Cluster Gr(3,6)

The flat family

In the paper we define a flat family associated to a maximal cone the the Gröbner cone of any weighted homogeneous ideal. We apply this construction to the ideal I^ex and the maximal cone C. The construction relies on a choice of matrix whose rows are ray generators for the cone (but the outcome is independent of this choice). We fix the matrix r to have rows r₁,...,r₁₆ as listed here: the cone C. For every element of the ideal I^ex its lift with respect to r is defined as follows:

Let t₁,...,t₁₆ be new variables associated to the rays. For m ∈ ℤ¹⁶_≥0 denote by t^m the monomial t₁^m₁...t₁₆^m₁₆. Further, denote by e₁,...,e₂₂ the standard basis of ℝ²². Now take f ∈ I^ex and recall that f is a polynomial in the variables p₁₂₃,...,p₄₅₆,X,Y. Let a₁,...,a_s ∈ ℝ²²_≥0 be the exponent vectors of non-zero monomials in f. Define

μ(f):= (min (r₁a₁,...,r₁a_s),...,min (r₁₆a₁,...,r₁₆a_s)).

Then the lift of f is defined as

f_r = f(t^re₁p₁₂₃,...,t^re₂₀p₄₅₆,t^re₁X,t^re₁Y)t^-μ(f).

It is a polynomial in p₁₂₃,...,p₄₅₆,X,Y with coefficients that are monomials in t₁,...,t₁₆. If f is an element of the reduced Gröbner basis then it has a unique monomial whose coefficient lies in ℂ. This monomial is exactly the initial form of f with respect to any element in the relative interior of C.

The lifted ideal I_r^ex is defined as the ideal generated by the lifts of all elements of I^ex. In the paper we show that equivalently I_r^ex is genetrated by the lifts of the elements of the reduced Gröbner basis. These can be found in this file: lifts.

Grassmannians and universal coefficients for cluster algebras

The flat family