# Shard polytopes

Abstract : 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 $B$.
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Preprints, Working Papers, ...

Cited literature [97 references]

https://hal.archives-ouvertes.fr/hal-02997662
Contributor : Vincent Pilaud <>
Submitted on : Tuesday, November 10, 2020 - 10:53:59 AM
Last modification on : Tuesday, December 8, 2020 - 10:55:51 AM

### File

shardsumotopes.pdf
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### Identifiers

• HAL Id : hal-02997662, version 1
• ARXIV : 2007.01008

### Citation

Arnau Padrol, Vincent Pilaud, Julian Ritter. Shard polytopes. 2020. ⟨hal-02997662⟩

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