Cluster Gr(3,6)

A maximal cone in the Gröbner fan of I^ex

We study the Gröbner fan of the ideal corresponding to the cluster embedding of Gr(3,6). It should be mentioned that we work with weights rather than term orders and that we use the minimum convention. It is an ideal inside a polynomial ring with 22 variables denoted by

p₁₂₃, p₁₂₄, p₁₂₅, p₁₂₆, p₁₃₄, p₁₃₅, p₁₃₆, p₁₄₅, p₁₄₆, p₁₅₆, p₂₃₄, p₂₃₅, p₂₃₆, p₂₄₅, p₂₄₆, p₂₅₆, p₃₄₅, p₃₄₆, p₃₅₆, p₄₅₆, X, Y.

We call it I^ex and a minimal generating set for it can be found here: I^ex. The Gröbner fan of I^ex, written GF(I^ex), contains a 6-dimensional linear subspace, called the lineality space. It is generated by the following elements (written with respect to the order of variables above):

l₁ = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1),
l₂ = (1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1),
l₃ = (1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1),
l₄ = (0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1),
l₅ = (0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1),
l₆ = (0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1).

Inside GF(I^ex) we identify a maximal cone C. Besides the lineality space C is generated by the following rays:

r₁ = (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
r₂ = (1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 2),
r₃ = (0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1),
r₄ = (1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2),
r₅ = (1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 1),
r₆ = (1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 1),
r₇ = (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
r₈ = (1, 2, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 2),
r₉ = (2, 3, 2, 1, 3, 2, 1, 3, 2, 1, 2, 1, 0, 2, 1, 0, 2, 1, 0, 2, 3, 3),
r₁₀ = (2, 3, 2, 1, 3, 2, 1, 3, 2, 2, 2, 1, 0, 2, 1, 1, 2, 1, 1, 2, 3, 4),
r₁₁ = (2, 3, 3, 2, 3, 2, 1, 3, 2, 2, 2, 1, 0, 2, 1, 1, 2, 1, 1, 2, 3, 4),
r₁₂ = (1, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2),
r₁₃ = (1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 2, 1),
r₁₄ = (1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1),
r₁₅ = (1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1),
r₁₆ = (1, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2).

A more copy-paste friendly version of the generators for the lineality space and the rays can be found here: rays and lineality space. To verify that the cone is maximal, we compute the initial ideal of I^ex with respect to the weight vector

w = (16, 19, 16, 16, 16, 11, 10, 19, 16, 16, 16, 10, 7, 16, 11, 10, 16, 10, 7, 16, 27, 27),

which is the sum of all the rays and therefore in the relative interior of C. The initial ideal has 54 generators that are monomials of degree 2. The generators as well as the computation in Macaulay2 can be found here: monomial ideal.

In fact, we obtained the cone C in a kind of backwards way: given the exchange relations of the cluster algebras A_3,6 we were looking for an initial ideal of I^ex that contains all exchange monomials (the first monomial of each 3-term polynomial as written in the file). So from every exchange relation we obtained two inequalities that the cone C should satisfy: the weight of each of the other two monomials is less or equal to the weight of the exchange monomial. For example, one exchange relation is

p₂₄₆p₃₅₆ - p₃₄₆p₂₅₆ - p₂₃₆p₄₅₆.

The monomial p₂₄₆p₃₅₆ is the exchange monomial. Let w_ijk denote the weight of a variable p_ijk. Then the above relation defines two inequalities:

w₂₄₆ + w₃₅₆ ≤ w₃₄₆ + w₂₅₆, and w₂₄₆ + w₃₅₆ ≤ w₂₃₆ + w₄₅₆.

We proceeded in this way with all exchange relations and defined a cone in polymake. The outcome is the cone C. Note that surprisingly the inequalities deduced from the exchange relations already determine C uniquely. We verified this afterwards using the same technique on the reduced Gröbner basis for I^ex with respect to C. The reduced Gröbner basis has 54 elements, 52 of which are exchange relations. They can be found in this file: reduced Gröbner basis.

Grassmannians and universal coefficients for cluster algebras

A maximal cone in the Gröbner fan of Iex

A maximal cone in the Gröbner fan of I^ex