The $S_3$-symmetric tridiagonal algebra

Abstract: The tridiagonal algebra is defined by two generators and two relations, called the tridiagonal relations. Special cases of the tridiagonal algebra include the $q$-Onsager algebra, the positive part of the $q$-deformed enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$, and the enveloping algebra of the Onsager Lie algebra. In this paper, we introduce the $S_3$-symmetric tridiagonal algebra. This algebra has six generators. The generators can be identified with the vertices of a regular hexagon, such that nonadjacent generators commute and adjacent generators satisfy a pair of tridiagonal relations. For a $Q$-polynomial distance-regular graph $\Gamma$ we turn the tensor power $V^{\otimes 3}$ of the standard module $V$ into a module for an $S_3$-symmetric tridiagonal algebra. We investigate in detail the case in which $\Gamma$ is a Hamming graph. We give some conjectures and open problems.