Kaan Akin

Schur Functors and Schur Complexes

The study of Schur functors has a relatively long history. Its main impetus derived from representation theory, originally in characteristic zero. Over the years, however, with the development of modular representations, and algebraic geometry over fields of positive characteristic, the need for a theory of universal polynomial functors increased and, since the mid-1960s, approaches to a characteristic-free treatment of Schur functors have been developing (see, for instance, the recent book of Green [ 1 l] in which the treatments by Carter and Lusztig [6], Higman [12], and Towber [20], among others, are discussed). Our own interest in such a treatment was awakened by the work of Lascoux [ 141 on resolutions of determinantal ideals. Although his thesis treated only the characteristic zero case, it suggested that a general and elementary theory of Schur functors could be developed using only the rudiments of multilinear algebra (involving the Hopf algebra structures of the symmetric, exterior, and divided power algebras). Moreover, this elementary development admitted of a natural generalization to the idea of Schur complexes, whose usefulness was demonstrated, for instance, in our construction of a universal minimal

Characteristic-free representation theory of the general linear group

On etudie systematiquement les Z-formes des representations rationnelles du groupe lineaire generale

Characteristic-free representation theory of the general linear group II. Homological considerations

In our first paper of this series Cl], we indicated how we were led to consider resolutions of Schur and Weyl modules of a particular form. In order to prove the existence of these resolutions, we were forced to enlarge the family of skew shapes to a class J containing new shapes whose corresponding Schur and Weyl modules had heretofore not been studied. With these shapes in hand, we proved in [ 1 ] the existence of some fundamental exact sequences and described how, from these exact sequences, we could use a mapping cone construction to build up the resolutions we were seeking. For this mapping cone construction, we needed maps, and to provide maps we needed projectivity of tensor products of divided powers. This led to the study of the Schur algebra and its decomposition into orthogonal idempotents. In Sections 1 and 2 we review the information about the Schur algebra that we need to carry out our program. Fortunately there is a very clear and detailed exposition of this subject in the notes of J. A. Green [S] from which we borrowed very heavily.’ In fact, the main function of the first two sections is to condense and translate into our notation and terminology the relevant sections of Green’s notes. In Sections 3 and 4 we define the family of shapes, J, that we will study,

Resolutions of determinantal ideals: The submaximal minors

For many years there has been considerable interest in finding a resolution of the ideal generated by the minors of order p of a generic m x n matrix. To put the problem more precisely, suppose R, is a commutative ring and X, are variables with 1 < i < m and 1 < j Q n. If we let R = R, [X,] be the polynomial ring over R,, then we have the “generic” matrix (X,) and we may form the ideal ZP in R generated by the p X p minors of this matrix. The problem, then, is to find an explicit free resolution of the ideal ZP over the ring R. It was proved by Eagon and Hochster [lo] that R/I, has a resolution of length (m-p + l)(n -p + l), but their proof consisted in showing that the ideal ZP is perfect; it did not provide a construction of the resolution. In fact, it is not known whether the Betti numbers of the ideal ZP depend on the characteristic of the ground ring R,. In [ 121, Lascoux succeeded in giving an explicit resolution provided that the ground ring R, contained the field, Q, of rational numbers. His construction rests heavily on the theory of Schur functors and the fact that in characteristic zero the Schur functors are the irreducible representations of the general linear group. Over the integers, however, the construction breaks down despite the fact that one can define the Schur functors over an arbitrary commutative ring (see [ 1,2, 14, 151 for various constructions of Schur functors). In analyzing the work of Lascoux and its subsequent reworking by Nielsen [ 131, some basic facts seemed to clamor for attention. One was that within a resolution of R/Z,, there appeared to be two types of boundary maps: one of degree 1 and one of degree p. The maps of degree 1 were maps between sums of Schur functors of fixed Durfee square k (see Section 2 for definitions), while the maps of degree p were from sums of Schur functors of

Representations, Resolutions and Intertwining Numbers

It may, at first glance, seem inappropriate to be talking about representations of the general linear group at a conference on commutative algebra. We would like, therefore, to offer some observations: one, historical; one, personal; and one, perhaps presumptuous, that might reconcile this apparent anomaly.