Shard polytopes

Arnau Padrol (1) , Vincent Pilaud (2, 3) , Julian Ritter (3)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 CNRS - Centre National de la Recherche Scientifique
3 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

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. In this paper, we provide an alternative simpler approach to realize this quotient fan 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, this Minkowski sum approach extends to type $B$.

Domaines: Combinatoire [math.CO]

Arnau Padrol, Vincent Pilaud, Julian Ritter. Shard polytopes. international mathematical research notices, In press, 2023 (9), pp.7686-7796. ⟨10.1093/imrn/rnac042⟩. ⟨hal-02997662⟩