Bidilatation of Small Littlewood-Richardson Coefficients

• Published in: arXiv.org 2022-06
• Main Authors: Chaput, Pierre-Emmanuel, Ressayre, Nicolas
• Format: Article
• Language: English
• Subjects: Tensors
• ISSN: 2331-8422

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 \(p, q \geq 0\).