The (k,n)(k,n)-hypersimplex is a convex polytope Δ(k,n)\Delta(k, n) (or Δ k,n\Delta_{k,n}) in n\mathbb{R}^n which is the convex hull of the (n k)\left(\array{n \\ k}\right) points e i 1+e i 2++e i ke_{i_1}+ e_{i_2}+ \cdots + e_{i_k}, 1i 1<i 2<<i kn1\leq i_1\lt i_2\lt \ldots \lt i_k\leq n, where the e je_{j} are the standard basis vectors. Δ(k,n)\Delta(k,n) may be seen as a convex hull of the barycenters of the (k1)(k-1)-dimensional faces of a(n1)a (n-1)-dimensional simplex. Special cases, Δ(1,n)\Delta(1,n) and Δ(n1,n)\Delta(n-1,n) are themselves simplices of dimension (n1)(n-1).

The combinatorics of hypersimplices extends the combinatorics of distinguished triangles and octahedra in the standard triangulated categories; in fact they are postulated in Maltsiniotis’s strong version of a triangulated category. The octahedron is a (2,4)(2,4)-hypersimplex. Higher hypersimplices were indeed obtained from A A_\infty-enrichments by Volodymyr Lyubashenko, so one could expect that they can also be obtained from the homotopy category of a stable (∞,1)-category. There are also connections between distinguished hypersimplices and Postnikov towers in triangulated categories.

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Last revised on August 8, 2012 at 18:17:06. See the history of this page for a list of all contributions to it.