Some unexpected properties of Littlewood-Richardson coefficients

Abstract

We are interested in identities between Littlewood-Richardson coefficients, and hence in comparing different tensor product decompositions of the irreducible modules of the linear group GL n (C). A family of partitions-called near-rectangular-is defined, and we prove a stability result which basically asserts that the decomposition of the tensor product of two representations associated to near-rectangular partitions does not depend on n. Given a partition $\lambda$, of length at most n, denote by V n ($\lambda$) the associated simple GL n (C)-module. We conjecture that, if $\lambda$ is near-rectangular and $\mu$ any partition, the decompositions of V n ($\lambda$) $\otimes$ V n ($\mu$) and V n ($\lambda$) * $\otimes$ V n ($\mu$) coincide modulo a mysterious bijection. We prove this conjecture if $\mu$ is also near-rectangular and report several computer-assisted computations which reinforce our conjecture.