Ring of Polytopes, Quasi-symmetric functions and Fibonacci numbers

الأصل · EN

In this paper we study the ring P of combinatorial convex polytopes. We introduce the algebra of operators D generated by the operators dₖ that send an n-dimensional polytope Pⁿ to the sum of all its (n-k)-dimensional faces. It turns out that D is isomorphic to the universal Leibnitz-Hopf algebra with the antipode χ(dₖ)=(-1)ᵏdₖ. Using the operators dₖ we build the generalized f-polynomial, which is a ring homomorphism from P to the ring [t₁,t₂,...][α] of quasi-symmetric functions with coefficients in Z[α]. 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(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 lα[α] such that F(lα(P))=f(P)*, where F is the Ehrenborg quasi-symmetric function.