Hypersimplex

Standard hypersimplices in ℝ3 |

|

Δ3,1
Hyperplane: x+y+z=1 | Δ3,2
Hyperplane: x+y+z=2

In polyhedral combinatorics, the hypersimplex Δd,k is a convex polytope that generalizes the simplex. It is determined by two integers d and k, and is defined as the convex hull of the d-dimensional vectors whose coefficients consist of k ones and d−k zeros. Equivalently, Δd,k can be obtained by slicing the d-dimensional unit hypercube [0,1]d with the hyperplane of equation x1+⋯+xd=k and, for this reason, it is a (d−1)-dimensional polytope when 0<k<d.[1]

## Contents

* 1 Properties
* 2 Alternative constructions
* 3 Examples
* 4 History
* 5 References
* 6 Further reading

## Properties

The number of vertices of Δd,k is (dk).[1] The graph formed by the vertices and edges of the hypersimplex Δd,k is the Johnson graph J(d,k).[2]

## Alternative constructions

An alternative construction (for k≤d/2) is to take the convex hull of all (d−1)-dimensional (0,1)-vectors that have either k−1 or k nonzero coordinates. This has the advantage of operating in a space that is the same dimension as the resulting polytope, but the disadvantage that the polytope it produces is less symmetric (although combinatorially equivalent to the result of the other construction).

The hypersimplex Δd,k is also the matroid polytope for a uniform matroid with d elements and rank k.[3]

## Examples

The hypersimplex Δd,1 is a (d−1)-simplex (and therefore, it has d vertices). The hypersimplex Δ4,2 is an octahedron, and the hypersimplex Δ5,2 is a rectified 5-cell.

Generally, the hypersimplex, Δd,k, corresponds to a uniform polytope, being the (k−1)-rectified (d−1)-dimensional simplex, with vertices positioned at the center of all the (k−1)-dimensional faces of a (d−1)-dimensional simplex.

Examples (d = 3...6) Name  | Equilateral
triangle | Tetrahedron
(3-simplex)  | Octahedron | 5-cell
(4-simplex)  | Rectified
5-cell | 5-simplex | Rectified
5-simplex  | Birectified
5-simplex

|
|
|
|
|
|
|
|

Δd,k = (d,k)
= (d,d − k) | (3,1)
(3,2)  | (4,1)
(4,3)  | (4,2)  | (5,1)
(5,4)  | (5,2)
(5,3)  | (6,1)
(6,5)  | (6,2)
(6,4)  | (6,3)
Vertices
(dk) | 3 | 4 | 6 | 5 | 10 | 6 | 15 | 20
d-coordinates  | (0,0,1)
(0,1,1)  | (0,0,0,1)
(0,1,1,1)  | (0,0,1,1)  | (0,0,0,0,1)
(0,1,1,1,1)  | (0,0,0,1,1)
(0,0,1,1,1)  | (0,0,0,0,0,1)
(0,1,1,1,1,1)  | (0,0,0,0,1,1)
(0,0,1,1,1,1)  | (0,0,0,1,1,1)
Image  |  |  |  |  |  |  |  |
Graphs |
J(3,1) = K2 |
J(4,1) = K3 |
J(4,2) = T(6,3) |
J(5,1) = K4 |
J(5,2)  |
J(6,1) = K5 |
J(6,2)  |
J(6,3)
Coxeter
diagrams  |

|  |

Schläfli
symbols | {3}
= r{3}  | {3,3}
= 2r{3,3}  | r{3,3} = {3,4}  | {3,3,3}
= 3r{3,3,3}  | r{3,3,3}
= 2r{3,3,3}  | {3,3,3,3}
= 4r{3,3,3,3}  | r{3,3,3,3}
= 3r{3,3,3,3}  | 2r{3,3,3,3}
Facets | { }  | {3}  | {3,3}  | {3,3}, {3,4}  | {3,3,3}  | {3,3,3}, r{3,3,3}  | r{3,3,3}

## History

The hypersimplices were first studied and named in the computation of characteristic classes (an important topic in algebraic topology), by (Gabrièlov Gelʹfand).[4][5]

## References

1. ↑ 1.0 1.1 Miller, Ezra; Reiner, Victor; Sturmfels, Bernd, Geometric Combinatorics, IAS/Park City mathematics series, 13, American Mathematical Society, p. 655, ISBN 9780821886953, https://books.google.com/books?id=W_SPdwfPTw8C&pg=PA655 .
2. ↑ Rispoli, Fred J. (2008), The graph of the hypersimplex, Bibcode: 2008arXiv0811.2981R .
3. ↑ "Cardinality homogeneous set systems, cycles in matroids, and associated polytopes", The Sharpest Cut: The Impact of Manfred Padberg and His Work, MPS/SIAM Ser. Optim., SIAM, Philadelphia, PA, 2004, pp. 99–120 . See in particular the remarks following Prop. 8.20 on p. 114.
4. ↑ Gabrièlov, A. M. (1975), "Combinatorial computation of characteristic classes. I, II", Akademija Nauk SSSR 9 (2): 12–28; ibid. 9 (1975), no. 3, 5–26 .
5. ↑ Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, New York, 1995, p. 20, doi:10.1007/978-1-4613-8431-1, ISBN 0-387-94365-X .

## Further reading

* Hibi, Takayuki; Solus, Liam (2016), "Facets of the r-stable (n,k)-hypersimplex", Annals of Combinatorics 20: 815–829, doi:10.1007/s00026-016-0325-x, Bibcode: 2014arXiv1408.5932H .

0.00

(0 votes)

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Hypersimplex&oldid=3907528"