bibkey: labelle2025toda authors: Labelle, A. year: 2025 title: On a specialization of Toda eigenfunctions doi: 10.48550/arXiv.2502.10655 url: https://arxiv.org/abs/2502.10655v3 claim: Conjecture 7.3 states that the polynomial (q)_α² J_α is unimodal. strata_touched:

• D5/S0/Certificates/Combinatorics/TodaSpecializationUnimodalityRefutation license: citation-only triage: anchor

Labelle’s Toda specialization

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DOI: 10.48550/arXiv.2502.10655. Primary version: https://arxiv.org/abs/2502.10655v3. The mathematical text is available at https://arxiv.org/html/2502.10655v3. Section 1, Definition 1.1 and equation (2), supplies the fermionic recursion and the displayed q-factor. Section 7, Conjecture 7.3, page 21, states the unimodality assertion in the paper’s general root-system setting.

Statement and scope

Section 1, page 1, states:

Let $\mathfrak{g}$ be a split semisimple Lie algebra over $\mathbb{Q}$ with Cartan subalgebra $\mathfrak{h}$ and let $\Phi \subseteq {\mathfrak{h}}^{\ast }$ be its root system. Fix a choice of positive roots ${\Phi }_{+}\subseteq \Phi$ and let ${\alpha }_1,\dots ,{\alpha }_n$ be the corresponding simple roots. … Let $(\cdot ,\cdot )$ be the invariant bilinear form on ${\mathfrak{h}}^{\ast }$, normalized so that short roots have length $2$. Let $d_i=\frac{({\alpha }_i,{\alpha }_i)}{2}$ and let $a_{ij}=\frac{({\alpha }_i,{\alpha }_j)}{d_i}$ be the entries of the Cartan matrix.

Definition 1.1, page 1, states:

Define ${\mathfrak{J}}_{\alpha }\in \mathbb{Q}(q)$, for $\alpha \in Q^{\ge 0}$, by ${\mathfrak{J}}_{\alpha }={\sum }_{0\le \beta \le \alpha }\frac{q^{\frac{(\beta ,\beta )}{2}}}{(q{)}_{\alpha -\beta }}{\mathfrak{J}}_{\beta }$ and ${\mathfrak{J}}_0=1,$ where $(q{)}_{\alpha }={\prod }_{i=1}^n{\prod }_{j=1}^{a_i}(1-q^{d_{i}j})$ for $\alpha =\sum a_i{\alpha }_i\in Q^{\ge 0}$.

Conjecture 7.3, page 21, states:

The polynomial $(q{)}_{\alpha }^2{\mathfrak{J}}_{\alpha }$ is unimodal.

The C₂ datum uses short-root Gram matrix [[2, −2], [−2, 4]], d = (1, 2), and α = 2α₁ + 2α₂. The certificate polynomial is 1 + q + 3q² + 2q³ + 5q⁴ + 2q⁵ + 3q⁶ + q⁷ + q⁸, whose coefficients contain the strict valley 3, 2, 5.

Equation (2) on page 1 moves the self-term to the left and gives, for α>0,

$${\mathfrak{J}}_{\alpha }=\frac{1}{1-q^{(\alpha ,\alpha )/2}}\sum _{0\le \beta <\alpha }\frac{q^{(\beta ,\beta )/2}}{(q{)}_{\alpha -\beta }}{\mathfrak{J}}_{\beta }.$$

The module defines this recursion in RatFunc ℚ on normalized finite-type symmetrizable Cartan data of every finite rank. Coordinates are functions Fin r → ℕ, and the Gram quadratic form is even on integral coordinates. The exponent is its exact integer half, used with integer powers in RatFunc ℚ. Unimodality uses the finite coefficient list through natDegree. The nine C₂ certificate identities are consequences of this recursion, rather than defining values of J. An existential polynomial with the indicated rational-function image uniquely represents the polynomial because the algebra map ℚ[X] → RatFunc ℚ is injective.