Ring of Polytopes, Quasi-symmetric functions and Fibonacci numbers

Abstract

In this paper we study the ring $\mathcal{P}$ of combinatorial convex polytopes. We introduce the algebra of operators $\mathcal{D}$ generated by the operators $d_k$ that send an $n$-dimensional polytope $P^n$ to the sum of all its $(n-k)$-dimensional faces. It turns out that $\mathcal{D}$ is isomorphic to the universal Leibnitz-Hopf algebra with the antipode $\chi(d_k)=(-1)^kd_k$. Using the operators $d_k$ we build the generalized $f$-polynomial, which is a ring homomorphism from $\mathcal{P}$ to the ring $\Qsym[t_1,t_2,...][\alpha]$ of quasi-symmetric functions with coefficients in $\mathbb Z[\alpha]$. The images of two polytopes coincide if and only if their flag $f$-vectors are equal. We describe the image of this homomorphism over the integers and prove that over the rationals it is a free polynomial algebra with dimension of the $n$-th graded component equal to the $n$-th Fibonacci number. This gives a representation of the Fibonacci series as an infinite product. The homomorphism is an isomorphism on the graded group $BB$ generated by the polytopes introduced by Bayer and Billera to find the linear span of flag $f$-vectors of convex polytopes. This gives the group $BB$ a structure of the ring isomorphic to $f(\mathcal{P})$. We show that the ring of polytopes has a natural Hopf comodule structure over the Rota-Hopf algebra of posets. As a corollary we build a ring homomorphism $l_{\alpha}\colon\mathcal{P}\to\mathcal{R}[\alpha]$ such that $F(l_{\alpha}(P))=f(P)^*$, where $F$ is the Ehrenborg quasi-symmetric function.