This led to the development of characteristic-free representation theory of the general linear group, to the general definition of Schur and Weyl modules (which are in a precise sense dual to each other; this will be explained in Section 5.8), and an attempt by Akin, Weyman and myself to replicate the Lascoux results for determinantal ideals in characteristic-free form ([4], [5]). An interesting early development along those lines was the discovery of the role Z-forms play in these results, and this led Akin and me to study the resolutions of Weyl modules in a serious systematic way ([3]). In Section 3, we will give a precise definition of Schur and Weyl modules, so that the terms used above will make sense; these are, in fact, the major objects of study in the rest of this article. In Section 4 we will consider resolutions of two-rowed Weyl modules associated to skew-shapes, and see the letter-place techniques in play. In particular, we’ll see how use of letter-place enables us to define a splitting homotopy for these resolutions ([11]). In Section 5, we’ll take the bull by the horns and give the general definition of letter-place algebras, indicate the proof of the basis theorem for them, and discuss