Existence of split bases for type II tridiagonal pairs with general shapes

Establish, for tridiagonal pairs of type II whose shape is not (1, 2, …, 2, 1), the existence of a split basis in which A acts as an upper triangular block bidiagonal matrix and A* acts as a lower triangular block bidiagonal matrix, analogous to the known existence in the (1, 2, …, 2, 1) case.

Background

The split basis is a canonical basis used in the analysis of TD-pairs, where A and A* take complementary triangular block-bidiagonal forms. It underpins explicit diagonalization and change-of-basis computations.

For type II TD-pairs with shape (1, 2, …, 2, 1), Ito and Sato proved the existence of such a split basis. The paper notes that extending this existence result to other shapes remains open, which is necessary to generalize the structural and spectral analysis of type II TD-pairs beyond this specific shape class.

References

In , it was also proven that there always exists such a split basis for all tridiagonal pairs of type II with shape $(1,\underbrace{2,2,\dots,2}_{\ell},1)$. For other shapes, similar results remain an open problem.

— Change of basis for the tridiagonal pairs of type II  (2503.01231 - Crampe et al., 3 Mar 2025) in Example (Subsection ‘Split basis’, Section 3)