The $$S_3$$-symmetric tridiagonal algebra

Source abstract

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 $$U_q({\widehat{\mathfrak {sl}}}_2)$$ U q ( sl ^ 2 ) , and the enveloping algebra of the Onsager Lie algebra. In this paper, we introduce the $$S_3$$ 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 $$\Gamma$$ Γ , we turn the tensor power $$V^{\otimes 3}$$ V ⊗ 3 of the standard module V into a module for an $$S_3$$ S 3 -symmetric tridiagonal algebra. We investigate in detail the case in which $$\Gamma$$ Γ is a Hamming graph. We give some conjectures and open problems.