Shape character formula conjecture for tridiagonal pairs

Prove that for every tridiagonal pair (A, A*) over the complex numbers with diameter d, there exist integers N ≥ 1 and nonnegative integers ell_1, …, ell_N such that the shape multiplicities rho_0, …, rho_d satisfy the polynomial identity sum_{i=0}^{d} rho_i λ^i = ∏_{p=1}^N (1 − λ^{ell_p+1})/(1 − λ), and consequently d = ell_1 + … + ell_N.

Background

In the theory of tridiagonal pairs (TD-pairs), the shape is the tuple of multiplicities (ρ_0, ρ_1, …, ρ_d) of the eigenspaces of the two diagonalizable linear transformations A and A*. Over the complex numbers, TD-pairs are sharp (ρ_0 = 1), and these multiplicities satisfy several symmetry constraints.

A central conjecture asserts that the generating polynomial of the shape factors as a product determined by an N-tuple of nonnegative integers (ell_1, …, ell_N). Establishing this formula would classify possible shapes of TD-pairs and relate the diameter d to the sum of these integers. Although the paper constructs families of TD-pairs consistent with such tuples, the general validity of the character formula for all TD-pairs remains conjectural.

References

It is conjectured that, for any TD-pair, there exist N integers \ell_1,\ell_2,\dots,\ell_N\geq 0, N\geq 1 such that the shape satisfies the character formula \begin{align}\label{eq:rhol} \sum_{i=0}{d} \rho_i \lambdai = \prod_{p=1}N \frac{1-\lambda{\ell_p+1}{1-\lambda}\,. \end{align} In particular, the diameter $d$ satisfies $d=\ell_1+\dots+ \ell_N$.

— Change of basis for the tridiagonal pairs of type II  (2503.01231 - Crampe et al., 3 Mar 2025) in Section 2 (Tridiagonal pairs of type II)