The Terwilliger Algebra Of H(2m,2)' From The Viewpoint Of Its Automorphism Group Action

The Terwilliger algebra of distance-regular graphs(T-algebra for short),also known as subcomponent algebra,is an important theory introduced by Terwilliger in the 1990s,which has a revolutionary effect on the classification of associative schemes and distanceregular graphs.When n is even and n?4,the Hamming scheme H(n,2)has two P-polynomial structures and two Q-polynomial structures.Let H(2m,2)' denote the distance-regular graph which is induced by the Hamming scheme H(2m,2)whose the second P-polynomial structure and the first Q-polynomial structure are only considered.Let X denote the vertex set of H(2m,2)' and let T:=T(x0)denote the Terwilliger algebra of H(2m,2)'with respect to a fixed vertex x0?X.In this thesis,we study the algebra T and obtain the following results.1.We prove that T coincides with the centralizer algebra of the stabilizer of x0 in the automorphism group of H(2m,2)' by considering the action of this automorphism group on X×X×X,and obtain a basis of the Terwilliger algebra.Moreover,we display three subalgebras of T and give one basis for every subalgebra.We also show that the Terwilliger algebra of graph H(2m,2)' is equal to that of hypercube Q2m.2.We describe an orthogonal basis for every homogeneous component of the standard module V:=CX.Then the decomposition of T is given by using these homogeneous components.