Shard polytopes

TL;DR

This paper introduces shard polytopes, a new Minkowski sum-based method to realize quotient fans associated with lattice congruences of the weak order, extending the construction to type B with notable combinatorial and geometric properties.

Contribution

It presents a simplified Minkowski sum approach to construct quotient fans and polytopes, extending the theory to type B.

Findings

Shard polytopes have remarkable combinatorial properties.

The Minkowski sum approach extends to type B.

Provides a simpler realization of quotient fans and polytopes.

Abstract

For any lattice congruence of the weak order on permutations, N. Reading proved that gluing together the cones of the braid fan that belong to the same congruence class defines a complete fan, called a quotient fan, and V. Pilaud and F. Santos showed that it is the normal fan of a polytope, called a quotientope. In this paper, we provide a simpler approach to realize quotient fans 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 $B$.