Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into $U({\mathfrak{sl}}_2)$

Abstract

From the viewpoint of Johnson graphs as slices of a hypercube, we derive a novel algebra homomorphism $♯$ from the universal Racah algebra $\Re$ into $U({\mathfrak{sl}}_2)$. We use the Casimir elements of $\Re$ to describe the kernel of $♯$. By pulling back via $♯$ every $U({\mathfrak{sl}}_2)$-module can be viewed as an $\Re$-module. We show that for any finite-dimensional $U({\mathfrak{sl}}_2)$-module $V$, the $\Re$-module $V$ is completely reducible and three generators of $\Re$ act on every irreducible $\Re$-submodule of $V$ as a Leonard triple. In particular, Leonard triples can be constructed in terms of the second dual distance operator of the hypercube $H(D,2)$ and a decomposition of the second distance operator of $H(D,2)$ induced by Johnson graphs.