The Hyperdeterminant and Triangulations of the 4-Cube

# The Hyperdeterminant and Triangulations of the 4-Cube

Debbie Grier, Peter Huggins, Bernd Sturmfels, and Josephine Yu

## The Newton Polytope N(D2222) of the 2x2x2x2 Hyperdeterminant

The following files are in POLYMAKE format. Namely, points are listed on individual lines with leading 1's, and inequalities of the form a^t x <= b are listed on individual lines as b -a1 -a2 ... -a16. Since there is a natural action, of the symmetry group B4 of the 4-cube on the the monomials of the hyperdeterminant (as well as the vertices and facets of the Newton polytope), we only list the lexicographically minimum representative of each orbit.

Exponent vectors of monomials occuring in the 2x2x2x2 hyperdeterminant D2222
Vertices of the Newton polytope N(D2222)
Facets of the Newton polytope N(D2222)
Lattice points in N(D2222) which are not exponent vectors of monomials in D2222

The following is a detailed description of monomial orbits appearing in the 2x2x2x2 hyperdeterminant.

Detailed description of monomial orbits in D2222

The format is

[[exponent vector], coefficient, dimension of face, orbit size].

For example, the row
[[0, 0, 0, 2, 0, 2, 3, 5, 7, 1, 1, 1, 1, 1, 0, 0], -2, 3, 192]

refers to an orbit of 192 monomials each having coefficient -2. Each sits in the relative interior of a 3-dimensional face of the Newton polytope of the 2x2x2x2-hyperdeterminant. A representative is a0011^2*a0101^2*a0110^3*a0111^5*a1000^7*a1001*a1010*a1011*a1100*a1101.

## The Secondary Polytope N(E2222) of the 4-cube

The following files are again in POLYMAKE format (see above), and only one representative is given from each orbit of the B4 group action.

The vertices of N(E2222) (or, equivalently, the GKZ vectors of the regular triangulations of the 4-cube)
Facets of N(E2222)