The $S_3$-symmetric tridiagonal algebra

TL;DR

This paper introduces the $S_3$-symmetric tridiagonal algebra with six generators, explores its structure, and applies it to the tensor powers of modules associated with certain distance-regular graphs, especially Hamming graphs.

Contribution

It defines a new algebraic structure with symmetric properties and connects it to graph theory, expanding the framework of tridiagonal algebras and their applications.

Findings

Defined the $S_3$-symmetric tridiagonal algebra with six generators.

Established a module structure for tensor powers of standard modules over distance-regular graphs.

Analyzed the case of Hamming graphs and proposed conjectures and open problems.

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.