The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in 3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let be a Galois ring of characteristic 4,X a cyclotomic scheme over with classD and the Terwilliger algebra ofX, then we show that most of the irreducible -modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible -modules using Jacobi sums.