Hypersimplex Explained

In polyhedral combinatorics, the hypersimplex is a convex polytope that generalizes the simplex. It is determined by two integers and , and is defined as the convex hull of the -dimensional vectors whose coefficients consist of ones and zeros. Equivalently, can be obtained by slicing the -dimensional unit hypercube with the hyperplane of equation and, for this reason, it is a -dimensional polytope when .^[1]

Properties

The number of vertices of

is .^[1] The graph formed by the vertices and edges of the hypersimplex is the Johnson graph .^[2]

Alternative constructions

An alternative construction (for

) is to take the convex hull of all -dimensional -vectors that have either or 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

is also the matroid polytope for a uniform matroid with elements and rank .^[3]

Examples

The hypersimplex

is a -simplex (and therefore, it has vertices).The hypersimplex is an octahedron, and the hypersimplex is a rectified 5-cell.

Generally, the hypersimplex,

, corresponds to a uniform polytope, being the -rectified -dimensional simplex, with vertices positioned at the center of all the -dimensional faces of a -dimensional simplex.

Examples

Name	Equilateral triangle	Tetrahedron (3-simplex)	Octahedron	5-cell (4-simplex)	Rectified 5-cell	5-simplex	Rectified 5-simplex	Birectified 5-simplex
	(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 \tbinom{d}{k}	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
valign=center	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	= r	= 2r	r =	= 3r	r = 2r	= 4r	r = 3r	2r
Facets					,		, r	r

History

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

Further reading

• .

Notes and References

1. .
2. .
3. . See in particular the remarks following Prop. 8.20 on p. 114.
4. .
5. .