TL;DR
This paper proves that for any lattice congruence of the weak order on permutations, the associated fan is the normal fan of a polytope, extending previous work on quotient fans.
Contribution
It establishes that the fan constructed from lattice congruences of the weak order is always realizable as the normal fan of a polytope.
Findings
The fan is the normal fan of a polytope for all lattice congruences.
Generalizes previous results on quotient fans and polytopal realizations.
Provides a new geometric interpretation of lattice congruences in the weak order.
Abstract
For any lattice congruence of the weak order on , N. Reading proved that glueing together the cones of the braid fan that belong to the same congruence class defines a complete fan. We prove that this fan is the normal fan of a polytope.
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