nLab
hypersimplex

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.

  • I.M.Gelfand, R.D.MacPherson, Geometry in Grassmannians and a generalization of the dilogarithm, Adv. Math., 44 A982), 279–312; Coll. pap. of I.M. Gelfand v. 3, Springer 1989, p. 492–525.

  • I. M. Gelfand, M. Kapranov, A. Zelevinsky, Discriminants, resultants, and multidimensional determinants, 1994.

  • Sergei Gelfand, Yuri Manin, Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original. Springer 1996. xviii+372 pp.; 2nd corrected ed. 2002.

  • Fred J. Rispoli, The graph of the hypersimplex, arxiv/0811.2981

  • T. Lam, A. Postnikov, Alcoved polytopes I, math.CO/0501246

  • Yu. Bespalov, V. Lyubashenko, O. Manzyuk, Pretriangulated A A_\infty-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)

  • A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1982). MR 86g:32015 (diagrams in Rem. 1.1.14 are hypersimplices)

  • Georges Maltsiniotis, Catégories triangulées supérieures, Pré-preprint ps (2005)

  • Jacob Lurie, Derived Algebraic Geometry I: Stable \infty-Categories, (arXiv, pdf)

Revised on August 8, 2012 18:17:06 by Andrew Stacey (192.76.7.219)