Toric faces of the cone C
The cluster algebra A
3,6 has 50 seeds consisting of four varying mutable cluster variables and six fixed frozen cluster variables.
The six frozen cluster variables contained in every seed are
p
123, p
234, p
345, p
456, p
156, p
126.
A list of all seeds in terms of their mutable cluster variables can be found here:
seeds.
It is known that every seed induces a toric degeneration of the Grassmannian Gr(3,6)
(see for example, Gross–Hacking–Keel–Kontsevich's paper "Canonical bases for cluster algebras" in J. Amer. Math. Soc. 31 (2018), no. 2, 497–608,
or also Rietsch–Williams paper "Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians" in Duke Math. J. 168 (2019), no. 18, 3437–3527,
or Bossinger–Fang–Fourier–Hering–Lanini's paper "Toric degenerations of Gr(2,n) and Gr(3,6) via plabic graphs" in Ann. Comb. 22 (2018), no. 3, 491–512).
In terms of Gröbner degenerations the toric degenerations can be obtained as follows.
We identify mutable cluster variables of A
3,6 with
rays of
the cone C:
r
1 ↔ p
125
r
2 ↔ p
134
r
3 ↔ p
124
r
4 ↔ p
145
r
5 ↔ p
135
r
6 ↔ p
136
r
7 ↔ p
146
r
8 ↔ p
256
r
9 ↔ p
356
r
10 ↔ p
346
r
11 ↔ y
r
12 ↔ p
245
r
13 ↔ p
235
r
14 ↔ x
r
15 ↔ p
236
r
16 ↔ p
246
Details on how we obtain this identification can be found in the paper.
It should just be mentioned at this point that we rely on the categorifications of the cluster algebras A
3,6 and the cluster algebra of type D
4 with universal coefficients
(see Nájera Chávez paper "A 2-Calabi-Yau realization of finite-type cluster algebras with universal coefficients" Math. Z. 291 (2019), no. 3-4, 1495–1523.)
Now we associate to every
seed a weight vector that is the sum of the corresponding rays.
By definition these weight vectors lie in the boundary of the cone C.
We compute the initial ideal of the ideal I
ex with respect to these weights.
The 50 initial ideals can be found in this file:
toric ideals.
Using Macaulay2 we verify that all 50 initial ideals are prime and generated by binomials, so they define
toric degenerations of the cluster-embedded Gr(3,6).
Moreover, one can observe that none of the initial ideals contains a non-zero polynomial with non-negative real coefficients.
This, by definition, means that the ideals are
totally positive (a notion defined in Speyer–Williams' paper "The tropical totally positive Grassmannian", J. Algebraic Combin. 22 (2005), no. 2, 189–210).
Moreover, the intersection of the cone C with the tropicalization of the ideal I
ex coincides with the totally positive part of this tropical variety.
This can be easily verified by the simple observation that the initial monomial of each element in the
reduced Gröbner basis is the unique monomial with positive coefficient.
We further compute the ideals obtained from the toric ideals by eliminating the variables X and Y. All of them are prime and they can be found here:
elinimation ideals