Reduction formulae of Littlewood-Richardson coefficients

There are two well-known reduction formulae by Griffiths-Harris for Littlewood-Richardson coefficients. Our observation is that some special cases of the factorization theorem of Littlewood-Richardson coefficients by King, Tollu and Toumazet give reduction formulae including the Griffiths-Harris formulae. We provide explicit statements of those reduction formulae in more general forms, and extend them to their conjugated forms also. Eight useful reduction formulae deleting one or two rows (columns) of each partition are listed up as results. As an application, we prove that if the Littlewood-Richardson coefficient is 1 and each partition has distinct parts, then one of two types of our reduction formulae is always applicable and hence we have an algorithm to test if the Littlewood-Richardson coefficient is 1. Furthermore, our conjecture is that one of four types of our reduction formulae is always applicable to all triples of partitions if the corresponding Littlewood-Richardson coefficient is I. (C) 2010 Elsevier Inc. All rights reserved.