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### Trop(Flag4)

The ideal corresponing the embedding of Flag4 into a product of Grassmannians Gr(1,4)xGr(2,4)xGr(3,4) and further with respect to the Plücker embedding of each Grassmannian into a product of projective spaces, is generated by

p3,4p1,2,4-p2,4p1,3,4+p1,4p2,3,4,
p3,4p1,2,3-p2,3p1,3,4+p1,3p2,3,4,
p2,4p1,2,3-p2,3p1,2,4+p1,2p2,3,4,
p1,4p1,2,3-p1,3p1,2,4+p1,2p1,3,4,
p4p1,2,3-p3p1,2,4+p2p1,3,4-p1p2,3,4,
p1,4p2,3-p1,3p2,4+p1,2p3,4,
p4p2,3-p3p2,4+p2p3,4,
p4p1,3-p3p1,4+p1p3,4,
p4p1,2-p2p1,4+p1p2,4,
p3p1,2-p2p1,3+p1p2,3

A more copy-paste friendly version in Macaulay2 code can be found here. In the follwing tables the entries are with respect to the order on Plücker coordinates:

p1, p2, p3, p4, p1,2, p1,3, p1,4, p2,3, p2,4, p3,4, p1,2,3, p1,2,4, p1,3,4, p2,3,4

No. ray generator
0 (9, -3, -3, -3, -2, -2, -2, 2, 2, 2, -1, -1, -1, 3)
1 (3, 3, -3, -3, 4, -2, -2, -2, -2, 4, -3, -3, 3, 3)
2 (3, 3, -3, -3, 2, -4, -4, -4, -4, 14, 3, 3, -3, -3)
3 (3, -1, -1, -1, 2, 2, 2, -2, -2, -2, -3, -3, -3, 9)
4 (3, -3, 3, -3, -2, 4, -2, -2, 4, -2, -3, 3, -3, 3)
5 (3, -3, 3, -3, -4, 2, -4, -4, 14, -4, 3, -3, 3, -3)
6 (3, -3, -3, 3, -2, -2, 4, 4, -2, -2, 3, -3, -3, 3)
7 (3, -3, -3, 3, -4, -4, 2, 14, -4, -4, -3, 3, 3, -3)
8 (-1, 3, -1, -1, 2, -2, -2, 2, 2, -2, -3, -3, 9, -3)
9 (-1, -1, 3, -1, -2, 2, -2, 2, -2, 2, -3, 9, -3, -3)
10 (-1, -1, -1, 3, -2, -2, 2, -2, 2, 2, 9, -3, -3, -3)
11 (-3, 9, -3, -3, -2, 2, 2, -2, -2, 2, -1, -1, 3, -1)
12 (-3, 3, 3, -3, -2, -2, 4, 4, -2, -2, -3, 3, 3, -3)
13 (-3, 3, 3, -3, -4, -4, 14, 2, -4, -4, 3, -3, -3, 3)
14 (-3, 3, -3, 3, -2, 4, -2, -2, 4, -2, 3, -3, 3, -3)
15 (-3, 3, -3, 3, -4, 14, -4, -4, 2, -4, -3, 3, -3, 3)
16 (-3, -3, 9, -3, 2, -2, 2, -2, 2, -2, -1, 3, -1, -1)
17 (-3, -3, 3, 3, 14, -4, -4, -4, -4, 2, -3, -3, 3, 3)
18 (-3, -3, 3, 3, 4, -2, -2, -2, -2, 4, 3, 3, -3, -3)
19 (-3, -3, -3, 9, 2, 2, -2, 2, -2, -2, 3, -1, -1, -1)

Trop(Flag4) contains a six-dimensional linear subspace generated by
(1, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1),
(0, 1, 0, 0, 0, -1, -1, 0, 0, -1, 0, 0, -1, 0),
(0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, -1, 0, 0),
(0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1),
(0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0),
(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1).

The polyhedral complex Trop(Flag4) consists of 78 maximal cones, each generated by three rays as follows below.

The intial ideals associated to the maximal cones can be found here: if they are prime, resp. here when they are not prime.

No. rays of maximal cone
C0 (0, 1, 2)
C1 (0, 1, 3)
C2 (0, 3, 4)
C3 (0, 3, 6)
C4 (0, 4, 5)
C5 (0, 6, 7)
C6 (0, 1, 8)
C7 (0, 2, 9)
C8 (0, 2, 10)
C9 (0, 5, 8)
C10 (0, 7, 8)
C11 (0, 4, 9)
C12 (0, 5, 10)
C13 (0, 7, 9)
C14 (0, 6, 10)
C15 (1, 2, 11)
C16 (1, 3, 11)
C17 (1, 2, 17)
C18 (1, 3, 17)
C19 (3, 4, 15)
C20 (3, 6, 13)
C21 (4, 5, 15)
C22 (6, 7, 13)
C23 (3, 4, 16)
C24 (3, 6, 19)
C25 (4, 5, 16)
C26 (6, 7, 19)
C27 (1, 8, 11)
C28 (2, 9, 11)
C29 (2, 10, 11)
C30 (1, 8, 17)
C31 (5, 8, 14)
C32 (7, 8, 12)
C33 (2, 9, 18)
C34 (4, 9, 15)
C35 (7, 9, 12)
C36 (2, 10, 18)
C37 (5, 10, 14)
C38 (6, 10, 13)
C39 (5, 8, 16)
C40 (7, 8, 19)
C41 (4, 9, 16)
C42 (5, 10, 16)
C43 (7, 9, 19)
C44 (6, 10, 19)
C45 (3, 11, 13)
C46 (3, 11, 15)
C47 (7, 12, 13)
C48 (5, 14, 15)
C49 (3, 13, 16)
C50 (3, 15, 19)
C51 (2, 17, 18)
C52 (3, 16, 17)
C53 (3, 17, 19)
C54 (8, 11, 12)
C55 (8, 11, 14)
C56 (9, 11, 12),
C57 (10, 11, 13)
C58 (9, 11, 15)
C59 (10, 11, 14)
C60 (8, 12, 16)
C61 (8, 14, 19)
C62 (9, 12, 16)
C63 (10, 13, 16)
C64 (9, 15, 19)
C65 (10, 14, 19)
C66 (8, 16, 17)
C67 (8, 17, 19)
C68 (9, 16, 18)
C69 (10, 16, 18)
C70 (9, 18, 19)
C71 (10, 18, 19)
C72 (11, 12, 13)
C73 (11, 14, 15)
C74 (12, 13, 16)
C75 (14, 15, 19)
C76 (16, 17, 18)
C77 (17, 18, 19)

The following symmetry classes are with respect to the action of the symmetric group S4 acting on the Plücker coordinates by permuting their indexing sets, and the action of Z2 by sending a Plücker coordinate pI to the Plücker coordinate p[n]-I. The fourth column indicates if the initial ideal associated to a maximal cone in this orbit is a prime ideal. Note that all initial ideals of maximal cones are generated by binomials.

Orbit size Cones prime
Orbit1 24C0, C4, C5, C15, C18, C19, C20, C25, C26, C30, C31, C32, C33, C34, C35, C36, C37, C38, C72, C73, C74, C75, C76, C77 yes
Orbit2 12 C1, C2, C3, C27, C41, C44, C54, C55, C62, C65, C68, C71 yes
Orbit3 12 C6, C11, C14, C16, C23, C24, C56, C59, C60, C61, C69, C70 yes
Orbit4 24 C7, C8, C9, C10, C12, C13, C28, C29, C39, C40, C42, C43, C45, C46, C49, C50, C52, C53, C57, C58, C63, C64, C66, C67 yes
Orbit5 6 C17, C21, C22, C47, C48, C51 no

The following table contains the F-vectors of the polytopes associated to a maximal prime cone (following the construction of Kaveh-Manon) in Trop(Flag4). The last column indicates combinatorially equivalences to string polytopes, the FFLV polytope, and the Gelfand-Tsetlin polytope (for weight rho). All polytopes live in a nine dimensional ambient space and are of dimension 6.

OrbitF-vector of associated polytope combinatorial equivalences
Orbit1 (42, 141, 202, 153, 63, 13) String2
Orbit2 (40, 132, 186, 139, 57, 12) String1 (Gelfand-Tsetlin)
Orbit3 (42, 141, 202, 153, 63, 13) String3 (&FFLV)
Orbit4 (43, 146, 212, 163, 68, 14) -

The above mentioned polytopes are listed in the next table. For the string polytopes we indicate the corresponding reduced expression of the symmetric group. Here si represents the simple transposition (i,i+1). The last column contains information about combinatorial equivalence classes. All polytopes are computed for the irreducible representation of highest weight rho, i.e. the sum of all fundamental weights, rep. half the sum of all simple roots. The polytopes live in a nine-dimensional space and are six dimensional. All polytopes below are normal.

polytopeF-vectorcombinatorial equivalence class
s1s2s1s3s2s1/GT (40, 132, 186, 139, 57, 12)String1
s2s1s2s3s2s1 (40, 132, 186, 139, 57, 12)String1
s2s3s2s1s2s3 (40, 132, 186, 139, 57, 12)String1
s3s2s3s1s2s3 (40, 132, 186, 139, 57, 12)String1
s1s2s3s2s1s2 (42, 141, 202, 153, 63, 13)String2
s3s2s1s2s3s2 (42, 141, 202, 153, 63, 13)String2
s2s1s3s2s3s1 (42, 141, 202, 153, 63, 13)String3
s1s3s2s3s1s2 (38, 133, 197, 152, 63, 13)String4
FFLV (42, 141, 202, 153, 63, 13)String3

We computed following a construction due to Caldero a weight vector for each string cone. They can be found in the thrid columns of the below table. The fourth and fifth columns contain information on the maximal cone of Trop(Flag4) that contains the corresponding weight vector. In the second column you can see if the string cone satisfies the weak Minkowski property or not.

reduced wordMPweight vectortropical coneprime
s1s2s1s3s2s1yes(0,32,24,7,0,16,6,48,38,30,0,4,20,52) C71yes
s2s1s2s3s2s1yes(0,16,48,7,0,32,6,24,22,54,0,4,36,28) C44yes
s2s3s2s1s2s3 yes(0, 4, 36, 28, 0, 32, 24, 6, 22, 54, 0, 16, 48, 7) C3yes
s3s2s3s1s2s3 yes(0, 4, 20, 52, 0, 16, 48, 6, 38, 30, 0, 32, 24, 7) C1yes
s1s2s3s2s1s2 yes(0, 32, 18, 14, 0, 16, 12, 48, 44, 27, 0, 8, 24, 56) C36yes
s3s2s1s2s3s2 yes(0, 8, 24, 56, 0, 16, 48, 12, 44, 27, 0, 32, 18, 14) C0yes
s2s1s3s2s3s1 yes(0, 16, 48, 13, 0, 32, 12, 20, 28, 60, 0, 8, 40, 22) C24yes
s1s3s2s3s1s2 no(0, 16, 12, 44, 0, 8, 40, 24, 56, 15, 0, 32, 10, 26) C17no

In the below column you can find weight vectors constructed from the FFLV polytope. This construction is due to Fang, Fourier, and Reineke. The last two columns indicate where the weight vectors can be found in Trop(Flag4).

typeweight vectortropical coneprime
minimal (0,2,2,1,0,1,1,2,1,2,0,1,1,1) C56 yes
regular (0,3,4,3,0,2,2,4,3,5,0,1,2,3)C56yes

Applying the procedure to Trop(Flag4) we obtain a buquet of three maximal prime cones in the new tropicalization projecting down to one non-prime maximal cone of the old tropicalization. The table below contains F-vectors of the polytopes associated to these three cones and information on combinatorial equivalences to known cones. Note that we recover the previously missing class String4 here.

new conesF-vector of associated polytopecombinatorial equivalence
cone1(38, 133, 197, 152, 63, 13)String4
cone2(38, 133, 197, 152, 63, 13)-
cone3(38, 133, 197, 152, 63, 13)String4