Hidden Structure of Jack Littlewood-Richardson Coefficients

TL;DR

This paper introduces novel polynomials related to Jack Littlewood-Richardson coefficients, explores their symmetry properties, and conjectures factorization and divisibility properties based on combinatorial and algebraic structures.

Contribution

It proposes that Jack Littlewood-Richardson coefficients are specializations of new polynomials with symmetry and factorization properties, advancing understanding of their structure.

Findings

Proved invariance of a polynomial under the automorphism group of Johnson graph for a specific partition triple.

Conjectured polynomial factorization on hyperplanes linked to Young graph relations.

Suggested divisibility of differences of adjacent coefficients by shared hook length.

Abstract

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.