Bidilatation of Small Littlewood-Richardson Coefficients

TL;DR

This paper investigates the behavior of Littlewood-Richardson coefficients under a specific dilatation, revealing a binomial pattern when the coefficients are equal to two, extending known conjectures and results.

Contribution

It proves that for coefficients equal to two, their dilatation follows a binomial coefficient pattern, generalizing previous conjectures about the case when the coefficient equals one.

Findings

For c^ν_{λ,μ}=2, the dilated coefficient c^{ν(p,q)}_{λ(p,q),μ(p,q)} equals the binomial coefficient (p+q choose q).

The result extends the Fulton conjecture to the case when the coefficient is two.

The paper introduces a new perspective on the dilatation behavior of Littlewood-Richardson coefficients.

Abstract

The Littlewood-Richardson coefficients $c^\nu_{\lambda,\mu}$ 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 $(\lambda, \mu, \nu)$ of length at most $n$. By the so-called Fulton conjecture, if $c^\nu_{\lambda,\mu}=1$ then $c^{k\nu}_{k\lambda,k\mu}= 1$, for any $k \geq 0$. Similarly, as proved by Ikenmeyer or Sherman, if $c^\nu_{\lambda,\mu}=2$ then $c^{k\nu}_{k\lambda,k\mu} = k + 1$, for any $k\geq 0$. Here, given a partition $\lambda$, we set $\lambda(p, q) = p(q\lambda')'$ , where prime denotes the conjugate partition. We observe that Fulton's conjecture implies that if $c^\nu_{\lambda,\mu}=1$ then $c^{\nu(p,q)}_{\lambda(p,q),\mu(p,q)}=1$, for any $p, q \geq 0$. Our main result is that if $c^\nu_{\lambda,\mu}=2$ then $c^{\nu(p,q)}_{\lambda(p,q),\mu(p,q)}$ is the binomial $\begin{pmatrix} p+q\\ q \end{pmatrix}$, for any $p, q \geq 0$.