Gram determinant factorization for Okada cell modules

Prove an explicit factorization of the Gram determinant of the invariant bilinear form on each Okada cell module V^S in terms of biserial clone Schur functions.

Background

The paper constructs a cellular basis for the Okada algebra Okada(X,Y) and associates a cell module VS to each Fibonacci set S. Each cell module carries a canonical invariant bilinear form, whose Gram determinant detects whether the form has a nonzero radical and consequently whether the cell module is simple.

The authors state that the expected factorization should be expressed through biserial clone Schur functions, extending analogous factorizations known for Temperley–Lieb, blob, and related diagram algebras. They identify this factorization as a conjectural part of the representation-theoretic development of Okada algebras.

References

The bilinear form φ_S attached to the cell module VS is addressed in~\cref{sec:clone} and we conjecture an explicit factorization of the associated Gram determinant \det GS_N in terms of biserial clone Schur functions in \ref{conj::SGram-Determinant}.

— Diagrammatic Okada monoid and cellularity of the Okada algebra  (2609.01440 - Hivert et al., 1 Sep 2026) in Section 1, subsection “Cellularity vis-à-vis Green structure”