Cluster Gr(3,6)

The cluster embedding of the Grassmannian Gr(3,6)

Consider the homogeneous coordinate ring A_3,6 of the Grassmannain Gr(3,6) with respect to its Plücker embedding. A description readable by Macaulay2 can be found here: A_3,6. It is a cluster algebra of finite cluster type D₄ with six frozen directions. This cluster algebra has 22 cluster variables:

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

The cluster variables of form p_ijk are the usual Plücker coordinates and the six frozen variables correspond to Plücker coordinates whose indexing sets are (cyclic) intervals:

p₁₂₃, p₂₃₄, p₃₄₅, p₄₅₆, p₁₅₆, p₁₂₆.

The additional two cluster variabels X and Y can be expressed as homogeneous binomials of degree two in the Plücker coordinates:

X = p₁₄₅p₂₄₆ - p₁₂₃p₄₅₆, and Y = p₁₂₅p₃₄₆ - p₁₂₆p₃₄₅.

We define a map from the polynomial ring in cluster variables ℂ[p₁₂₃,..., p₄₅₆, X, Y] to A_3,6 sending each variables to its corresponding coset in A_3,6. The map is surjective and we denote its kernel by I^ex. The ideal in Macaulay2 friendly form is here: I^ex. Eliminating the variables X and Y from I^ex gives back the the Plücker ideal defining Gr(3,6) (see the file A_3,6). The quotient of ℂ[p₁₂₃,..., p₄₅₆, X, Y] by I^ex (we denote it by A from now on) is isomorphic to A_3,6. Moreover, this is an isomorphism of graded algeberas: A has the non-standard grading where the Plücker coordinates are of degree 1 and the variables X and Y of degree 2. The vanishing set V(I^ex) is a weighted projective variety inside the weighted projective space ℙ(d) where

d=(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2)

As weighted projective variety V(I^ex) and Gr(3,6) are isomorphic. We therefore call V(I^ex) the cluster embedding of Gr(3,6). In fact, the ideal I^ex contains all 52 exchange relations of the cluster algebra A_3,6, you can find them here: exchange relations. The first monomial of each 3-term exchange relation (as they are written in the file) is what we call the exchange monomial. The other two terms are cluster monomials. However, the exchange relations do not generate the ideal I^ex, as can be seen from comparing the minimal generating set given in I^ex to the exchange relations.

Grassmannians and universal coefficients for cluster algebras

The cluster embedding of the Grassmannian Gr(3,6)