Hidden Structure of Jack Littlewood-Richardson Coefficients

We argue that Jack Littlewood-Richardson coefficients $g_{\mu \nu }^{\lambda }(\alpha )$ are specialisations of certain novel polynomials. For the triple of partitions $(\mu ,\nu ,\lambda )=(21,21,321)$, we prove the corresponding polynomial is invariant under $S_6\times {\mathbb{Z}}_2$, which is identified as the automorphism group of the Johnson graph $J(6,3)$. We conjecture that these polynomials exhibit a factorization property on certain hyperplanes, which is a consequence of compatibility relations between polynomials associated to adjacent triples in the Young graph. As a consequence of this, we conjecture that the difference of adjacent Jack Littlewood-Richardson coefficients is divisible by the shared hook length.