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On Lee association schemes over \(\mathbb{Z}_4\) and their Terwilliger algebra. (English) Zbl 1352.05197
Summary: Let \(F = \{0, 1, 2, 3 \}\) and define the set \(K = \{K_0, K_1, K_2 \}\) of relations on \(F\) such that \((x, y) \in K_i\) if and only if \(x - y \equiv \pm i\pmod 4\). Let \(n\) be a positive integer. We consider the Lee association scheme \(L(n)\) over \(\mathbb{Z}_4\) which is the extension of length \(n\) of the initial scheme \((F, K)\). Let \(\mathcal{T}\) denote the Terwilliger algebra of \(L(n)\) with respect to the zero codeword of length \(n\). We show that \(\mathcal{T}\) is generated by a homomorphic image of the universal enveloping algebra of the Lie algebra \(\mathfrak{sl}_3(\mathbb{C})\) and the center \(Z(\mathcal{T})\). Furthermore, we determine the irreducible modules for \(\mathcal{T}\) using the Schur-Weyl duality.

MSC:
05E30 Association schemes, strongly regular graphs
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A04 Linear transformations, semilinear transformations
20C30 Representations of finite symmetric groups
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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