On the Terwilliger Algebra and Quantum Adjacency Algebra of the Shrikhande Graph
J. Morales, T. Palma (pp. 31-39)
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 distanceregular graphs using the notion of quasiisomorphism 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.