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Toric Bruhat interval polytopes. (English) Zbl 1467.52017

Summary: For two elements \(v\) and \(w\) of the symmetric group \(\mathfrak{S}_n\) with \(v \leq w\) in Bruhat order, the Bruhat interval polytope \(Q_{v , w}\) is the convex hull of the points \((z(1), \ldots, z(n)) \in \mathbb{R}^n\) with \(v \leq z \leq w\). It is known that the Bruhat interval polytope \(Q_{v , w}\) is the moment map image of the Richardson variety \(X_{w^{- 1}}^{v^{- 1}}\). We say that \(Q_{v , w}\) is toric if the corresponding Richardson variety \(X_{w^{- 1}}^{v^{- 1}}\) is a toric variety. We show that when \(Q_{v , w}\) is toric, its combinatorial type is determined by the poset structure of the Bruhat interval \([v, w]\) while this is not true unless \(Q_{v , w}\) is toric. We are concerned with the problem of when \(Q_{v , w}\) is (combinatorially equivalent to) a cube because \(Q_{v , w}\) is a cube if and only if \(X_{w^{- 1}}^{v^{- 1}}\) is a smooth toric variety. We show that a Bruhat interval polytope \(Q_{v , w}\) is a cube if and only if \(Q_{v , w}\) is toric and the Bruhat interval \([v, w]\) is a Boolean algebra. We also give several sufficient conditions on \(v\) and \(w\) for \(Q_{v , w}\) to be a cube.

MSC:

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52C45 Combinatorial complexity of geometric structures
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