Trop(Flag4)
The ideal corresponing the embedding of Flag
4 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
p
3,4p
1,2,4-p
2,4p
1,3,4+p
1,4p
2,3,4,
p
3,4p
1,2,3-p
2,3p
1,3,4+p
1,3p
2,3,4,
p
2,4p
1,2,3-p
2,3p
1,2,4+p
1,2p
2,3,4,
p
1,4p
1,2,3-p
1,3p
1,2,4+p
1,2p
1,3,4,
p
4p
1,2,3-p
3p
1,2,4+p
2p
1,3,4-p
1p
2,3,4,
p
1,4p
2,3-p
1,3p
2,4+p
1,2p
3,4,
p
4p
2,3-p
3p
2,4+p
2p
3,4,
p
4p
1,3-p
3p
1,4+p
1p
3,4,
p
4p
1,2-p
2p
1,4+p
1p
2,4,
p
3p
1,2-p
2p
1,3+p
1p
2,3
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:
p
1, p
2, p
3, p
4, p
1,2, p
1,3, p
1,4, p
2,3, p
2,4, p
3,4, p
1,2,3, p
1,2,4, p
1,3,4, p
2,3,4
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(Flag
4) 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(Flag
4) 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 S
4 acting on the Plücker coordinates by permuting their indexing sets, and the action of Z
2 by sending a Plücker coordinate p
I 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 | 24 | C0, 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(Flag
4). 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.
Orbit | F-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 s
i 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.
polytope | F-vector | combinatorial 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(Flag
4) 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 word | MP | weight vector | tropical cone | prime |
s1s2s1s3s2s1 | yes | (0,32,24,7,0,16,6,48,38,30,0,4,20,52) | C71 | yes |
s2s1s2s3s2s1 | yes | (0,16,48,7,0,32,6,24,22,54,0,4,36,28) | C44 | yes |
s2s3s2s1s2s3
| yes | (0, 4, 36, 28, 0, 32, 24, 6, 22, 54, 0, 16, 48, 7) | C3 | yes |
s3s2s3s1s2s3
| yes | (0, 4, 20, 52, 0, 16, 48, 6, 38, 30, 0, 32, 24, 7) | C1 | yes |
s1s2s3s2s1s2
| yes | (0, 32, 18, 14, 0, 16, 12, 48, 44, 27, 0, 8, 24, 56) | C36 | yes |
s3s2s1s2s3s2
| yes | (0, 8, 24, 56, 0, 16, 48, 12, 44, 27, 0, 32, 18, 14) | C0 | yes |
s2s1s3s2s3s1
| yes | (0, 16, 48, 13, 0, 32, 12, 20, 28, 60, 0, 8, 40, 22) | C24 | yes |
s1s3s2s3s1s2
| no | (0, 16, 12, 44, 0, 8, 40, 24, 56, 15, 0, 32, 10, 26) | C17 | no |
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(Flag
4).
type | weight vector | tropical cone | prime |
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) | C56 | yes |
Applying the procedure to Trop(Flag
4) 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 cones | F-vector of associated polytope | combinatorial equivalence |
cone1 | (38, 133, 197, 152, 63, 13) | String4 |
cone2 | (38, 133, 197, 152, 63, 13) | - |
cone3 | (38, 133, 197, 152, 63, 13) | String4 |