On the Terwilliger Algebra and Quantum Adjacency Algebra of the Shrikhande Graph

Article Details

John Vincent S. Morales, john.vincent.morales@dlsu.edu.pj, Mathematics Department, De La Salle Lipa, Lipa City
Tessie M. Palma, , Mathematics and Statistics Department, De La Salle University, Manila

Journal: Manila Journal of Science
Volume 13 Issue 1 (Published: 2020-01-01)

Abstract

Let 𝑋 denote the vertex set of the Shrikhande graph. Fix 𝑥 ∈ 𝑋. Associated with 𝑥 is the Terwilliger algebra 𝑇=𝑇(𝑥) of the Shrikhande graph, a semisimple subalgebra of Mat𝑋(ℂ). There exists a subalgebra 𝑄= 𝑄 (𝑥) of 𝑇 that is generated by the lowering, flat, and raising matrices in 𝑇. The algebra 𝑄 is semisimple and is called the quantum adjacency algebra of the Shrikhande graph. Terwilliger & Zitnik (2019) investigated how 𝑄 and 𝑇 are related for arbitrary distance regular graphs using the notion of quasi isomorphism between irreducible 𝑇-­modules. Using their results, together with description of the irreducible 𝑇-­modules of the Shrikhande graph by Tanabe (1997), we show in this paper that for the Shrikhande graph, we have 𝑄≠ 𝑇.

Keywords: Terwilliger algebra, quantum adjacency algebra, Shrikhande graph, distance-­regular graph

DOI: https://www.dlsu.edu.ph/wp-content/uploads/pdf/research/journals/mjs/MJS13-2020/volume-1/MJS13-3-2020-morales-et-al.pdf
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