Universal coefficients for A3,6
The main result regarding the Grassmannian Gr(3,6) in our paper is that the quotient of ℂ[t1,...,t16][p123,...,p456,X,Y] by the lifted ideal Irex is isomorphic to the cluster algebra A3,6 endowed with universal coefficients, called A3,6univ.We therefore proceed by explaining how to describe A3,6univ using the Quiver Mutation App. A3,6univ, just as A3,6, is a cluster algebra of cluster type D4 with the same 22 (mutable and frozen) cluster variables:
p123, p124, p125, p126, p134, p135, p136, p145, p146, p156, p234, p235, p236, p245, p246, p256, p345, p346, p356, p456, X, Y.
Additionally, A3,6univ has coefficients: one for every mutable cluster variable (so 16 in total). For computations all we need is an initial seed for A3,6univ and we obtain in using Reading's description of universal coefficients. (This is equivalent to Fomin–Zelevinsky's original definition, but more handy for computations; see Reading's paper "Universal geometric cluster algebras" Math. Z. 277 (2014), no. 1-2, 499–547) We start by choosing an initial quiver Q for A3,6:
Then, we compute the g-vectors for all mutable cluster variables with respect to the opposite seed. Every g-vector now corresponds to one coefficient and as we are in geometric type we can realize the coefficients as frozen directions. The outcome is the quiver Q with 16 additional frozen vertices whose arrows are indicated by the g-vectors. You can download the quiver here: Quniv. Note that this is in .qmu-format which can be opened directly in the Quiver Mutation App. The frozen variables numbered 1,...,16 correspond to the coefficients. Using the x-variables function of the Quiver Mutation App one can now recursively verify that the exchange relations of A3,6univ coincide with the lifts of exchange relations in A3,6 (see lifted relations). The labelling of the coefficients is aligned with the labelling of the rays here.