The Littlewood-Richardson rule for Coschur modules

Abstract

In the last 15 years, many mathematicians have turned their attention to the problem of extending to more general settings (fields of arbitrary characteristic, commutative rings) the representation theory of the general linear and symmetric groups over fields of characteristic zero. In this undertaking, the first problem is, perhaps, that of finding suitable generalizations of those modules that, in the classical theory, yield the irreducible representations first discovered and classified by Schur in his Dissertation of 1901 [24]. Over fields of characteristic zero, the more popular constructions of these modules, in terms of symmetry classes of tensors, are those due to Schur [25] and Weyl [31]; as is well known, these contructions are equivalent since they give rise to modules that, even though they are different as subspaces of the tensors space, turn out to be isomorphic with respect to the action of the group. Recently, it has been recognized that such constructions can be adequately adapted in order to make sense over arbitrary commutative rings; Akin, Buchsbaum and Weyman [ 1 ] have, therefore, introduced the so-called Schur and Coschur functors that turn out to be the natural generalizations to commutative rings of the constructions of Schur and Weyl, respectively. These functors are universally free functors and they give rise to modules-Schur and Coschur modules-which, even though they are isomorphic over fields of characteristic zero, are far from being isomorphic over the integers. Furthermore, Schur and Coschur modules are indecomposable, but they are not, in general, irreducible over commutative rings. Working from a slightly different point of view, Carter and Lusztig [7] discovered a class of modules which are irreducible over fields of positive characteristic; these modules were later elegantly described by Clausen [S] as the images of a natural operator-the Capelli operator-whose domain and codomain are the so-called Weyl modules of the second and first kind, which turn out to be the same as Coschur and Schur modules, respectively. Alongside the classification of the irreducible modules lies the second 143