The -hypersimplex is a convex polytope (or ) in which is the convex hull of the points , , where the are the standard basis vectors. may be seen as a convex hull of the barycenters of the -dimensional faces of -dimensional simplex. Special cases, and are themselves simplices of dimension .
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 -hypersimplex. Higher hypersimplices were indeed obtained from -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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Fred J. Rispoli, The graph of the hypersimplex, arxiv/0811.2981
T. Lam, A. Postnikov, Alcoved polytopes I, math.CO/0501246
Yu. Bespalov, Volodymyr Lyubashenko, O. Manzyuk, Pretriangulated -categories, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 (ps.gz) (chapter 13)
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