Bidilatation of Small Littlewood-Richardson Coefficients

The Littlewood-Richardson coefficients $c^ν_{λ,μ}$ are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group GL$(n, {\mathbb C})$. They are parametrized by the triples of partitions $(λ, μ, ν)$ of length at most $n$. By the so-called Fulton conjecture, if $c^ν_{λ,μ}=1$ then $c^{kν}_{kλ,kμ}= 1$, for any $k \geq 0$. Similarly, as proved by Ikenmeyer or Sherman, if $c^ν_{λ,μ}=2$ then $c^{kν}_{kλ,kμ} = k + 1$, for any $k\geq 0$. Here, given a partition $λ$, we set $λ(p, q) = p(qλ')'$ , where prime denotes the conjugate partition. We observe that Fulton's conjecture implies that if $c^ν_{λ,μ}=1$ then $c^{ν(p,q)}_{λ(p,q),μ(p,q)}=1$, for any $p, q \geq 0$. Our main result is that if $c^ν_{λ,μ}=2$ then $c^{ν(p,q)}_{λ(p,q),μ(p,q)}$ is the binomial $\begin{pmatrix} p+q\\ q \end{pmatrix}$, for any $p, q \geq 0$.

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