Gamma vectors as inverted Chebyshev expansions, type A to B transformations, and connections to algebraic structures

TL;DR

This paper explores the relationship between gamma vectors, Chebyshev polynomial expansions, and algebraic structures, revealing new connections in polynomial root theory, combinatorics, and Hopf algebras.

Contribution

It introduces a novel inverted Chebyshev expansion for gamma vectors of reciprocal polynomials and links it to geometric and algebraic combinatorics, including Coxeter complexes and Hopf algebras.

Findings

Gamma vector is given by an inverted Chebyshev basis expansion.

Characterization of real-rootedness via reciprocal polynomials and Chebyshev expansions.

Connections established between gamma vectors, simplicial complexes, and algebraic structures.

Abstract

Given a reciprocal/palindromic polynomial of even degree, we show that the gamma vector is essentially given by an inverted Chebyshev polynomial basis expansion. As an immediate consequence, we characterize real-rootedness of a linear combination of Chebyshev polynomials in terms of real-rootedness of that of the reciprocal polynomial built out of an inverted scaled tuple of the coefficients with one fixed and the rest divided by 2. It can be taken as a counterpart for arbitrary dimensions of a recent result of Bel-Afia--Meroni--Telen on hyperbolicity of Chebyshev curves with respect to the origin. In general, Chebyshev varieties serve as a counterpart of toric varieties in sparse polynomial root finding. Apart from this, the inverted Chebyshev expansion also yields connections between intrinsic properties of the gamma vector construction and the geometric combinatorics of simplicial complexes and posets. We find this by applying work of Hetyei on Tchebyshev subdivisions and Tchebyshev posets. In particular, we find that the gamma vector transformation is closely related to $f$-vectors of simplicial complexes resulting from successive edge subdivisions that transform the type A Coxeter complex to the type B Coxeter complex. Lifting to this to a modification of $cd$-indices, we show that the gamma vector inverted Chebyshev polynomial expansion lifts to a sum of (subdivisions of) cross polytopes which can be computed using (topological) descent statistics. While there are many examples where gamma positivity involving descent statistics, it is interesting to note we only assume the input polynomial is reciprocal/palindromic. Finally, Chebyshev polynomials of the second kind from derivatives give connections to Hopf algebras and quasisymmetric functions along with Lefschetz-type maps induced by $\mathfrak{sl}_2(\mathbb{C})$-representations.