Opposite Littlewood-Richardson sequences and their matrix realizations

AbstractWe introduce the concept of opposite Littlewood-Richardson sequences. Given the partitions a0,a1,…,at and the nonnegative integers m1 ⩽ … ⩽ mt, that concept gives a necessary and sufficient condition for the existence of a sequence of matrices A0, B1,…, Bt with invariant partitions a0, (1m1),…, (1mt) such that ai is the invariant partition of A0B1 … Bi for i = 1,…, t, and (1m1) + ··· + (1mt) is the invariant partition of B1B2 … Bt. We also present an explicit construction of a sequence of matrices which realizes a previously given opposite Littlewood-Richardson sequence. This result is a generalization of a well-known result of T. Klein and J. A. Green on extensions of p-modules