nLab hypersimplex

Idea

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

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 $(2,4)$-hypersimplex. Higher hypersimplices were indeed obtained from $A_\infty$-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.

References

• 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.

• I. M. Gelfand, R. M. Goresky, R. D. MacPherson, V. V. Serganova?, Combinatorial geometries, convex polyhedra, and Schubert cells, Advances in Math. 63 (1987) 301–316 doi

• 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

• Thomas Lam, Alexander Postnikov, Alcoved polytopes I, math.CO/0501246

• Yu. Bespalov, Volodymyr Lyubashenko, O. Manzyuk, Pretriangulated $A_\infty$-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)

• Alexander Beilinson, Joseph Bernstein, Pierre 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)