Removahedral congruences versus permutree congruences

TL;DR

This paper explores the geometric realization of permutree fans as removahedra, revealing unique properties and providing a complete description of their type cone, thus advancing the understanding of permutree and quotient fan structures.

Contribution

It demonstrates that permutree fans are uniquely realizable as removahedra and characterizes their type cone, extending the classical associahedron construction.

Findings

Permutree fans are the only quotient fans realized by removahedra.

Any permutree fan can be realized from any braid fan realization.

Complete description of the type cone of permutree fans.

Abstract

The associahedron is classically constructed as a removahedron, i.e. by deleting inequalities in the facet description of the permutahedron. This removahedral construction extends to all permutreehedra (which interpolate between the permutahedron, the associahedron and the cube). Here, we investigate removahedra constructions for all quotientopes (which realize the lattice quotients of the weak order). On the one hand, we observe that the permutree fans are the only quotient fans realized by a removahedron. On the other hand, we show that any permutree fan can be realized by a removahedron constructed from any realization of the braid fan. Our results finally lead to a complete description of the type cone of the permutree fans.