A simple approach to matrix realizations for Littlewood—Richardson sequences

Abstract

In this paper we present a simple and explicit construction for matrix realizations of Littlewood-Richardson sequences as defined in MacDonald (I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, London, 1979). A Littlewood-Richardson sequence is a sequence of partitions that determine, among other things, the isomorphism types of a module, submodule, and quotient module over a discrete valuation ring of characteristic zero. A matrix realization provides a method for using a Littlewood-Richardson sequence to build matrices over such rings whose invariant factors are prescribed by these modules. Earlier constructions (O. Azenhas, E. Marques de Sa’, Linear and Multilinear Algebra 27 (1990) 229–242; O. Azenhas, Linear Algebra Appl. 225(1995) 91–116) were based on different definitions for such sequences and utilized conjugate partitions in their constructions. Our results avoid their use of conjugate partitions and are direct and explicit.