Construction of dual bases for Fuss-Catalan modules

Abstract

We introduce a systematic method to derive the dual bases associated with certain cell modules of the Fuss–Catalan algebras with respect to the canonical bilinear form. Our approach is based on a detailed and explicit analysis of the corresponding Gram matrices, together with the computation of their inverses. By exploiting the cellular structure of the algebra, we obtain concrete formulas for the dual basis elements, expressed directly in terms of diagrammatic data. This construction provides an effective computational framework inside the cell modules and clarifies the role played by the bilinear form in their representation theory. The resulting dual basis proves to be a powerful tool for structural investigations of Fuss–Catalan algebras. In particular, it plays a fundamental role in the explicit determination of central primitive idempotents and facilitates the computation of characters associated with simple modules. Moreover, the method developed here highlights the close relationship between Gram matrix degeneracy, parameter specialisation, and the appearance of reducible cell modules. These results offer new insights into the internal structure of Fuss–Catalan algebras and provide a foundation for further applications to related diagram algebras and their representation theory.