Irreducible T-module Research Articles

A Generalization of the Terwilliger Algebra

P. M. Terwilliger (1992, J. Algebraic Combin.1, 363–388) considered the C-algebra generated by a given Bose Mesner algebra M and the associated dual Bose Mesner algebra M*. This algebra is now known as the Terwilliger algebra and is usually denoted by T. Terwilliger showed that each vanishing intersection number and Krein parameter of M gives rise to a relation on certain generators of T. These relations determine much of the structure of T, thought not all of it in general. To illuminate the role these relations play, we consider a certain generalization T of T. To go from T to T, we replace M and M* with a pair of dual character algebras C and C*. We define T by generators and relations; intuitively T is the associative C-algebra with identity generated by C and C* subject to the analogues of Terwilliger's relations. T is infinite dimensional and noncommutative in general. We construct an irreducible T-module which we call the primary module; the dimension of this module is the same as that of C and C*. We find two bases of the primary module; one diagonalizes C and the other diagonalizes C*. We compute the action of the generators of T on these bases. We show T is a direct sum of two sided ideals T0 and T1 with T0 isomorphic to a full matrix algebra. We show that the irreducible module associated with T0 is isomorphic to the primary module. We compute the central primitive idempotent of T associated with T0 in terms of the generators of T.

Bipartite Distance-regular Graphs, Part II

This is a continuation of “Bipartite Distance-regular Graphs, Part I”. We continue our study of the Terwilliger algebra T of a bipartite distance-regular graph. In this part we focus on the thin irreducible T-modules of endpoint 2 and on those distance-regular graphs for which every irreducible T-module of endpoint 2 is thin.

The Terwilliger algebras of bipartite P- and Q-polynomial schemes

Let Y denote a D-class symmetric association scheme with D ⩾ 3, and suppose Y is bipartite P- and Q-polynomial. Let T denote the Terwilliger algebra with respect to any vertex x. We prove that any irreducible T-module W is both thin and dual-thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T-module is determined by two parameters, the endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the irreducible T-modules occur in the standard module. Using this recurrence, we produce formulas for the multiplicities of the irreducible T-modules with endpoint at most four.