Lifted generalized permutahedra and composition polynomials

TL;DR

This paper introduces a lifting construction for generalized permutahedra, connecting them to multiplihedra and nestomultiplihedra, and explores associated composition polynomials with positive integer coefficients.

Contribution

It defines a new lifting method for generalized permutahedra, relates it to important polytopes in homotopy theory, and studies the properties of resulting composition polynomials.

Findings

Lifting generalized permutahedra yields multiplihedra and nestomultiplihedra.

Constructs a subdivision indexed by compositions with polynomial volume pieces.

Composition polynomials have positive integer coefficients and may be unimodal.

Abstract

Generalized permutahedra are the polytopes obtained from the permutahedron by changing the edge lengths while preserving the edge directions, possibly identifying vertices along the way. We introduce a "lifting" construction for these polytopes, which turns an $n$-dimensional generalized permutahedron into an $(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff's multiplihedron from homotopy theory, and to the more general "nestomultiplihedra," answering two questions of Devadoss and Forcey. We construct a subdivision of any lifted generalized permutahedron whose pieces are indexed by compositions. The volume of each piece is given by a polynomial whose combinatorial properties we investigate. We show how this "composition polynomial" arises naturally in the polynomial interpolation of an exponential function. We prove that its coefficients are positive integers, and present evidence suggesting that they may also be unimodal.