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By means of higher Gr¨obner theory (see for example, [KM19, §8]), we consider the initial ideal inMv (I) ⊂ S of I with respect to Mv (see Definition 4). Our main result is the following theorem. It is formulated in greater detail in Theorem 1 below.

d Theorem. Let v : A \{0} → Z be a full-rank valuation with full rank weighting matrix Mv ∈ Zd×(k1+···+ks) for the presentation S/I of A. Then

S(A, v) is generated by {v(¯xij )}i∈[s],j∈[ki] if and only if inMv (I) is prime. The theorem has some very interesting implications in view of toric degenerations and Newton– s d−s Okounkov bodies. Consider a valuation of form v : A \{0} → Z≥0 × Z with v(f) = (deg f, ·) for all f ∈ A. Without loss of generality by [IW18, Remark 2.6] we may assume that any full-rank homogeneous valuation is of this form. The Newton–Okounkov cone C(A, v) ⊂ Rs × Rd−s is the cone over its image. The Newton–Okounkov body ∆(A, v) is then the intersection of C(A, v) with the subspace {(1,..., 1)}× Rd−s, for details consider Definition 3. Newton–Okounkov bodies are in general pretty wild objects, they are convex bodies but need neither be polyhedral nor finite. However, if S(A, v) is finitely generated they are rational polytopes (this case for s = 1 is treated by Anderson).

Recall, that if S(A, v) is generated by {v(¯xij )}i∈[s],j∈[ki] (i.e. {x¯ij }i∈[s],j∈[ki] is a Khovanskii basis for (A, v)) this has a number of useful consequences. For example, in this case the associated graded algebra of v can be presented as S/ inMv (I). The following corollary is another such consequence and crucial for our application to flag varieties:

d d×(k +···+k ) Corollary. Let v : A \{0} → Z be a full-rank valuation with Mv ∈ Z 1 s the weighting matrix of v for the presentation S/I of A. Assume additionally inMv (I) is prime, hence S(A, v) is 2 generated by {v(¯xij )}i∈[s],j∈[ki]. Then the Newton–Okounkov polytope is the Minkowski sum :

∆(A, v) = conv(v(¯x1j ))j∈[k1 ] + ··· + conv(v(¯xsj ))j∈[ks]. We continue our study by considering monomial maps as appear for example in [MS18]. Let φv : S → C[y1,...,yd] be the homomophism defined by sending a generator xij to the monomial in yk’s with exponent vector v(¯xij ). Its kernel ker(φv) ⊂ S is a toric ideal, see (3.5). k +···+k Further analyzing our weighting matrices, we associate to each a weight vector. Let wv ∈ Z 1 s be a weight vector associated to Mv satisfying inMv (I) = inwv (I) (see Lemma 3). The following lemma reveals the relation between wv and the toric ideal ker(φv), see also Lemma 2.

d Lemma. For every full-rank valuation v : A \{0} → Z we have inwv (I) ⊂ ker(φv). In particular, inwv (I) is monomial-free and wv is contained in the tropicalization of I (in the sense of [MS15], see (3.2)). We apply our results to two classes of valuations.

Grassmannians and valuations from plabic graphs. We consider a class of valuations on the homogeneous coordinate rings of Grassmannians defined in [RW19]. In the context of cluster algebras and cluster duality for Grassmannians, they associate a full-rank valuation vG to every plabic graph G with certain properties [Pos06]. [n] For k

2For two polytopes A, B ⊂ Rd their Minkowski sum is defined as A + B := {a + b | a ∈ A, b ∈ B}⊂ Rd. 1. Introduction 3 and show that, in these cases, they lie in the tropical Grassmannian [SS04], i.e. the tropicalization of n Ik,n. With our methods, we show that for Gr2(C ) their weight yields the same toric degeneration as Rietsch–Williams’ Newton–Okounkov polytope (see Proposition 3 for details):

n Proposition. For every plabic weight vector wG for Gr2(C ) we have inMG (I2,n) = inwG (I2,n). In particular, for the associated graded of vG we have grvG (A2,n)= C[pij ]ij / inwG (I2,n). For more general Grassmannians Theorem 3 below shows that this is not always the case.

Flag varieties and string valuations. We consider string valuations [Kav15,FFL17] on the homo- 3 geneous coordinate ring of the full flag variety Fℓn . They were defined to realize string parametriza- tions [Lit98,BZ01] of Lusztig’s dual canonical basis in terms of Newton–Okounkov cones and polytopes. Consider the algebraic group SLn and its Lie algebra sln over C. Fix a Cartan decomposition and n−1 take Λ =∼ Z to be the weight lattice. It has a basis of fundamental weights ω1,...,ωn−1 and every n−1 dominant integral weight, i.e. λ ∈ Z≥0 yields an irreducible highest weight representation V (λ) of sln. The Weyl group of sln is the symmetric group Sn. By w0 ∈ Sn we denote its longest element. n−1 For every reduced expression w0 of w0 ∈ Sn and every dominant integral weight λ ∈ Z≥0 , there n(n−1) 2 exists a string polytope Qw0 (λ) ⊂ R . Its lattice points parametrize a basis for V (λ). The string polytope for the weight ρ = ω1 + ··· + ωn−1 is the Newton–Okounkov polytope for the string valuation v w0 on the homogeneous coordinate ring of Fℓn. We embed Fℓn into a product of projective spaces as follows: first, consider the embedding into the n n product of Grassmannians Gr1(C )×···×Grn−1(C ). Then concatenate with the Pl¨ucker embeddings n (n)−1 Grk(C ) ֒→ P k for every 1 ≤ k ≤ n − 1. This yields the (multi-)homogeneous coordinate ring An of Fℓn as C[pJ |J ⊂ [n]]/In. Our main result applied to string valuations yields the following, for more details see Theorem 2. Theorem. v Let w0 be a reduced expression of w0 ∈ Sn and consider the string valuation w0 . If inMv (In) is prime, then w0

v [n] v [n] Qw0 (ρ) = conv( w0 (¯pJ )) + ··· + conv( w0 (¯pJ )) . J∈( 1 ) J∈(n−1)

A central question concerning string polytopes is the following: fix a reduced decomposition w0 and let λ = a1ω1 + ...an−1ωn−1 with ai ∈ Z≥0. Is the string polytope Qw0 (λ) equal to the Minkowski sum a1Qw0 (ω1)+ ··· + an−1Qw0 (ωn−1) of fundamental string polytopes? If equality holds for all λ, we say w0 has the Minkowski property. In [BLMM17] the authors define w weight vectors w0 for every w0. They conjecture a relation between the Minkowski property of w0 and w the weight vector w0 lying in a maximal prime cone of the tropical flag variety, i.e. the tropicalization of In. A corollary of our main theorem proves an even stronger version of their conjecture. It can be summarized as follows (for details see Corollary 4): Corollary. w Let w0 be a reduced expression of w0 ∈ Sn and consider the weight vector w0 . Then in (I ) is prime if and only if w has the Minkowski property. ww0 n 0 The paper is structured as follows. We recall preliminaries on valuations, toric degenerations and Newton–Okounkov bodies in §2. We then turn to quasi-valuations and weighting matrices in §3 and prove our main result Theorem 1. We make the connection to weight vectors and tropicalization and prove the above mentioned result Lemma 2. In §4 we apply our results to string valuations for flag varieties and in §5 to valuations from plabic graphs for Grassmannians. The Appendix contains background information for §5. Acknowledgements. The results of this paper were mostly obtained during my PhD at the University of Cologne under Peter Littelmann’s supervision. I am deeply grateful for his support

3 n We consider the full flag variety of type A, i.e. Fℓn := {{0} ⊂ V1 ⊂···⊂ Vn−1 ⊂ C | dim Vi = i}. It can also be realized as SLn/B, where B are upper triangular matrices in SLn. 2. Notation 4 and guidance throughout my PhD. I would like to further thank Xin Fang, Kiumars Kaveh, Fatemeh Mohammadi and Bea Schumann for inspiring discussions. Moreover, I am grateful to Alfredo N´ajera Ch´avez for helpful comments during the preparation of this manuscript and to two anonymous referees who helped improve and clarify the statements.

2 Notation

We recall basic notions on valuations and Newton–Okounkov polytopes as presented in [KK12]. Let A be the homogeneous coordinate ring of a projective variety or more generally the (multi-)homogeneous coordinate ring of a subvariety of a product of projective spaces, which can be described as below. k1−1 ks−1 The total coordinate ring of P ×···× P for k1,...,ks ≥ 1 is S := C[xij |i ∈ [s], j ∈ [ki]]. It s s is graded by Z≥0 as follows. Let {ǫi}i∈[s] denote the standard basis of Z . The degree of coordinates is s k +···+ks given by deg xij := ǫi ∈ Z (see e.g. [CLS11, Example 5.2.2]) for all i ∈ [s], j ∈ [ki]. For u ∈ Z 1 u uij Zs let x denote the monomial i∈[s],j∈[ki] xij ∈ S. We fix the lexicographic order on and consider u f = aux ∈ S. Then Q u deg f := max lex{deg x | au 6=0}. P Let X be subvariety of Pk1−1 ×···× Pks−1 and A its homogeneous coordinate ring of Krull-dimension s d. Then A = S/I for some prime ideal I ⊂ S that is homogeneous with respect to the Z≥0-grading. s s The Z -grading on S induces a Z -grading on A, which we denote A = s Am. We call a ≥0 ≥0 m∈Z≥0 grading of this form a positive (multi-)grading. L To define a valuation on A we fix a linear order ≺ on the additive abelian group Zd. Definition 1. A map v : A \{0} → (Zd, ≺) is a valuation4, if it satisfies for f,g ∈ A \{0},c ∈ C∗ (i) v(f + g)  min{v(f), v(g)}, (ii) v(fg)= v(f)+ v(g), and (iii) v(cf)= v(f). If we replace (ii) by v(fg)  v(f)+v(g) then v is called a quasi-valuation (also called loose valuation in [Tei03]). Let v : A \{0} → (Zd, ≺) be a valuation. The image {v(f) | f ∈ A \{0}} ⊂ Zd forms an additive semi-group. We denote it by S(A, v) and refer to it as the value semi-group. The rank of the valuation is the rank of the sublattice generated by S(A, v) in Zd. We are interested in valuations of full rank, i.e. rank(v)= d. d One naturally defines a Z -filtration on A by Fva := {f ∈ A \{0}|v(f)  a}∪{0} (and similarly Fv≻a). The associated graded algebra of the filtration {Fva}a∈Zd is

grv(A) := Fva/Fv≻a. (2.1) d aM∈Z d If the filtered components Fva/Fv≻a are at most one-dimensional for all a ∈ Z , we say v has one- dimensional leaves. If v has full rank by [KM19, Theorem 2.3] it also has one-dimensional leaves. Moreover, in this case there exists an isomorphism C[S(A, v)] =∼ grv(A) by [BG09, Remark 4.13]. To define a Z≥0-filtration on A induced by v we make use of the following standard trick that con be found in [Bay82, Proposition 1.8] or [Cal02, Lemma 3.2].

d d Lemma 1. Let F be a finite subset of Z . Then there exists a linear form e : Z → Z≥0 such that for all m, n ∈ F we have m ≺ n ⇒ e(m) >e(n) (note the switch!).

Assume S(A, v) is finitely generated, more precisely assume it is generated by {v(¯xij )}i∈[s],j∈[ki]. 5 In this case {x¯ij }i∈[s],j∈[ki] is called a Khovanskii basis for (A, v). Now choose a linear form as in the

4We assume the image of v lies in Zd for simplicity. Without any more effort, we could replace it by Qd. The same is true for all (quasi-)valuations considered in the rest of the paper. 5This term was introduced in [KM19] generalizing the notion of SAGBI basis. 3. Quasi-valuations with weighting matrices 5

d lemma for F = {v(¯xij )}i∈[s],j∈[ki] ⊂ Z . We construct a Z≥0-filtration on A by F≤m := {f ∈ A \{0} | e(v(f)) ≤ m}∪{0} for m ∈ Z≥0. Define similarly F

grv(A) =∼ F≤m/F

For f ∈ A \{0} denote by f its image in the quotient F≤e(v(f))/F

Definition 2. The Rees algebra associated with the valuation v and the filtration {F≤m}m is the flat C[t]-subalgebra of A[t] defined as

m Rv,e := (F≤m)t . (2.3) m M≥0

It has the properties that Rv,e/tRv,e =∼ grv(A) and Rv,e/(1 − t)Rv,e =∼ A. In particular, if A is Z≥0- graded, it defines a flat family over A1 (the coordinate on A1 given by t). The generic fiber is isomorphic to Proj(A) and the special fiber is the toric variety Proj(grv(A)).

It is desirable to have a valuation that encodes the grading of A: a valuation v on A = s Am m∈Z≥0 is called homogeneous, if for f ∈ A with homogeneous decomposition f = s amfm we have m∈Z≥0 L P v(f) = min ≺{v(fm) | am 6=0}.

The valuation is called fully homogeneous, if v(f) = (deg f, ·) ∈ Zs × Zd−s for all f ∈ A. Given a full rank homogeneous valuations by [IW18, Remark 2.6] without loss of generality one may assume it is fully homogeneous. Introduced by Lazarsfeld-Mustat¸˘a[LM09] and Kaveh-Khovanskii [KK12] we recall the definition of Newton–Okounkov body. Definition 3. Let v : A \{0} → (Zs × Zd−s, ≺) be a fully homogeneous valuation. The Newton– Okounkov cone is

C(A, v) := cone(S(A, v) ∪{0}) ⊂ Rd. (2.4)

The Newton–Okounkov body of (A, v) is now defined as

∆(A, v) := C(A, v) ∩{(1,..., 1)}× Rd−s. (2.5)

For Z≥0-graded A, Anderson showed in [And13] that if grv(A) is finitely generated, ∆(A, v) is a rational polytope. Moreover, it is the polytope associated to the normalization of the toric variety Proj(grv(A)). Dealing with polytopes throughout the paper we need the notion of Minkowski sum. For two polytopes A, B ⊂ Rd it is defined as

A + B := {a + b | a ∈ A, b ∈ B}⊂ Rd. (2.6)

d−s s Consider polytpes of form Pv(λ) := C(A, v)∩({λ}×R ) for λ ∈ R≥0. It is a central question whether Pv(ǫ1)+ ··· + Pv(ǫs) = ∆(A, v). Our main result (Theorem 1 below) treats this question.

3 Quasi-valuations with weighting matrices

We briefly recall some background on higher-dimensional Gr¨obner theory and quasi-valuations with weighting matrices as in [KM19, §3.1&8.1]. For classical results in Gr¨obner theory we rely on [Eis95, 3. Quasi-valuations with weighting matrices 6

§15]. Then we define for a given valuation an associated quasi-valuation with weighting matrix. The central result of this paper is Theorem 1. It is proved in full generality below, and applied to specific classes of valuations in §4 and §5. As above let A be a finitely generated algebra and domain of Krull-dimension d. We assume A has a presentation of form π : S → A, such that A = S/ ker(π). Here S denotes the total coordinates ring k1−1 ks−1 s of P ×···×P with Z≥0-grading as defined above. Let I := ker(π) be homogeneous with respect s to the Z≥0-grading. To simplify notation, let π(xij )=:x ¯ij for i ∈ [s], j ∈ [ki] and n := k1 + ··· + ks. For simplicity we always consider (quasi-)valuations with image in Zd, but without much more effort all results extend to the case of Qd.

u n u u1 un d×n Definition 4. Let f = aux ∈ S with u ∈ Z , where x = x1 ··· xn . For M ∈ Z and a linear order ≺ on Zd we define the initial form of f with respect to M as P m inM (f) := amx . (3.1) Mm=minX≺{Mu|au6=0} We extend this definition to ideals I ⊂ S by defining the initial ideal of I with respect to M as inM (I) := hinM (f) | f ∈ Ii⊂ S.

s Note that, by definition, if the ideal I is homogeneous with respect to the Z≥0-grading, then so is every initial ideal of I. Using the fundamental theorem of tropical geometry [MS15, Theorem 3.2.3] we recall the related notion of tropicalization of an ideal I ⊂ S:

n trop(I)= {w ∈ R | inw(I) is monomial-free} . (3.2)

By the Structure Theorem [MS15, Theorem 3.3.5] we can choose a fan structure on trop(I) in such a way that it becomes a subfan of the Gr¨obner fan of I: w and v lie in the relative interior of a cone ◦ C ⊂ trop(I) (denoted w, v ∈ C ) if inw(I) = inv(I). For this reason we adopt the notation inC(I), ◦ which is defined as inw(I) for arbitray w ∈ C . Further, we call a cone C ⊂ trop(I) prime, if inC (I) is a prime ideal. Coming back to the more general case of weighting matrices, we say that M ∈ Zd×n lies in the Gr¨obner region GRd(I) of an ideal I ⊂ S, if there exists a monomial order6 < on S such that

in<(inM (I)) = in<(I).

Such a monomial order is called compatible with M. If the ideal I ⊂ S is (multi-)homogeneous with s d×n d respect to the Z≥0-grading, then by [KM19, Lemma 8.7] we have Q ⊂ GR (I). To a given matrix M ∈ Zd×n one associates a quasi-valuation as follows. As above, fix a linear order ≺ on Zd.

u d Definition 5. Let f˜ = aux ∈ S and define v˜M : S \{0} → (Z , ≺) by v˜M (f˜) := min ≺{Mu | au 6= d 0}. By [KM19, Lemma 3.2], there exists a quasi-valuation vM : A \{0} → (Z , ≺) given for f ∈ A by P

vM (f) := max ≺{v˜M (f˜) | f˜ ∈ S, π(f˜)= f}.

It is called the quasi-valuation with weighting matrix M. We denote the associated graded algebra (defined analogously as in (2.1) for valuations) of the quasi-valuation vM by grM (A). It has the property ∼ grM (A) = S/ inM (I). (3.3)

s In particular, grM (A) inherits the Z≥0-grading of S, as I is homogeneous with respect to this grading. From the definition, it is usually hard to explicitly compute the values of a quasi-valuation vM . The following proposition makes it more computable, given that M lies in the Gr¨obner region of

6 u m The initial form of an element f = aux ∈ S with respect to a monomial order < is the monomial amx of f m which satisfies am 6= 0 and x is minimal respect to < among all monomials of f with non-zero coefficient. P 3. Quasi-valuations with weighting matrices 7

d I. Recall, that if C< ⊂ GR (I) is a maximal cone with associated monomial ideal in<(I), then α α B := {x¯ | x 6∈ in<(I)} is a vector space basis for A, called standard monomial basis. The monomials α x 6∈ in<(I) are called standard monomials. In general, a vector space basis B ⊂ A is called adapted d d to a valuation v : A \{0} → (Z , ≺), if Fva ∩ B is a vector space basis for Fva for every a ∈ Z . Proposition. ([KM19, Proposition 3.3]) Let M ∈ GRd(I) and B ⊂ A be a standard monomial basis for the monomial order < on S compatible with M. Then B is adapted to vM . Moreover, for every α α element f ∈ A written as f = x¯ aα withx ¯ ∈ B and aα ∈ C we have

P vM (f) = min ≺{Mα | aα 6=0}. (3.4)

k1+···+ks Remark 1. The proposition implies that vM is homogeneous: for α ∈ Z≥0 define mα := k1 ks s α α ( j=1 α1j ,..., j=1 αsj ) ∈ Z≥0. Thenx ¯ ∈ Amα for allx ¯ ∈ B. For f ∈ A the unique expression α f = x¯ aα is also its homogeneous decomposition and by the proposition P P α P vM (f) = min ≺{vM (¯x ) | aα 6=0}.

From our point of view, quasi-valuations with weighting matrices are not the primary object of interest. In most cases we are given a valuation v : A \{0} → (Zd, ≺) whose properties we would like to know. In particular, we are interested in the generators of the value semi-group and if there are only finitely many. The next definition establishes a connection between a given valuation and weighting matrices. It allows us to apply techniques from Kaveh-Manon for quasi-valuations with weighting matrices to other valuations of our interest. Definition 6. Given a valuation v : A \{0} → (Zd, ≺). We define the weighting matrix of v associated with the presentation S/I of A by

d×n Mv := (v(¯xij ))i∈[s],j∈[ki] ∈ Z .

That is, the columns of Mv are given by the images v(¯xij ) for i ∈ [s], j ∈ [ki].

Note we slightly abuse notation and write Mv instead of Mv(S/I) as we fix the presentation of A from the start. The following corollary is obtained by an argument very similar to the proof of [KM19, Proposition 4.2]. We therefore leave its proof to the reader.

d×n Corollary 1. Let M ∈ Z be of full rank with d the Krull-dimension of A. If inM (I) is prime, then vM is a valuation whose value semi-group S(A, vM ) is generated by {vM (¯xij )}i∈[s],j∈[ki]. In particular, the associated Newton–Okounkov body is given by

s ∆(A, vM )= conv(vM (¯xij ) | j ∈ [ki]). i=1 X To a full rank valuation v : A \{0} → (Zd, ≺) we associate a homomorphism of polynomial rings, called monomial map of v (see e.g. [MS18]):

v(¯xij ) φv : S → C[y1,...,yd] by φv(xij ) := y .

The image im(φv) is naturally isomorphic to the semi-group algebra C[S(v(¯xij )ij )], where S(v(¯xij )ij ) d is the semi-group generated by {v(¯xij )|i ∈ [s], j ∈ [ki]}⊂ Z . We have

S/ ker(φv) =∼ C[S(v(¯xij )ij )] ⊂ C[S(A, v)] =∼ grv(A). (3.5)

Therefore, ker(φv) is a binomial prime ideal. The height of ker(φv) is n − rank(S(v(¯xij )ij )) = n − rank(Mv).

d Lemma 2. For every full rank valuation v : A \{0} → (Z , ≺) we have inMv (I) ⊂ ker(φv). 3. Quasi-valuations with weighting matrices 8

s ui n Proof. Let f ∈ I and consider inMv (f) = i=1 cix with ci ∈ C and ui ∈ Z≥0 for all i. So Mui = ui uj ui uj Muj =: a, i.e. v(¯x ) = v(¯x ) andx ¯ , x¯ ∈ Fva for all 1 ≤ i, j ≤ s. As v has one-dimensional s ui P leaves, we have i=1 cix¯ ∈ Fv≻a, which implies it is zero in grv(A). Hence, by (3.5) we have inMv (f) ∈ ker(φv). P Before stating the main theorem relating a given valuation v with the (quasi-)valuation with weight- ing matrix Mv we prove the following proposition that is used in the proof. s Proposition 1. Assume I is homogeneous with respect to the Z≥0-grading and Krull dimension of d d×n A = S/I is d. Let v : A \{0} → (Z , ≺) be a full rank valuation with Mv ∈ Z the weighting matrix of v. Then

v = vMv ⇔ inMv (I) is prime and rank(Mv)= d.

Proof. “⇒” Assume v = vM , then C[S(A, v)] = S/ inMv (I) by (3.3). In particular, inMv (I) is binomial and prime. Further, rank(Mv) = rank(S(A, v)) = Krull dimension of A.

“⇐” Assume inMv (I) is prime and Mv has rank equal to the Krull dimension of A. By [KM19, Theorem

2.17] we obtain v = vMv if and only if inMv (I) = ker φv. By Lemma 2 we have inMv (I) ⊂ ker φv. As rank(Mv) = rank(S(A, v)) both ideals have the same height and are therefore equal. s Theorem 1. Assume I is homogeneous with respect to the Z≥0-grading and Krull dimension of A = S/I is d. Let v : A \{0} → (Zd, ≺) be a full rank valuation with associated weighting matrix d×n Mv ∈ Z . Then,

S(A, v) is generated by {v(¯xij )}i∈[s],j∈[ki] ⇔ inMv (I) is prime and rank(Mv)= d.

Both directions of the proof rely on the equality v = vMv . The equality is false in general, a counterexample is given in §4 Example 3. For “⇐” the equality follows from Proposition 1 while for “⇒” it follows from a direct computation.

Proof. As I is homogeneous with respect to a positive grading, Mv lies in the Gr¨obner region of I. Let

B ⊂ A be the standard monomial basis adapted to vMv .

“⇐” Assume inMv (I) is prime and Mv has full rank. Then we obtain by Proposition 1 S(A, v) = ∼ S(A, vMv ) = S/ inMv (I). Hence, by Corollary 1 S(A, v) is generated by {v(¯xij )}ij . α α α “⇒” Assume S(A, v) is generated by {v(¯xij )}i∈[s],j∈[ki]. We have vMv (¯x )= Mvα = v(¯x ) forx ¯ ∈ B

by (3.4). We need to prove v = vMv then inMv (I) is prime and rank(Mv) = Krull dimension of A by Proposition 1. This follows from: α β α β Claim: vMv (¯x ) 6= vMv (¯x ) forx ¯ , x¯ ∈ B with α 6= β. α β α β Proof of claim: Assume there existx ¯ , x¯ ∈ B with α 6= β and vMv (¯x )= vMv (¯x ). Then α α β β v(¯x )= Mvα = vMv (¯x )= vMv (¯x )= Mvβ = v(¯x ). This implies that v(¯xα +x ¯β) ≻ v(¯xα) = v(¯xβ ). In particular, v(¯xα +x ¯β) does not lie in the semi-group span hv(¯xij ) | i ∈ [s], j ∈ [ki]i = S(A, v), a contradiction by assumption. Knowing the generators of S(A, v) explicitly has a number of crucial consequences for applications.

Given the theorem, they now follow directly from inMv (I) being prime: s Corollary 2. Assume I is homogeneous with respect to the Z≥0-grading and generated by elements d d×n f ∈ I with deg f >ǫi for all i ∈ [s]. Let v : A \{0} → (Z , ≺) be a full rank valuation with Mv ∈ Z of full rank. Assume additionally that inMv (I) is prime. Then v is homogeneous and ∼ (i) grv(A) = S/ inMv (I),

(ii) {x¯ij }i∈[s],j∈[ki] is a Khovanskii basis for (A, v), and s (iii) ∆(A, v)= i=1 conv(v(¯xij ) | j ∈ [ki]). P 4. Application: (tropical) flag varieties and string cones 9

3.1 From weighting matrix to weight vector and tropicalization In this section we summarize how to pass from a weighting matrix to a weight vector which is desirable for applications. We assume the ideal I ⊂ S is homogeneous with respect to the Zd-grading on S and that A = S/I has Krull dimension d. We consider weighting matrices that lie in the Gr¨obner region of I and for simplicity we assume they have integer entries.

d×n d Definition 7. Let M ∈ Z and denote by M1,...,Mn its columns. A linear map e : Z → Z is called an order preserving projection with respect to M and I, if for eM := (e(M1),...,e(Mn)) we have inM (I) = ineM (I). The following lemma a reformulation of classical results in Gr¨obner theory (see e.g. [Bay82, Cal02, KM19]). We include it for completeness in view of applications. Lemma 3. For every M ∈ Zd×n the order preserving projections with respect to M and I are organized ◦ d n in a polyhedral cone σM (I) ⊂ R . Moreover, there exist w ∈ R with inw(I) = inM (I).

Proof. Let {R1,...,Rs} be a reduced Gr¨obner basis for inM (I). Assume the initial form of Rl is of l l q l uj l l weight Mml for l ∈ [s]. So we have Rl = inM (Rl)+ j=1 aj x with Mml ≺ Muj for all j ∈ [q ]. We define the (open) polyhedral cone of order preserving projections P ◦ d l l σM := {e ∈ R | he,M(ml − uj )i < 0 for all l ∈ [s], j ∈ [q ]}. (3.6)

◦ ◦ Applying the linear map defined by M we obtain a set EM := {eM | e ∈ σM } and

n l l ◦ EM ⊂{w ∈ R | hw,ml − uji < 0 for all l ∈ [s], j ∈ [q ]} =: CM . (3.7)

◦ ◦ By Lemma 1, σM is nonempty, hence CM is nonempty and by definition an open cone in the Gr¨obner fan of I. Corollary 3. Consider v : A \{0} → (Zd, ≺) a full rank valuation with associated weighting matrix d×n Mv ∈ Z . Then inMv (I) is monomial-free and CMv is a cone in trop(I).

Proof. By Lemma 2 we have inMv (I) is contained in a binomial prime ideal.

4 Application: (tropical) flag varieties and string cones

In this subsection we focus on a particular valuation on the Cox ring of the full flag variety. The valuation was defined in the context of representation theory by [FFL17] (see also [Kav15]). It is of particular interest, as it realizes Littelmann’s string polytopes [Lit98, BZ01] as Newton–Okounkov polytopes. Applying Theorem 1 to this valuation, we solve a conjecture by [BLMM17] relating the Minkowski property of string cones (see Definition 8) to the tropical flag variety. n Consider Fℓn, the full flag variety of flags of vector subspaces of C . Its dimension as a projective n(n−1) n n variety is N := 2 . We embed it into the product of Grassmannians Gr1(C ) ×···× Grn−1(C ). Further embedding each Grassmannian via its Pl¨ucker embedding into projective space we obtain n n (1)−1 (n−1)−1 n n Fℓn ֒→ P ×···×P . Set K := 1 +···+ n−1 . In this way, we get the (multi-)homogeneous coordinate ring An as a quotient of the total coordinate ring of the product of projective spaces ∼

n−1 S := C[pJ | 0 6= J ( [n]]. Namely An = S/In. In this case S is multigraded by Z≥0 : deg(pJ ) := ǫk n−1 n−1 for |J| = k and {ǫk}k=1 standard basis of Z . The variables pJ are called Pl¨ucker variables and their cosetsp ¯J ∈ An are Pl¨ucker coordinates. The ideal In ⊂ S is generated by the quadratic Pl¨ucker n−1 relations, see [MS05, Theorem 14.6]. It is homogeneous with respect to the Z≥0 -grading induced from S. The tropical flag variety, denoted trop(Fℓn) (as defined in [BLMM17]) is the tropicalization of the ideal In ⊂ S. 4. Application: (tropical) flag varieties and string cones 10

Realizing Fℓn = SLn/B, where B ⊂ SLn are upper triangular matrices, one can make use of the representation theory of SLn to define valuations on An. We summarize the necessary representation- theoretic background below. Let Sn denote the symmetric group and w0 ∈ Sn its longest element. By w0 we denote a reduced expression si1 ...siN of w0 in terms of simple transpositions si := (i,i +1). Let sln be the Lie algebra − + of SLn and fix a Cartan decomposition sln = n ⊕ h ⊕ n into lower, diagonal and upper triangular A n traceless matrices. Let R = {εi −εj}i6=j be the root system of type n−1, where {εi}i=1 is the standard n 7 − basis of R . Every εi − εi+1 defines an element fi ∈ n . n−1 Denote the weight lattice by Λ =∼ Z . It is spanned by the fundamental weights ω1,...,ωn−1 + n−1 + and we set Λ := Z≥0 with respect to the basis of fundamental weights. For every λ ∈ Λ we have an irreducible highest weight representation V (λ) of sln. It is cyclically generated by a highest weight − vector vλ (unique up to scaling) over the universal enveloping algebra U(n ). In particular, for every mi1 miN reduced expression w0 = si1 ··· siN the set Sw0 := {fi1 ··· fiN (vλ) ∈ V (λ) | mij ≥ 0} is a spanning set for the vector space V (λ).

k n Example 1. Given a fundamental weight ωk, we have V (ωk) = C . We can choose the highest n n weight vector as vωk := e1 ∧···∧ ek, where {ei}i=1 is a basis of C . In this context Pl¨ucker coordinates V ∗ ∗ p¯{j1,...,jk} ∈ An are identified with dual basis elements (ej1 ∧···∧ ejk ) ∈ V (ωk) . Littelmann [Lit98] introduced in the context of quantum groups and crystal bases the so called

(weighted) string cones and string polytopes Qw0 (λ). The motivation is to find monomial bases for the irreducible representations V (λ). Littelmann identifies a linearly independent subset of the spanning set N Sw0 by introducing the notion of adapted string (see [Lit98, p. 4]) referring to a tuple (a1,...,aN ) ∈ Z≥0. a1 aN His basis for V (λ) consists of those elements fi1 ··· fiN (vλ) for which (a1,...,aN ) is adapted. + For a fixed reduced expression w0 of w0 ∈ Sn and λ ∈ Λ he gives a recursive definition of the the N N string polytope Qw0 (λ) ⊂ R ([Lit98, p. 5], see also [BZ01]). The lattice points Qw0 (λ) ∩ Z are the N + adapted strings for w0 and λ. The string cone Qw0 ⊂ R is the convex hull of all Qw0 (λ) for λ ∈ Λ . The weighted string cone is defined as

n−1+N Q + w0 := conv λ∈Λ {λ}× Qw0 (λ) ⊂ R . [ By definition, one obtains the string polytope from the weighted
string cone by intersecting it with the hyperplanes {λ}× RN . In this context, a central question is the following:

n−1 + Given w0, does Qw0 (λ)= a1Qw0 (ω1)+ ··· + an−1Qw0 (ωn−1) hold for all λ = k=1 akωk ∈ Λ ? P Definition 8. A reduced expression w0 of w0 ∈ Sn has the (strong) Minkowski property, if Qw0 (λ)= n−1 + a1Qw0 (ω1)+ ··· + an−1Qw0 (ωn−1) holds for all λ = k=1 akωk ∈ Λ . If Qw (ρ)= Qw (ω1)+ ··· + Qw (ωn−1) for ρ := ω1 + ··· + ωn−1, we say w satisfies MP. 0 0 0 P 0 String polytopes can be realized as Newton–Okounkov polytopes, as was done by [Kav15, FFL17]. The corresponding valuations are defined on the coordinate ring of G//U := Spec(C[G]U ), where U ⊂ B are upper triangular matrices with 1’s on the diagonal. By [VP89] C[G//U] is isomorphic to the Cox ring Cox(Fℓn) of the flag variety. As a basis for the Picard group we fix the homogeneous line bundle associated to the fundamental weights Lω1 ,...,Lωn−1 (see e.g. [Bri04, p.15]). This way we obtain 0 ∗ Cox(Fℓn)= H (Fℓn,Lλ) =∼ V (λ) , + + λM∈Λ λM∈Λ where the isomorphism is due to the Borel-Weil theorem. For every w0 there exists a fully homogeneous v + N § valuation w0 : Cox(Fℓn)\{0} → Λ ×Z called string valuation. By [FFL17, 11] it has the properties

+ C(Cox(Fℓ ), v )= Q and Pv (λ)= Q (λ) for all λ ∈ Λ . (4.1) n w0 w0 w0 w0

7 − The element fi ∈ n is the elementary matrix with only non-zero entry 1 in the (i + 1,i)-position. 4. Application: (tropical) flag varieties and string cones 11

(n)−1 ( n )−1 As described above, we fixed Fℓn ֒→ P 1 ×···× P n−1 and want to consider the string valuations on the homogeneous coordinate ring An. The following proposition is therefore very useful. As a consequence of the standard monomial theory for flag varieties it follows from [LB09, Theorem n −1 n −1 12.8.3]. We give the proof for completion. To simplify notation let Z := P(1) ×···×P(n−1) . Recall that Schubert varieties are of form X(w) := BwB/B ⊂ SLn/B for w ∈ Sn and that for w = w0sk they are divisors in Fℓn.

Proposition 2. We have An =∼ Cox(Fℓn).

(n)−1 Proof. Let Lk be the ample generator of Pic(P k ). By [LB09, Theorem 11.2.1] it pulls back to the n−1 . line bundle Lωk of the Schubert divisor X(w0sk) ⊂Fℓn ֒→ Z and Pic(Fℓn) is generated by {Lωk }k=1 n−1 n−1 For every λ = k=1 akωk ∈ Z≥0 by [LB09, Theorem 12.8.3] we have a surjection

P 0 a1 an−1 0 H (Z,L1 ⊗···⊗ Ln−1 ) → H (Fℓn,Lλ).

0 a1 an−1 Note that S = n−1 H (Z,L ⊗···⊗ Ln ), hence we have a surjection π : S → Cox(Fℓn). Its a∈Z≥0 1 −1 0 a1 an−1 kernel is the multi-homogeneousL ideal generated by elements f ∈ H (Z,L1 ⊗···⊗ Ln−1 ) vanishing n−1 ∼ on Fℓn for varying (a1,...,an−1) ∈ Z≥0 . In particular, ker(π)= In and Cox(Fℓn) = S/In = An. v From now on we consider for a fixed reduced expression w0 = si1 ...siN the valuation w0 on An. v We explain how to compute w0 on Pl¨ucker coordinates, following [FFL17]. We use the total order ≺ N N N N N on Z defined by m ≺ n if and only if, i=1 mi < i=1 ni or i=1 mi = i=1 ni and m>lex n. Now 8 vw can be computed explicitly on Pl¨ucker coordinates by [FFL17, Proposition 2]: 0 P P P P v m N f m w0 (¯pj1,...,jk ) = min ≺{ ∈ Z≥0 | (e1 ∧···∧ ek)= ej1 ∧···∧ ejk },

f m m1 mN n− for {j1,...,jk}⊂ [n], where = fi1 ··· fiN ∈ U( ). § v N We want to apply the results from 3 to the given valuations of form w0 : An \{0} → (Z , ≺). v N×K Let therefore Mw0 := ( w0 (¯pJ ))06=J([n] ∈ Z be the matrix whose columns are given by the images v N of Pl¨ucker coordinates under w0 . Consider as in [BLMM17] the linear form e : Z → Z given by

N−1 N−2 −e(m) := 2 m1 +2 m2 + ... +2mN−1 + mN .

Definition 9. v For a fixed reduced expression w0 the weight of the Pl¨ucker variable pJ is e( w0 (¯pJ )). w v K We define the weight vector w0 := (e( w0 (¯pJ )))06=J([n] ∈ R . w The weight vectors w0 are computed in [BLMM17] for n = 4 and n = 5. The authors verify that they lie in the tropical flag variety and conjecture this is true in general.

Lemma 4. For every reduced expression w we have in (I ) = inw (I ). In particular, w ∈ 0 Mw0 n w0 n w0 trop(Fℓn). Proof. The linear map e : ZN → Z is constructed using the recursive recipe given in [Cal02, Proof of Lemma 3.2]. Caldero uses it to define the filtration {Fm}m∈Z on An. In Proposition 3.1 and N+(n−1) Corollary 3.2 of [Cal02] he shows that this Z-filtration and the Z -filtration {Fv } have the w0 v same associated graded algebra, namely C[S(An, w0 )]. To be precise, given a relation in An coming v from a lifting of a minimal relation in S(An, w0 ) Caldero shows that in the associated graded algebras with respect to {F } and {Fv } the same initial relation holds. The generators of I are naturally m m∈Z w0 n v liftings of minimal relations in S(An, w0 ) as they are of minimal total degree 2. So in particular, in (I ) = in (I ). The rest follows from Corollary 3. Mw0 n ww0 n

8 n− k n k The action of on C necessary to make explicit computations is given by fi,j (ei1 ∧···∧ eik ) = s=1 ei1 ∧ ···∧ (fi,j ei ) ∧···∧ ei , where fi,j ei = δi,i ej+1 and fi = fi,i+1. s k V s s P 4. Application: (tropical) flag varieties and string cones 12

w MP weight vector −w in (I ) prime 0 w0 ww0 4 String 1: s1s2s1s3s2s1 yes (0, 32, 24, 7, 0, 16, 6, 48, 38, 30, 0, 4, 20, 52) yes s2s1s2s3s2s1 yes (0, 16, 48, 7, 0, 32, 6, 24, 22, 54, 0, 4, 36, 28) yes s2s3s2s1s2s3 yes (0, 4, 36, 28, 0, 32, 24, 6, 22, 54, 0, 16, 48, 7) yes s3s2s3s1s2s3 yes (0, 4, 20, 52, 0, 16, 48, 6, 38, 30, 0, 32, 24, 7) yes String 2: s1s2s3s2s1s2 yes (0, 32, 18, 14, 0, 16, 12, 48, 44, 27, 0, 8, 24, 56) yes s3s2s1s2s3s2 yes (0, 8, 24, 56, 0, 16, 48, 12, 44, 27, 0, 32, 18, 14) yes String 3: s2s1s3s2s3s1 yes (0, 16, 48, 13, 0, 32, 12, 20, 28, 60, 0, 8, 40, 22) yes String 4: s1s3s2s1s3s2 no (0, 16, 12, 44, 0, 8, 40, 24, 56, 15, 0, 32, 10, 26) no

Table 1: Isomorphism classes of string polytopes (up to unimodular equivalence) for n = 4 and ρ w depending on w0, the property MP, the weight vectors w0 as in Definition 9, and primeness of the initial ideals in (I ). For details see [BLMM17, §5] ww0 4

Example 2. v Consider the reduced expressionw ˆ 0 = s1s2s1s3s2s1 for w0 ∈ S4. We compute wˆ 0 (¯p13)= 4 (0, 1, 0, 0, 0, 0) and the weight ofp ¯13: e(0, 1, 0, 0, 0, 0) = −(1 · 2 ) = −16. Similarly, we obtain weights for all Pl¨ucker variables and (ordered lexicographically by their indexing sets)

w − w0 = (0, 32, 24, 7, 0, 16, 6, 48, 38, 30, 0, 4, 20, 52).

Table 1 contains all weight vectors (up to sign) for Fℓ4 constructed this way. Lemma 5. For every reduced decomposition w0 of w0 ∈ Sn we have rank(Mw0 )= N + n − 1, i.e Mw0 is of full rank.

Proof. We first restrict our attention to a submatrix of Mw0 having only those columns corresponding v [n] to w0 (¯pI ) for I ∈ k , denote it by Mw0 ||I|=k for any k ∈ [n − 1]. This matrix corresponds to a full- rank valuation on the homogeneous coordinate ring of the Grassmannian under the Pl¨ucker embedding.
So, rank(Mw0 ||I|=k)= k(n − k)+1. In particular, we deduce for n ≥ 4

n−1

rank(Mw0 ) = min (k(n − k)+1),N + n − 1 = N + n − 1. (k ) X=1 Direct computations reveals the statement is also true for n ≤ 3.

Based on computational evidence for n ≤ 5 the authors of [BLMM17] state the following conjecture relating the Minkowski property with the tropical flag variety. Conjecture 1. w For n ≥ 3 and w0 a reduced expression of w0 ∈ Sn we have w0 ∈ trop(Fℓn). Moreover, w if w0 lies inside the relative interior of a maximal prime cone of trop(Fℓn), then w0 satisfies MP. We already proved the first part of the conjecture in Lemma 4. The second part is a consequence of Theorem 1 as we explain below.

Theorem 2. Let w be a reduced expression of w ∈ S and consider w ∈ RK . If in (I ) is 0 0 n w0 ww0 n v v prime, then S(An, w0 ) is generated by { w0 (¯pJ ) | 0 6= J ( [n]}, i.e. the Pl¨ucker coordinates form a v Khovanskii basis for (An, w0 ). Moreover, we have

v [n] v [n] Qw0 (ρ) = conv( w0 (¯pJ )) + ··· + conv( w0 (¯pJ )) . J∈( 1 ) J∈(n−1) 5. Application: Rietsch-Williams valuation from plabic graphs 13

Proof. We have M is of full rank and in (I ) = in (I ). So, in (I ) is prime by assumption. w0 ww0 n Mw0 n Mw0 n v v We apply Theorem 1 and deduce that S(An, w0 ) is generated by { w0 (¯pJ ) | 0 6= J ( [n]}. Then

v v [n] v [n] ∆(An, w0 ) = conv( w0 (¯pJ )) + ··· + conv( w0 (¯pJ )) . J∈( 1 ) J∈(n−1)

§ v By [FFL17, 11] Qw0 (ρ) = ∆(An, w0 ), so the claim follows. As a corollary of the above theorem, we prove an even stronger version of the [BLMM17]-conjecture. Corollary 4. w K Let w0 be a reduced expression of w0 ∈ Sn and consider w0 ∈ R . Then in (I ) is prime ⇔ w has the (strong) Minkowski property. ww0 n 0 v v Proof. ”⇒” By Theorem 2, S(An, w0 ) is generated by { w0 (¯pJ )}J([n], which are ray generators of v v C(An, w0 ). Further, the generators are of form w0 (¯pJ ) = (ωk, ·) for all k ∈ [n − 1] and |J| = k. In n−1 + particular, if λ = k=1 akωk ∈ Λ we obtain by (4.1) P n−1 Qw0 (λ)= k=1 akQw0 (ωk). P ”⇐” Assume w0 has the strong Minkowski property. Then, by the reverse argument from above, v v S(An, w0 ) is generated by { w0 (¯pJ )}J([n]. Applying Theorem 1, which we can do by the proof of Theorem 2, it follows that in (I ) is prime. ww0 n

Example 3. Corollary 4 implies what we have seen from computations already, namely that w0 = s1s3s2s3s1s2 ∈ String 4 does not satisfy MP. The reason is that the element

v v v w0 (¯p2p¯134 +¯p1p¯234)  min ≺{ w0 (¯p2p¯134), w0 (¯p1p¯234)} v v v is missing as a generator for S(A4, w0 ). As w0 (¯p2p¯134) = w0 (¯p1p¯234) = (1, 0, 1, 1, 0, 0), we deduce v w0 (¯p2p¯134 +p ¯1p¯234) ≻ (1, 0, 1, 1, 0). Hence, this element can not be obtained from the images of v v Pl¨ucker coordinates under w0 and therefore w0 (¯p2p¯134 +p ¯1p¯234) has to be added as a generator for v v v S(A4, w0 ). In this example, and M are in fact different:

vM (¯p2p¯134 +p ¯1p¯234)=(1, 0, 1, 1, 0, 0), while v(¯p2p¯134 +¯p1p¯234) ≻ (1, 0, 1, 1, 0, 0) =: a.

In particular, vM does not have 1-dimensional leaves as the quotient FvM a/FvM ≻a is two-dimensional. For v this is not the case, as p2p134 = p1p234 in grv(A4).

5 Application: Rietsch-Williams valuation from plabic graphs

In this section we apply Theorem 1 from §3 to the valuation vG defined by Rietsch-Williams for Grassmannians using the cluster structure and Postnikov’s plabic graphs in [RW19]. We identify a class of plabic graphs with non-integral associated Newton–Okounkov polytope. The same combinatorial + n objects are used in [BFF 18] to define weight vectors. For Gr2(C ) we show that these weight vectors n yield the same toric degeneration of Gr2(C ) as the corresponding Newton–Okounkov polytope. n (n)−1 Consider the Grassmannian with its Pl¨ucker embedding Grk(C ) ֒→ P k . In this setting, its homogeneous coordinate ring Ak,n is a quotient of the polynomial ring in Pl¨ucker variables pJ with [n] [n] J ⊂ [n] of cardinality k, denoted J ∈ k . We quotient by the Pl¨ucker ideal Ik,n ⊂ C[pJ |J ∈ k ], a homogenous prime ideal generated by all Pl¨ucker relations. More precisely, for K ∈ [n] and
k−1
[n] #{l∈L|jj} L ∈ k+1 let sgn(j; K,L) := (−1) . Then the associated Pl¨ucker relation
(see e.g. [MS15, §4.3]) is of form

RK,L := j∈L sgn(j; K,L)pK∪{j}pL\{j}. (5.1) P 5. Application: Rietsch-Williams valuation from plabic graphs 14

Then Ak,n = C[pJ ]J /Ik,n. By [Sco06], Ak,n has the structure of a cluster algebra [FZ02]: clusters are sets of algebraically independent algebra generators of Ak,n over an ambient field of rational functions. Together with certain combinatorial data (e.g. a quiver) a cluster forms a seed. They are related by mutation, a procedure creating new seeds from a given one, that recovers the whole algebra after possibly infinitely many recursions. In particular, certain subsets of Pl¨ucker coordinates p¯J ∈ Ak,n are clusters. These special clusters are encoded by combinatorial objects called plabic graphs9 [Pos06]. We recall plabic graphs in the Appendix A.1 below. n For every plabic graph G (or more generally every seed) for Grk(C ) in [RW19] they define a d n valuation vG : Ak,n \{0} → Z where d := k(n − k) = dim Grk(C ). The images of Pl¨ucker coordinates vG (¯pJ ) can be computed using the combinatorics of the plabic graph G. Please consider Appendix A.2 for the precise definition of the valuation and how to compute it. In what follows we use the terminology summarized there without further explanation.

1 ∅

2

5 4 3

rec Figure 1: The plabic graph G of type π3,5 with perfect orientation and source set {1, 2}. Faces are labelled by Young tableaux as described in §A.1. n Let G be a reduced plabic graph for Grk(C ) with perfect orientation chosen such that [k] is the source set, see e.g. Figure 1. Consider the weighting matrix MG := MvG of vG as defined in [n] Appendix A.2. The columns of MG are vG (¯pJ ) for J ∈ k and the rows M1,...,Md+1 are indexed by the faces of the plabic graph G. Denote the boundary faces of G by F ,...,Fn, where Fi is adjacent
1 to the boundary vertices i and i + 1. Hence, Fn = F∅ is the face that does not contribute to the image of vG as Mn = (0,..., 0) and we omit this row of M. Order the rows of MG such that Mi is the row corresponding to the face Fi in G. In the following lemma we compute the columns of M corresponding to boundary faces of G explicitly.

[n] Lemma 6. Let r ∈ [n − 1] and J = {j1,...,jk} ∈ k with j1 < ...js ≤ k < js+1 < ··· < jk. Set th [k] \{j ,...,js} = {i ,...,ik s} with i < ···

(Mr)J = #{l | r ∈ [il, jk−l+1 − 1]}, where [il, jk−l+1] is the cyclic interval in Z/nZ.

Proof. Let f = {pj1 ,..., pjk } be a flow (see Definition 14) from [k] to J, where pji denotes the path with sink ji. The paths pjr for r ≤ s are “lazy paths”, starting and ending at jr without moving. Let [k] \{j1,...,js} = {i1,...,ik−s} with i1 < ··· < ik−s. Hence, for k − l +1 > s we have non-trivial paths pjk−l+1 with source il and sink jk−l+1. To its left are all boundary faces Fr with r in the cyclic interval [il, jk−l+1 − 1]. Recall, the tropicalization of an ideal defined in (3.2). The tropical Grassmannain [SS04] refers n (n) to the tropicalization of the Pl¨ucker ideal Ik,n ⊂ C[pJ ]J , we denote it by trop(Grk(C )) ⊂ R k . It n (k) contains an n-dimensional linear subspace LIk,n := {w ∈ R | inw(Ik,n)= Ik,n}, called lineality space, see [SS04, page 393]. In the following corollary we treat columns of M as weight vectors for Pl¨ucker variables.

9 To be precise, we consider only reduced plabic graphs of trip permutation πn−k,n, for details see Appendix A.1. 5. Application: Rietsch-Williams valuation from plabic graphs 15

Corollary 5. For all r ∈ [n − 1] we have Mr ∈ LIk,n .

[n] [n] Proof. Consider a Pl¨ucker relation RK,L with K ∈ k−1 and L ∈ k+1 of form (5.1). Every term in ′ [n] RK,L equals ±pJ pJ′ for some J, J ∈ k . We can rewrite
the formula
for (Mr)J as follows
|J \ [r − 1]|, if r < k, (M ) = r J |J ∩ [r +1,n]|, if r ≥ k.  ′ ′ ′ ′ ′ Let J = {j1,...,jk}, J = {j1,...,jk} and define the sequence S := (j1,...,jk, j1,...,jk). For any set N we denote by S ∩ N the sequence obtained from S when deleting all entries that are not elements of N. Similarly, let S \ N be the sequence obtained from S when deleting all entries that do belong to N. Further, for any sequence S′ let |S′| denote its length. Then |S \ [r − 1]|, if r < k, (M ) + (M ) ′ = r J r J |S ∩ [r +1,n]|, if r ≥ k.  Note that the right hand side only depends on the entries of S regardless of the ordering. Further, ′ for all pairs J, J corresponding to monomials in RK,L the sequence S is the same up to reordering.

Hence, inMr (RK,L)= RK,L for all r ∈ [n − 1].

Recall the plabic weight vector wG from Definition 15. The following proposition establishes the connection to what we have seen in §3. In terms of the weighting matrix MG, we observe w G = Ej interior face of G Mj , where the sum contains exactly those Mj correspondingP to interior faces of G.

vGrec e35 e25 e45 e15 e12 e23 e34 gGrec f35 f25 f45 f15 f12 f23 f34 p¯12 0000000 p¯12 0000100 p¯13 0000100 p¯13 0 -1 0 1 0 1 0 p¯14 0000110 p¯14 -1 0 0 1 0 0 1 p¯15 0000111 p¯15 0001000 p¯23 0001100 p¯23 0000010 p¯24 0001110 p¯24 -1 1 0 0 0 0 1 p¯25 0001111 p¯25 0100000 p¯34 0101210 p¯34 0000001 p¯35 0101211 p¯35 1000000 p¯45 1101221 p¯45 0010000

rec Table 2: The images of Pl¨ucker coordinates under the valuation vGrec for G as in Figure 1 on the left and under the valuation gGrec on the right. See §A.2&A.3 for details.

Example 4. Consider the plabic graph G = Grec with perfect orientation from Figure 1 and source [5] set [2]. We compute degG(pJ ) and vG (¯pJ ) for all J ∈ 2 . Order the faces of G by

F35 = , F25 = , F45 = ∅, F15 = , F
12 = , F23 = and F34 = .

For example, consider J = {2, 4}. There are two flows, f1 and f2 from [2] to J = {2, 4}. Both consist of only one path from 1 to 4. One of them, say f1, has faces labelled by , , , to its left while the other f2 has faces , , , and to its left. Then with respect to the above order of coordinates (corresponding to faces of G) on Z7 we have

wt(f1)=(0, 0, 0, 1, 1, 1, 0) and wt(f2)=(0, 1, 0, 1, 1, 1, 0).

As degG (f1) = 0 and degG (f2) = 1, we have vG (¯p24) = (1, 0, 1, 1, 0, 0) and degG(p24) = 0. All other vG (¯pJ ) and degG (pJ ) can be recovered from Table 2. 5. Application: Rietsch-Williams valuation from plabic graphs 16

1

En−1 F1 E E E ... 5 4 3 2 F2 Fn−1 F4 F3 ... n n − 1 5 4 3

rec n Figure 2: The plabic graph G for Gr2(C ).

n Proposition 3. For every plabic graph G for Gr2(C ) we have inMG (I2,n) = inwG (I2,n). In general, for k ≥ 3 and n ≥ 6 the statement of the proposition is false, see Example 5 and Theorem 3 below.

rec Proof. Fix the plabic graph G = G . We show that for every Pl¨ucker relation Ri,j,k,l := pij pkl − pikpjl + pilpjk with 1 ≤ i

vGrec (¯p1i) = (0,...... , 0), wGrec (¯p1i)= 0, vGrec (¯p2i) = (0,...... , 0), wGrec (¯p2i)= 0, vGrec (¯pij ) = (1,..., 1, 0,..., 0), wGrec (¯pij ) = i − 2, where i < j and the 1’s in vGrec (¯pij ) correspond to the faces E3,...,Ei. In particular, for Rijkl with

1 ≤ i

1 2 6 1 ∅

∅

6 3 5 2

5 4 4 3

6 Figure 3: Two plabic graphs for Gr3(C ) for which inMG (I3,6) is not prime (see [RW19, §9]).

6 Example 5. For Gr3(C ) we consider the plabic graph G on the left in Figure 3 and compute the 9 images of Pl¨ucker coordinates under vG : A3,6 \{0} → Z (see Table 3 below). With respect to the order on variables as indicated in Table 3 we fix the lexicographic order on Z9. Take the following four-term Pl¨ucker relation written as a sum of two binomials:

(p123p456 − p124p356) + (p125p346 − p126p345) =: f1 + f2 5. Application: Rietsch-Williams valuation from plabic graphs 17

We compute the flow polynomials for f¯1 and f¯2 and obtain: vG(f¯1) = vG(f¯2) = (1, 2, 3, 2, 1, 1, 1, 3, 1). Comparing with the valuations of Pl¨ucker coordinates we deduce vG(f¯i) does not lie in the semigroup- span of {vG(¯pijk) | 1 ≤ i

k − 2 k − 1

Fk−2 ...... A Fk−1

B

D C k Fk Fk+2 Fk+1 k + 3 k + 2 k + 1

Figure 4: On the left: the local structure of a hexagonal plabic graph. On the right: a possible labelling with perfect orientation of a hexagonal plabic graph.

Theorem 3. Let G be a hexagonal plabic graph. Then inMG (Ik,n) is not prime. Moreover, inMG (Ik,n) 6= inwG (Ik,n) and ∆(Ak,n, vG ) is not integral.

Proof. We will show that the value semigroup S(Ak,n, vG ) associated with the valuation of a plabic [n] graph as on the right side of Figure 4 is not generated by {vG (¯pJ ) | J ∈ k } and how this is enough to deduce the claims of the Theorem. Assuming this statement is true by the isomorphism of [BCMN19] d
(see §A.3) the same is true for the valuation gG : Ak,n \{0} → Z sending every Pl¨ucker coordinate to its g-vector (see §A.3). While vG depends on the perfect orientation given to G, the valuation gG purely depends on the combinatorial type of G (on the associated quiver to be precise). If G′ is a hexagonal plabic graph obtained from G by permuting the labelling of the boundary vertices this induces a natural isomorphism between S(Ak,n, gG′ ) =∼ S(Ak,n, gG). Combining with [BCMN19] we obtain

S(Ak,n, vG′ ) =∼ S(Ak,n, gG′ ) =∼ S(Ak,n, gG) =∼ S(Ak,n, vG ).

So without loss of generality that G is perfectly oriented and of form as on the right in Figure 4. Consider the Pl¨ucker relation

f1 + f2 := (p[k]p[k−3]∪{k+1,k+2,k+3} − p[k−1]∪{k+1}p[k−3]∪{k,k+2,k+3})

+ (p[k−1]∪{k+2}]p[k−3]∪{k,k+1,k+3} − p[k−1]∪{k+3}p[k−3]∪{k,k+1,k+2}).

¯ [n] Claim: The image of f1 under vG does not lie in the semigroup span of {vG (¯pJ ) | J ∈ k }.
A. Appendix for §5 18

We compute (the lowest degree terms of) the flow polynomials

p¯[k] = 1,

p¯[k−1]∪{k+1} = xFk (1 + xC ),

p¯[k−1]∪{k+2} = xC xFk xFk+1 ,

p¯[k−1]∪{k+3} = xC xFk xFk+1 xFk+2 , 2 2 2 p¯[k−3]∪{k,k+1,k+2} = xAxB xC xFk−2 xFk−1 xFk xFk+1 , 2 2 2 p¯[k−3]∪{k,k+1,k+3} = xAxB xC xFk−2 xFk−1 xFk xFk+1 xFk+2 (1 + xD + h.o.t.), 2 2 2 2 p¯[k−3]∪{k,k+2,k+3} = xAxB xC xDxFk−2 xFk−1 xFk xFk+1 xFk+2 (1 + h.o.t.), 2 2 3 2 p¯[k−3]∪{k+1,k+2,k+3} = xAxB xC xDxFk−2 xFk−1 xFk xFk+1 xFk+2 .

Ordering the variables corresponding to faces of G by A,B,C,D,Fk−2, Fk−1, Fk, Fk+1, Fk+2 followed by the ones not displayed in Figure 4 we obtain

vG (¯p[k]p¯[k−3]∪{k+1,k+2,k+3}) = (1, 1, 2, 1, 1, 2, 3, 2, 1, 0,..., 0),

vG (¯p[k−1]∪{k+1}p¯[k−3]∪{k,k+2,k+3}) = (1, 1, 2, 1, 1, 2, 3, 2, 1, 0,..., 0),

vG(¯p[k−1]∪{k+2}]p¯[k−3]∪{k,k+1,k+3}) = (1, 1, 3, 0, 1, 2, 3, 2, 1, 0,..., 0),

vG (¯p[k−1]∪{k+3}p¯[k−3]∪{k,k+1,k+2}) = (1, 1, 3, 1, 1, 2, 3, 2, 1, 0,..., 0),

vG (f¯1)= vG (f¯2) = (1, 1, 3, 1, 1, 2, 3, 2, 1, 0,..., 0).

If vG(f¯1) would lie in the semigroup span of the values of Pl¨ucker coordinates, by Corollary 5 it has to be of the form ¯ vG (f1)= vG (¯pI1 )+ vG (¯pI2 ), [n] for I1, I2 ∈ k such that I1 ∩ I2 = [k − 3] and I1 ∪ I2 = [k + 3]. The values of such Pl¨ucker coordinates resemble the ones in the case of Gr (C6) from Table 3. This can be seem by making the following
3 identifications: identify faces of the plabic graph G and the one on the left in Figure 3:

←→ A, ←→ B, ←→ C, ←→ D, ←→ Fk−2,..., ←→ Fk+2, and identify Pl¨ucker coordinatesp ¯i,j,l ←→ p¯[k−3]∪{i+k−3,j+k−3,l+k−3} for 1 ≤ i

Theorem 1 it follows that inMG (Ik,n) is not prime. Further, we have inMG (f1 + f2) = f1 and ¯ inwG (f1 + f2) = f1 + f2. Hence, inMG (Ik,n) 6= inwG (Ik,n). Lastly, the ray spanned by vG (f1) gives a vertex of ∆(Ak,n, vG ) that is of form

1 1 3 1 1 3 1 , , , , , 1, , 1, , 0,..., 0 , 2 2 2 2 2 2 2   which makes ∆(Ak,n, vG ) non-integral.

A Appendix for §5 A.1 Plabic graphs We review the definition of plabic graphs due to Postnikov [Pos06]. This section is closely oriented towards [RW19] and [BFF+18]. A. Appendix for §5 19

p¯123 00000 0000 p¯234 11100 0010 p¯124 00100 0000 p¯235 11110 0010 p¯125 00110 0010 p¯236 11111 0010 p¯126 00111 0010 p¯245 11210 0010 p¯134 01100 0010 p¯246 11211 0010 p¯135 01110 0010 p¯256 11221 0121 p¯136 01111 0010 p¯345 12210 1120 p¯145 01210 0010 p¯346 12211 1120 p¯146 01211 0010 p¯356 12221 1121 p¯156 01220 0121 p¯456 12321 1121

Table 3: The images of Pl¨ucker coordinates under the valuation vG for the plabic graph G as on the left in Figure 3.

Definition 11. A plabic graph G is a planar bicolored graph embedded in a disk (up to homotopy). It has n boundary vertices numbered 1,...,n in a clockwise order. Boundary vertices lie on the boundary of the disk and are not coloured. Additionally, there are internal vertices coloured black or white. Each boundary vertex is adjacent to a single internal vertex. For our purposes we assume that plabic graphs are connected and that every leaf of a plabic graph is a boundary vertex. We first recall the four local moves on plabic graphs.

(M1) If a plabic graph contains a square of four internal vertices with alternating colours, each of which is trivalent, then the colours can be swapped. So every black vertex in the square becomes white and every white vertex becomes black (see Figure 5).

(M2) If two internal vertices of the same colour are connected by an edge, the edge can be contracted and the two vertices can be merged. Conversely, any internal black or white vertex can be split into two adjacent vertices of the same colour (see Figure 5).

Figure 5: Square move (M1), and merging vertices of same colour (M2)

(M3) If a plabic graph contains an internal vertex of degree 2, it can be removed. Equivalently, an internal black or white vertex can be inserted in the middle of any edge (see Figure 6). (R) If two internal vertices of opposite colour are connected by two parallel edges, they can be reduced to only one edge. This can not be done conversely (see Figure 6).

Figure 6: Insert/remove degree two vertex (M3), and reducing parallel edges (R)

The equivalence class of a plabic graph G is defined as the set of all plabic graphs that can be obtained from G by applying (M1)-(M3). If in the equivalence class there is no graph to which (R) can be applied, we say G is reduced. From now on we only consider reduced plabic graphs. A. Appendix for §5 20

Definition 12. Let G be a reduced plabic graph with boundary vertices v1,...,vn labelled in a clockwise order. We define the trip permutation πG as follows. We start at a boundary vertex vi and form a path along the edges of G by turning maximally right at an internal black vertex and maximally left at an internal white vertex. We end up at a boundary vertex vπ(i) and define πG = [π(1),...,π(n)] ∈ Sn. It is a fact that plabic graphs in one equivalence class have the same trip permutation. Further, it was proven by Postnikov in [Pos06, Theorem 13.4] that plabic graphs with the same trip permu- tation are connected by moves (M1)-(M3) and are therefore equivalent. Let πn−k,n = (k +1, k + 2, . . . , n, 1, 2,...,k). From now on we focus on plabic graphs G with trip permutation πG = πn−k,n.

Each path vi to vπn−k,n(i) defined above, divides the disk into two regions. We label every face in the region to the left of the path by i. After repeating this for every 1 ≤ i ≤ n, all faces have a labelling by an k-element subset of [n]. Every such k-element subset defines a Young diagram that fits into an (n − k) × k-rectangle of boxes. We denote by PG the set of all such subsets (resp. their associated Young diagrams) for a fixed plabic graph G. The cardinality of PG is d + 1. A face of a plabic graph is called internal, if it does not intersect with the boundary of the disk. Other faces are called boundary faces. Following [RW19] we define an orientation on a plabic graph. This is the first step in establishing the flow model introduced by Postnikov, which we use to define plabic degrees on the Pl¨ucker coordinates. Definition 13. An orientation O of a plabic graph G is called perfect, if every internal white vertex has exactly one incoming arrow and every internal black vertex has exactly one outgoing arrow. The set of boundary vertices that are sources is called the source set and is denoted by IO.

Postnikov showed in [Pos06] that every reduced plabic graph with trip permutation πn−k,n has a perfect orientation with source set of order k. See Figure 1 for a plabic graph with trip permutation π3,5. Index the standard basis of ZPG = Zd+1 by the faces of the plabic graph G, where d = k(n − k). Given a perfect orientation O on G, every directed path p from a boundary vertex in the source set to PG a boundary vertex that is a sink, divides the disk in two parts. The weight wt(p) ∈ Z≥0 has entry 1 in the position corresponding to a face F of G, if F is to the left of p with respect to the orientation. The degree degG (p) ∈ Z≥0 is defined the number of internal faces to the left of the path. The boundary face between the boundary vertices k and k + 1 never lies to the left of any path and therefore is also referred to as F∅.

Definition 14. For a set of boundary vertices J with |J| = |IO|, we define a J-flow as a collection of self-avoiding, vertex disjoint directed paths with sources IO − (J ∩ IO) and sinks J − (J ∩ IO). Let

IO − (J ∩ IO) = {j1,...,jr} and f = {pj1 ,..., pjr } be a flow where each path pji has sink ji. Then the weight of the flow is wt(f) := wt(pj1 )+ ··· + wt(pjr ). Similarly, we define the degree of the flow as degG(f) = degG (pj1 )+ ··· + degG (pjr ). By FJ we denote the set of all J-flows in G with respect to O.

A.2 Valuation and plabic degree n In [RW19] Rietsch-Williams use the cluster structure on Grk(C ) (due to Scott, see [Sco06]) to define n a valuation on C(Grk(C )) \{0} for every seed. In fact, a plabic graph G defines a seed in the corresponding cluster algebra. A combinatorial algorithm associates a quiver with G (see e.g. [RW19, Definition 3.8]). The corresponding cluster is a set of Pl¨ucker coordinatesp ¯J , where J is a face label in G as described above. Let Ak,n := C[Gr(k,n)]. We recall the definition of the valuation vG from [RW19, §8]. By [Pos06, §6] ∗ PG n n×k for every plabic graph G there exists a map ΦG : (C ) → Grk(C ) sending (xµ)µ∈PG to A ∈ C , where #{i′∈[k]|i

Here xwt(p) denotes the monomial with exponent vector wt(p) ∈ {0, 1}PG . Fix an order on the PG coordinates {xµ}µ∈PG and let ≺ be the total order on Z be the corresponding lexicographic order ∗ ±1 [n] (see [RW19, Definition 8.1]). The pullback of ΦG satisfies ΦG (¯pJ ) ∈ C[xµ |µ ∈ PG] for J ∈ k . Moreover, every polynomial in Pl¨ucker coordinates has a unique Laurent polynomial expression in s mi PG
{xµ}µ∈PG . Suppose f ∈ Ak,n has expression i=1 aix for ai ∈ C and mi ∈ Z . Then the lowest n PG term valuation vG : C(Grk(C )) \{0} → (Z , ≺) is defined as P

vG(f) := min lex{mi | ai 6=0}.

f For a rational function h = g it is defined as vG (h) := vG (f) − vG (g). For Pl¨ucker coordinates the [n] images of vG can be computed explicitly using the combinatorics of the plabic graph. For J ∈ k let fJ ∈FJ be the flow with deg (fJ ) = min {deg (f) | f ∈FJ }. Then on a Pl¨ucker coordinatep ¯J ∈ Ak,n G ≺ G
the valuation vG is given by

PG vG (¯pJ ) = wt(fJ ) ∈ Z . (A.1)

In fact, PG contains one label µ whose corresponding face never contributes to the weight of any flow. With our convention it is the boundary face between the vertices k and k + 1, call it F∅. Therefore, PG −∅ d we can omit the corresponding variable and have vG : Ak,n \{0} → Z =∼ Z . In particular, this n implies that the rank of vG is d = dim(Grn(C )) which is one less than the Krull-dimension of Ak,n. In order to have a valuation that fits into our framework, we slightly modify vG and define

d+1 vˆG : Ak,n \{0} → Z given by vˆG(f) = (deg f, vG (f)). (A.2)

(d+1)× n d× n ˆ (k) (k) Note that vG is a full-rank valuation on Ak,n. Further, MvˆG ∈ Z differs from MG ∈ Z by n (k) the row (1,..., 1) ∈ Z . As Ik,n is homogeneous, seen as a weight vector, we have (1,..., 1) ∈ LIk,n . In particular,

inM (Ik,n) = inMv (Ik,n). (A.3) G ˆG

In [BFF+18] they define closely related to the valuation the following notion of degree for Pl¨ucker (n) variables in C[pJ ]J and associate a weight vector in Z k . For an example, consider Example 4 above.

[n] Definition 15. For J ∈ k and a plabic graph G, the plabic degree of the Pl¨ucker variable pJ is defined as
degG (pJ ) := min{degG(f) | f ∈FJ } ∈ Z≥0.

n (k) It gives rise to the plabic weight vector wG ∈ Z defined by (wG )J := degG (pJ ). By [PSW09, Lemma 3.2] and its proof, the plabic degree is independent of the choice of the perfect orientation. We therefore fix the perfect orientation by choosing the source set IO = [k]. The following proposition guarantees that the degree (and the valuation) are well-defined. It is a reformulation of the original statement adapted to our notion degree. Proposition. ([RW19, Corollary 12.4]) There is a unique J-flow in G with respect to O with degree equal to degG (pJ ).

A.3 [GHKK18]’s g-vector valuation vs. [RW19]’s valuation Gross, Hacking, Keel and Kontsevich construct vector space bases for cluster algebras in [GHKK18]. Among other powerful applications their so-called theta basis can be used to construct toric degenera- tions for partially compactified cluster varieties. For the Grassmannian their toric degeneration can be formulated in terms of valuations and Newton- Okounkov bodies. They rely on Fomin and Zelevinsky’s principal coefficients (introduced in [FZ07]) A. Appendix for §5 22 and the associated multiweights for cluster monomials called g-vectors. Generalized g-vectors are introduced in [GHKK18, Definition 5.10] for elements of the theta basis. For any seed s the assignment d of its g-vector to a theta basis element extends to a full-rank valuation gs : Ak,n \{0} → Z . For us the following straight forward lemma is important.

Lemma 7. For a plabic graph G consider the set of Pl¨ucker coordinatedp ¯J , where J ∈PG − ∅. Then the matrix with rows gG(¯pJ ) is (up to permutation of the rows) the identity matrix. The following result is a direct consequence of the main theorem in [BCMN19]:

∗ d d Theorem ([BCMN19]). For every plabic graph G there exists a linear mapp ¯G : R → R inducing a unimodular equivalence between ∆(Ak,n, vG ) =∼ ∆(Ak,n, gG).

rec 5 7 Example 6. We consider the plabic graph G for Gr2(C ) as in Figure 1. For a lattice N =∼ Z we rec ∗ fix an ordered basis {e35,e25,e45,e12,e23,e34} corresponding to the faces of G . Let M := N be the ∗ dual lattice with dual basis (in order) {f35,f25,f45,f12,f23,f34}. Thenp ¯Grec : N ⊗Z R → M ⊗Z R with respect to our chosen bases is given by the matrix

0 1 −1 0 0 −1 1 −10 0 1 −1 1 0 1 0 1 −1 0 0 −1 0 −10 1 0 0 0  0 1 0 −1 1 −1 0  1 −10 0 0 1 −1  −1000001    The Newton–Okounkov cone C(A2,5, vGrec ) is cut out by the tropicalized Marsh-Rietsch superpotential rec W = W1 + ··· + W5 (defined in [MR13]). Expressed in the Pl¨ucker coordinates corresponding to G it is of form

p¯34 p¯23p¯45 p¯12p¯45 p¯25 p¯15 p¯12p¯35 p¯25 p¯23p¯45 p¯35 W1 = + + , W2 = q , W3 = + , W4 = + , W5 = . p¯35 p¯35p¯25 p¯15p¯25 p¯12 p¯25 p¯23p¯25 p¯35 p¯34p¯35 p¯45

∗ fij The convention to make sense ofp ¯ as written above isp ¯ij := z . Note that the image of vGrec lies inside a hyperplane defined by e45 = 0. By [RW19] the Newton–Okounkov polytope associated to vGrec : A2,5 \{0} → N is

trop trop trop trop trop ∆(A2,5, vGrec )= {W1 ≥ 0}∩{W2 ≥−1}∩{W3 ≥ 0}∩{W4 ≥ 0}∩{W5 ≥ 0}.

trop For example, W5 = f35 − f45 is the normal vector for the inequality e35 − e45 ≥ 0. The lattice points of ∆(A2,5, vGrec ) can be found on the left in Table 2. The tropicalized Gross-Hacking-Keel-Kontsevich potential WGHKK = ϑ1 + ··· + ϑ5 cuts out the Newton–Okounkov cone C(A2,5, gGrec ). Expressed in our seed we have

−e15 −e25 −e35 −e12 −e23 −e25 −e34 −e25 −e45 ϑ1 = z (1 + z (1 + z )), ϑ2 = z , ϑ3 = z (1 + z ), ϑ4 = z (1 + z ), ϑ5 = z .

∗ The image of ∆(A2,5, vGrec ) underp ¯Grec is

trop trop trop trop trop P = {ϑ1 ≥ 0}∩{ϑ2 ≥−1}∩{ϑ3 ≥ 0}∩{ϑ4 ≥ 0}∩{ϑ5 ≥ 0}.

It is the Newton–Okounkov polytope for a linearly equivalent divisor. The Newton–Okounkov polytope trop ∆(A2,5, gGrec ) associated to gGrec : A2,5 \{0} → M equals {WGHKK ≤ 0}∩{ fij = 1}. It is unimodularly equivalent to P . The polytope ∆(A2,5, gGrec ) is sent to P by the a linear map qGrec : P M ⊗Z R → M ⊗Z R defined by 1 0 2 000 0 0 1 −1000 0 −2 −2 −2 −1 0 −1 −1 0 1 0 001 0 .  −1 −1 −1000 1  −1 0 −1000 0  1 1 1 100 0    6. References 23

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