The fixed space of the ribbon-to-homogeneous transition matrix of noncommutative symmetric functions: a proof of a conjecture recorded by J. M. Campbell

Abstract

We prove a conjecture recorded by J. M. Campbell in 2018 in the OEIS entry A001405 (the central binomial coefficients binomial(n, floor(n/2))). Let NSym be the algebra of noncommutative symmetric functions of Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon, and let C_n(R, H) be the transition matrix from the ribbon basis to the complete homogeneous basis of its degree-n component, a square matrix of order 2^(n-1) whose rows and columns are indexed by the compositions of n. The conjecture states that the kernel of C_n(R, H) - I has dimension binomial(n-1, floor((n-1)/2)), that is, A001405(n-1). We prove it for every n >= 1. The proof has three steps. First, indexing the compositions of n by their descent sets (subsets of {1, ..., n-1}), the matrix C_n(R, H) is the Moebius matrix of the Boolean lattice, and hence the (n-1)-fold Kronecker power of the 2 x 2 unipotent matrix [[1, 0], [-1, 1]]. Second, this Kronecker power is the exponential of the nilpotent operator D = sum of E^(i), where E = [[0, 0], [-1, 0]] acts in the i-th tensor factor, and the fixed space of C_n(R, H) is the kernel of D; up to conjugation, D is the action of a nilpotent element e of the Lie algebra sl_2(C) on the (n-1)-fold tensor power of its two-dimensional representation. Third, the kernel of e on a direct sum of irreducible representations of sl_2(C) has dimension equal to the number of summands, and a count of weights (equivalently, iterating the Clebsch-Gordan rule) shows that the (n-1)-fold tensor power of the two-dimensional representation has binomial(n-1, k) - binomial(n-1, k-1) summands of highest weight n-1-2k for 0 <= k <= (n-1)/2, which telescopes to binomial(n-1, floor((n-1)/2)) summands in all. The dimension does not depend on the conventions used to write the transition matrix (ordering of the compositions, transposition, or the inverse matrix expressing H in terms of R). The argument also gives the full Jordan form of C_n(R, H): binomial(n-1, k) - binomial(n-1, k-1) Jordan blocks of size n - 2k, for instance type (4, 2, 2) for n = 4. A second remark obtains the same count from the full-rank property of the up and down operators of the Boolean lattice (Gottlieb, Kantor). As of October 6, 2026 the statement was still marked as conjectural in the entry, and we have not found a proof in the literature. The proof is a direct application of the representation theory of sl_2(C); its interest lies in the identification of the transition matrix with a tensor power of a Jordan block. The note also explains the methodology of the author's project on open statements in the OEIS (translation of the statements into formulas for the reasoning engine SyntheticMind, with time budgets by type of computation; a log of obstacles that decides which methods to implement next; recursive splitting into subgoals; independent verification; a review by a second AI system) and the steps that led to this proof. The use of AI is described in the paper. Files: Blanco_Gomez_2026_Campbell_ribbon_matrix_EN.pdf is the paper; Blanco_Gomez_2026_Campbell_ribbon_matrix_ES.pdf is the Spanish version (same content); verify_campbell.py is the independent verification script (Python, with NumPy for the modular ranks; it builds the transition matrix from the ribbon expansion for n <= 12, checks its Kronecker structure, computes dim ker(C_n(R,H) - I) exactly for n <= 8 and modulo two primes for 9 <= n <= 12, obtaining the central binomial coefficients, checks the telescoping sum, and verifies the commutative image of the ribbon expansion through the Jacobi-Trudi determinant; it runs in about a minute and prints ALL CHECKS PASSED).