A study of q-analogues of symmetric spaces in the framework of association schemes

关于P-和Q-多项式方案的Terweiger代数，猜想一个特征标公式对其不可约表示成立，而它们的不可约表示又具有对应于特征标公式分解的结构。这一猜想对于第一类情形是已知的，对于其余情形，即第二类、第三类情形是开放的。在本研究中，我们证明了第二类、第三类TD-对是L对的某种张量积，从而肯定地解决了这一猜想。结果表明，一个特征标公式不仅适用于类型I的TD对，而且也适用于类型II、III的TD对。

It is conjectured about the Terwilliger algebras of P- and Q-polynomoial schemes that a character formula holds for their irreducible representations, and in turn that their irreducible representations have a structure that corresponds to the decomposition of the character formula. This conjecture was known to be true for the case of type I and left open for the remaining cases, i.e., the cases of type II, III. In this study, we have settled the conjecture affirmatively by showing TD-pairs of type II, III are certain kind of tensor products of L-pairs. As a result, it is shown that a character formula holds not only for TD-pairs of type I, but also for those of type II, III.