On Lee association schemes over Z4Z4 and their Terwilliger algebra

Let F={0,1,2,3}F={0,1,2,3} and define the set K={K0,K1,K2}K={K0,K1,K2} of relations on F   such that (x,y)∈Ki(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)L(n) over Z4Z4 which is the extension of length n   of the initial scheme (F,K)(F,K). Let TT denote the Terwilliger algebra of L(n)L(n) with respect to the zero codeword of length n  . We show that TT is generated by a homomorphic image of the universal enveloping algebra of the Lie algebra sl3(C)sl3(C) and the center Z(T)Z(T). Furthermore, we determine the irreducible modules for TT using the Schur–Weyl duality.