The S₃-symmetric tridiagonal algebra

2024/06/30 by Paul Terwilliger, Terwilliger, Paul · 1 citation
Mathematics · Computer Science · #Advanced Topics in Algebra #Matrix Theory and Algorithms #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2407.00551

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\mathfraksl₂), and the enveloping algebra of the Onsager Lie algebra. In this paper, we introduce the S₃-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^{⊗ 3} of the standard module V into a module for an S₃-symmetric tridiagonal algebra. We investigate in detail the case in which Γ is a Hamming graph. We give some conjectures and open problems.

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