The $S_3$-symmetric tridiagonal algebra

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 $Γ$ 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 $Γ$ is a Hamming graph. We give some conjectures and open problems.

💡 Research Summary

The paper introduces a new algebraic structure called the S₃‑symmetric tridiagonal algebra (denoted T), which extends the classical tridiagonal algebra by incorporating the full symmetry of the symmetric group S₃. While the original tridiagonal algebra is generated by two elements A and A* subject to two “tridiagonal relations,” the S₃‑symmetric version has six generators \