Polytopes, Hopf algebras and Quasi-symmetric functions

Abstract

In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group \(\mathcal{P}\) generated by all combinatorial polytopes. There are two natural bilinear operations on this group defined by a direct product \(\times \) and a join \(\divideontimes\) of polytopes. \((\mathcal{P},\times)\) is a commutative associative bigraded ring of polynomials, and \(\mathcal{RP}=(\mathbb Z\varnothing\oplus\mathcal{P},\divideontimes)\) is a commutative associative threegraded ring of polynomials. The ring \(\mathcal{RP}\) has the structure of a graded Hopf algebra. It turns out that \(\mathcal{P}\) has a natural Hopf comodule structure over \(\mathcal{RP}\). Faces operators \(d_k\) that send a polytope to the sum of all its \((n-k)\)-dimensional faces define on both rings the Hopf module structures over the universal Leibnitz-Hopf algebra \(\mathcal{Z}\). This structure gives a ring homomorphism \(\R\to\Qs\otimes\R\), where \(\R\) is \(\mathcal{P}\) or \(\mathcal{RP}\). Composing this homomorphism with the characters \(P^n\to\alpha^n\) of \(\mathcal{P}\), \(P^n\to\alpha^{n+1}\) of \(\mathcal{RP}\), and with the counit we obtain the ring homomorphisms \(f\colon\mathcal{P}\to\Qs[\alpha]\), \(f_{\mathcal{RP}}\colon\mathcal{RP}\to\Qs[\alpha]\), and \(\F^*:\mathcal{RP}\to\Qs\), where \(F\) is the Ehrenborg transformation. We describe the images of these homomorphisms in terms of functional equations, prove that these images are rings of polynomials over \(\mathbb Q\), and find the relations between the images, the homomorphisms and the Hopf comodule structures. For each homomorphism \(f,\;f_{\mathcal{RP}}\), and \(\F\) the images of two polytopes coincide if and only if they have equal flag \(f\)-vectors. Therefore algebraic structures on the images give the information about flag \(f\)-vectors of polytopes.