Trop(Flag<sub>4</sub>)

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

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,
p₄p_1,2,3-p₃p_1,2,4+p₂p_1,3,4-p₁p_2,3,4,
p_1,4p_2,3-p_1,3p_2,4+p_1,2p_3,4,
p₄p_2,3-p₃p_2,4+p₂p_3,4,
p₄p_1,3-p₃p_1,4+p₁p_3,4,
p₄p_1,2-p₂p_1,4+p₁p_2,4,
p₃p_1,2-p₂p_1,3+p₁p_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:

p₁, p₂, p₃, p₄, 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(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 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(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.

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(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 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(Flag₄).

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(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 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

Computing toric degenerations of flag varieties

Trop(Flag4)

Trop(Flag₄)