On Lee association schemes over [formula omitted] and their Terwilliger algebra

Abstract

Let F={0,1,2,3} and define the set K={K0,K1,K2} of relations on F such that (x,y)∈Ki if and only if x−y≡±i(mod 4). Let n be a positive integer. We consider the Lee association scheme L(n) over Z4 which is the extension of length n of the initial scheme (F,K). Let T denote the Terwilliger algebra of L(n) with respect to the zero codeword of length n. We show that T is generated by a homomorphic image of the universal enveloping algebra of the Lie algebra sl3(C) and the center Z(T). Furthermore, we determine the irreducible modules for T using the Schur–Weyl duality.