Shard polytopes

For any lattice congruence of the weak order on permutations, N. Reading proved that glueing together the cones of the braid fan that belong to the same congruence class defines a complete fan, called quotient fan, and V. Pilaud and F. Santos showed that it is the normal fan of a polytope, called quotientope. We provide an alternative simpler approach based on Minkowski sums of elementary polytopes, called shard polytopes, which have remarkable combinatorial and geometric properties. In contrast to the original construction of quotientopes, our approach extends to type B.