The Littlewood-Richardson rule for Coschur modules (Q1112175)

Coschur modules are the generalizaion to commutative rings of the modules constructed by Schur and Weyl in the classical theory of the general linear and symmetric groups over a field of characteristic zero. The present paper is based on the theory of Rota and Stein which deals with the tensor product of a symmetric algebra and an exterior algebra, called the letterplace algebra. The main objective of this paper is to give a universal version of the Littlewood-Richardson rule for the Coschur modules. To each Young shape \(\sigma\), the authors associate a submodule of the letterplace algebra which is called Weyl module. This submodule is spanned by the determinants of bitableaux whose right tableau is of Deruyts type (that is, it contains only negative symbols and its ith column contains only the ith symbol of the alphabet), while its left tableau is any tableau of shape \(\sigma\) containing only positive symbols. The authors show that the Weyl module relative to a given Young shape is isomorphic naturally to the Coschur module relative to the conjugate shape and then they get the universal version of the Littlewood-Richardson rule for the tensor product of two Coschur modules by the corresponding result for Weyl modules. The authors also introduce a further class of modules, called Deruyts modules, which arises from a common generalization of both Weyl and Schur modules. The tensor product of two Weyl modules is isomorphic naturally to a submodule called the Skew submodule of a Deruyts module. They also show that this Skew submodule has a basis consisting of so called standard Yamanouchi shuffle elements by which they get a filtration of this module whose associated graded object is the direct sum of Weyl modules described by the classical Littlewood-Richardson rule.