On cell modules of symmetric cellular algebras

Let A be a symmetric cellular algebra with cell datum (Λ, M, C, i) and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda_1=\{\lambda \in \Lambda_0 \mid W(\lambda) \, {\rm is \, simple}\}}$$\end{document}. We prove that Λ1 consists of two parts: one gives a lower bound for the cardinality of the set of cell modules with zero bilinear forms and the other parametrizes all the projective cell modules. Moreover, it is proved in Li (arxiv: math0911.3524, 2009) that the dual basis of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{C_{S, T}^{\lambda} \mid \lambda \in \Lambda, S,T \in M(\lambda)\}}$$\end{document} is again cellular. In this paper, we will study the cell modules defined by dual basis. In particular, we study the dual basis of the Murphy basis.