Opposite Littlewood-Richardson sequences and their matrix realizations (Q1899378)

In 1961 J. A. Green exhibited a close connection between the structure of modules over a local principal ideal domain \({\mathcal R}_p\) and the Littlewood-Richardson rule for multiplying Schur functions. (See Section II.3 of I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Clarendon Press, 2nd edition (1995; Zbl 0824.05059).) The author develops this theory in terms of sequences of matrices over \({\mathcal R}_p\) which realise Young tableaux. If \(M\) is a finite dimensional module whose type is the partition \(\lambda = (\lambda_1, \dots, \lambda_n)\), then the Smith normal form associates to \(M\) the diagonal matrix with entries \(p^{\lambda_1}, \dots, p^{\lambda_n}\), where \((p)\) is the maximal ideal of \({\mathcal R}_p\). Thus elementary modules \((pM = 0)\) of dimension \(r\) as vector spaces over the residue field \(k = {\mathcal R}_p/(p)\) have type \((1^r)\). Green's theory relates the composition series \((p^iN)\) of a submodule \(N \subseteq M\) with a Littlewood-Richardson tableau for the skew diagram \(\lambda' - \mu'\), where \(\mu\) is the type of \(M/N\) and the primes denote conjugate partitions. Earlier work of the author and \textit{E. Marques de Sá} [Linear Multilinear Algebra 27, No. 4, 229-242 (1990; Zbl 0704.15009)] gives an explicit matrix version of the work of Green and T. Klein on extension of such modules. This relates to skew tableaux whose weight (or content) is a partition. The present paper considers the dual situation where the weight is a reversed partition, i.e. the parts are taken in increasing order. This leads to appropriate notions of dual tableaux and opposite Littlewood-Richardson sequences. Algorithms are given for transforming a tableau of given shape and weight into one of the same shape with reversed weight, and for construction of a sequence of matrices which realises a tableau satisfying the opposite Littlewood-Richardson rule. These are used to prove a matrix realisation theorem.