The structure of nonthin irreducible \(T\)-modules of endpoint 1: Ladder bases and classical parameters
For \(Y = (X, \{R_i\}_{0 \leq i \leq d})\) a commutative association scheme, Terwilliger introduced a non-commutative algebra \(T\), the Terwilliger algebra, which reflects the combinatorial structure of \(Y\) [see \textit{P. Terwilliger}, J. Algebr. Comb. 1. No. 4, 363-388 (1992; Zbl 0785.05089)]. There is hope that these algebras contribute significantly to the classification of \(P\)- and \(Q\)-polynomial schemes. Terwilliger determined for the latter class of schemes a subclass of the irreducible \(T\)-modules, the so-called thin modules [see the reference above and J. Algebr. Comb. 2, No. 1, 73-103 (1993; Zbl 0785.05090); J. Algebr. Comb. 2, No. 2, 177-210 (1993; Zbl 0785.05091)]. The \(P\)- and \(Q\)-polynomial schemes which also have nonthin modules are the Doob and the classical forms schemes of diameter \(\geq 6\) [see the references above]. The irreducible \(T\)-modules of the Doob schemes were classified by \textit{K. Tanabe} [J. Algebr. Comb. 6, No. 2, 173-195 (1997; Zbl 0868.05056)]. In the present paper the authors investigate the irreducible \(T\)-modules of endpoint \(1\) of the classical forms schemes. They show that these modules always have a nice basis which they call ``ladder basis.'' They determine the ladder bases explicitly for the alternating, bilinear and Hermitian forms schemes and find also the matrices of the actions of the adjacency matrix \(A_1\) of the relation \(R_1\) on these modules.
scientific article; zbMATH DE number 1139617