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 $\text{GL}_n({\mathbb C})$. 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 $\text{GL}_n({\mathbb C})$-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.