Chebyshev quotients, Demazure multiplicities, and Dyck-path models

Abstract: We study Chebyshev quotients that arise in the representation theory of Lie algebras, specifically within the theory of Demazure flags for fusion products of $\mathfrak{sl}_2[t]$-modules. Motivated by a recent formula that expresses certain Demazure multiplicities as coefficients of such quotients, we prove a general eventual non-negativity theorem: each quotient either terminates or has strictly positive coefficients for sufficiently large degrees, which we in turn interpret in terms of matchings and bounded walks. In several natural infinite families, these are unsigned bounded Dyck path models, giving both a structural explanation for the observed positivity phenomenon and concrete combinatorial models for key families of Demazure multiplicities. The theorems in this paper were autonomously produced and formalized in Lean/Mathlib by AxiomProver from natural-language statements.

• The paper proposes a novel analytic framework that links Chebyshev quotient generating functions with Demazure multiplicity polynomials in sl2[t] representations.
• It establishes a structural dichotomy where coefficients either terminate due to cancellation or become strictly positive based on Chebyshev polynomial root analysis.
• Explicit combinatorial models using path graph matchings and bounded (Dyck) lattice walks are provided, with main theorems formally verified by AI.

Chebyshev Quotients, Demazure Multiplicities, and Combinatorial Dyck Path Models

Overview

This paper develops a systematic analysis of Chebyshev-quotient generating functions arising in the representation theory of current algebras, particularly in the context of Demazure flags for fusion products of $\mathfrak{sl}_2[t]$-modules. Motivated by formulas that express Demazure multiplicities as coefficients of rational functions involving Chebyshev polynomials, the authors establish a robust structural dichotomy: the coefficient sequences of these quotients either terminate or are eventually strictly positive. The paper provides explicit, notably signed combinatorial interpretations for these coefficients, utilizing the language of path graph matchings and bounded lattice walks. Furthermore, for key infinite families, the combinatorics simplify to unsigned bounded Dyck path models, directly encoding numerical Demazure multiplicities.

An additional notable aspect is that all the main theorems were autonomously formalized and proved by the AxiomProver AI, with formal verification in Lean/Mathlib.

Chebyshev-Quotient Formula and Demazure Multiplicities

A central analytic object is the graded multiplicity polynomial for Demazure flag filtrations,

$$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$

where $\xi$ is a partition and $V(\xi)$ is the associated fusion product. The focus is on the specialization at $q=1$, yielding integer multiplicities relevant both for explicit representation-theoretic decomposition and for analytic study.

The underlying Chebyshev-quotient formula is expressed (see Proposition~1.1 in the paper) as

$$V_\mu^{\xi\to m}(1) = \left[ x^{(|\xi| - \mu)/2} \right] \frac{p_{m-\mu_0-1}(x) p_\xi(x)}{p_m(x)^{\mu_1 + 1}}$$

where $p_r(x)$ is a family of polynomials defined by:

$$p_0(x) = p_1(x) = 1, \quad p_{r+1}(x) = p_r(x) - x p_{r-1}(x) \quad (r \geq 1)$$

and $p_\xi(x) = \prod_i p_{\xi_i}(x)$, with $\mu$ written uniquely as $$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$0, $$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$1.

This compact rational function serves as a generating function whose relevant coefficient yields the desired numerical multiplicity.

Eventual Positivity Dichotomy

A principal contribution is a structural dichotomy for the coefficient sequence of the Chebyshev quotient $$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$2:

• Termination: If enough polynomial cancellations occur—namely, if the number of size-$$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$3 parts in $$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$4 is at least $$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$5—then $$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$6 is a genuine polynomial and the coefficient sequence terminates.
• Strict Eventual Positivity: Otherwise, the Chebyshev quotient is genuinely rational with simple positive roots (analyzed via roots of $$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$7) and, for sufficiently large degree, all coefficients are strictly positive.

The dichotomy is established using detailed root analysis of $$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$8, linking coefficients via Cauchy's formula to residues at the smallest positive root, ensuring positivity after a finite range.

Explicit Signed Combinatorial Models

To provide concrete and computationally effective interpretations, the authors derive explicit signed combinatorial expansions for the coefficients of the general Chebyshev quotient.

Each numerator factor $$V_n^{\xi\to m}(q) = \sum_{p \ge 0} [V(\xi) : \tau_p^* D(m, n)]\, q^p$$9 corresponds (via classical results on matching polynomials) to the signed count of matchings in a path graph of length $\xi$0. Each denominator factor $\xi$1 is analyzed as a generating function for bounded walks (so-called full-height strip walks) in a lattice path model.

This leads to a precise formula for the coefficient $\xi$2 as the signed count of tuples, with the sign determined by the combined parity of the sizes of the matchings:

• Tuples of matchings in specific path graphs and bounded walk ensembles, with total combinatorial "weight" summing to $\xi$3.

The combinatorial model elegantly links the Chebyshev quotient under study directly to structures amenable to enumeration and explicit computation.

Unsigned Dyck Path Models in Key Families

For natural classes of partitions $\xi$4 (notably those with only small parts and large enough denominators), the complicated signed combinatorial expressions simplify to unsigned counts. In these cases, the Chebyshev quotient can be factorized such that all signs cancel, yielding direct enumeration in terms of bounded Dyck paths:

• The coefficients $\xi$5 count $\xi$6-tuples of Dyck paths confined under certain height, first-step, and last-step constraints, subject to a total semilength condition.
• Three infinite families are identified where this construction applies, parametrized by partitions of the form
  • $\xi$7,
  • $\xi$8, and
  • $\xi$9,
  • with constraints on $V(\xi)$0 and the $V(\xi)$1.

This provides explicit, positive formulae for relevant Demazure multiplicities, making complex representation-theoretic multiplicity questions explicitly combinatorial.

Chebyshev Roots, Bounded Walks, and Transfer-Matrix Methods

The analysis makes essential use of the roots of the Chebyshev-type polynomials, which are shown to be real, simple, and positive. Moreover, the polynomials $V(\xi)$2 coincide (up to a change of variable) with the matching polynomials of path graphs, allowing the transfer of combinatorial interpretations from spectral graph theory to the representation-theoretic context.

The generating functions for bounded lattice walks (strip walks) are obtained via transfer-matrix arguments and Cramer's rule for path graph adjacency matrices, producing continued fraction expansions and explicit generating series.

Implications and Future Directions

The findings have several implications:

• Computability: The combinatorial models make large classes of Demazure multiplicities explicitly computable.
• Conceptual Transparency: They reveal structural reasons for observed positivity and explain phenomena arising in affine fusion module representation theory in terms of standard combinatorial families.
• AI-Mathematics Synergy: The autonomous production and formal verification of the main theorems demonstrates the maturity of AI-assisted mathematical research, addressing both the formal rigor and the automation of human-level mathematical proofs.

One open direction concerns the statistic on the Dyck-path model that would recover the fully graded multiplicity polynomials (not just the specialization at $V(\xi)$3). While similar results for admissible Dyck paths exist in the literature (e.g., via the co-major index), the present path model does not yet have an identified appropriate statistic, constituting a compelling problem for future investigation.

Conclusion

This work transforms the Chebyshev quotient formula for Demazure multiplicities for $V(\xi)$4 fusion products from a compact, abstract character-theoretic expression to a structural and combinatorial description, complete with explicit positivity criteria and combinatorial objects amenable to direct enumeration. The results refine and generalize prior representation-theoretic descriptions by identifying eventual sign behavior and, in important special cases, reducing to explicit Dyck-path enumeration. The interplay of combinatorics, Lie theory, and formal AI-driven theorem proving marks an advance in both methodology and conceptual understanding of current algebra module multiplicities.