Newton-Okounkov bodies of flag varieties and combinatorial mutations

A Newton-Okounkov body is a convex body constructed from a polarized variety with a higher rank valuation on the function field, which gives a systematic method of constructing toric degenerations of polarized varieties. Its combinatorial properties heavily depend on the choice of a valuation, and it is a fundamental problem to relate Newton-Okounkov bodies associated with different kinds of valuations. In this paper, we address this problem for flag varieties using the framework of combinatorial mutations which was introduced in the context of mirror symmetry for Fano manifolds. By applying iterated combinatorial mutations, we connect specific Newton-Okounkov bodies of flag varieties including string polytopes, Nakashima-Zelevinsky polytopes, and FFLV polytopes.


Introduction
A Newton-Okounkov body ∆(X, L, v) is a convex body constructed from a polarized variety (X, L) with a higher rank valuation v on the function field C(X), which generalizes the notion of Newton polytopes for toric varieties to arbitrary projective varieties. It was introduced by Okounkov [37,38,39] and afterward developed independently by Kaveh-Khovanskii [29] and by Lazarsfeld-Mustata [33]. A remarkable fact is that the theory of Newton-Okounkov bodies gives a systematic method of constructing toric degenerations (see [2,Theorem 1] and [23,Corollary 3.14]). Since combinatorial properties of ∆(X, L, v) heavily depend on the choice of a valuation v, it is a fundamental problem to give concrete relations among Newton-Okounkov bodies associated with different kinds of valuations. In the case of flag varieties and Schubert varieties, their Newton-Okounkov bodies realize the following representation-theoretic polytopes: (i) Berenstein-Littelmann-Zelevinsky's string polytopes [28], (ii) Nakashima-Zelevinsky polytopes [19], (iii) FFLV (Feigin-Fourier-Littelmann-Vinberg) polytopes [13,31], where the attached references are the ones giving realizations as Newton-Okounkov bodies. The set of lattice points in every polytope of (i)-(iii) parametrizes a specific basis of an irreducible highest weight module of a semisimple Lie algebra. In particular, string polytopes and Nakashima-Zelevinsky polytopes give polyhedral parametrizations of crystal bases (see [5,34,35,36]). Our aim in the present paper is to relate these polytopes by applying iterated combinatorial mutations.
Combinatorial mutations for lattice polytopes were introduced by Akhtar-Coates-Galkin-Kasprzyk [1] in the context of mirror symmetry for Fano manifolds. The original motivation in [1] is to classify Fano manifolds by using combinatorial mutations. A Laurent polynomial f ∈ C[z ±1 1 , . . . , z ±1 m ] is said to be a mirror partner of an m-dimensional Fano manifold X if the period π f of f coincides with the quantum period G X of X (see [1,7] and references therein for more details). Note that Laurent polynomials having the same period are not unique. In order to relate Laurent polynomials having the same period, combinatorial mutations are useful. Indeed, it is proved in [1,Lemma 1] that if Laurent polynomials f and g are connected by iterated combinatorial mutations, then the period of f is equal to that of g. The notion of combinatorial mutations for lattice polytopes just rephrases that for Laurent polynomials in terms of their Newton polytopes. Notation 1.1. We adopt the standard notation in toric geometry. Let N Z m be a Z-lattice of rank m, and M := Hom Z (N, Z) Z m its dual lattice. We write N R := N ⊗ Z R and M R := M ⊗ Z R. Denote by ·, · : M R × N R → R the canonical pairing.
The combinatorial mutation in [1] is an operation for lattice polytopes in N R . For a rational convex polytope ∆ ⊆ M R with a unique interior lattice point a, we define its dual ∆ ∨ to be the polar dual of its translation: A combinatorial mutation on ∆ ∨ corresponds to a piecewise-linear operation on ∆ − a, which is extended to the whole of M R . We call it a combinatorial mutation in M R . We consider this framework when ∆ is a Newton-Okounkov body of a flag variety.
To state our results more explicitly, let G be a simply-connected semisimple algebraic group over C, B a Borel subgroup of G, W the Weyl group, and P + the set of dominant integral weights. We denote by X(w) ⊆ G/B the Schubert variety corresponding to w ∈ W , by L λ the globally generated line bundle on X(w) associated with λ ∈ P + , and by ρ ∈ P + the half sum of the positive roots. Let R(w) be the set of reduced words for w ∈ W , and w 0 ∈ W the longest element. The Schubert variety X(w 0 ) corresponding to w 0 coincides with the full flag variety G/B. Let ∆ i (λ) (resp., ∆ i (λ)) denote the string polytope (resp., the Nakashima-Zelevinsky polytope) associated with i ∈ R(w) and λ ∈ P + . In order to relate string polytopes and Nakashima-Zelevinsky polytopes by combinatorial mutations, we use the theory of cluster algebras. Cluster algebras were introduced by Fomin-Zelevinsky [16,17] to develop a combinatorial approach to total positivity and to the dual canonical basis. Fock-Goncharov [15] introduced a cluster ensemble (A, X ) which gives a more geometric point of view to the theory of cluster algebras. Gross-Hacking-Keel-Kontsevich [22] developed the theory of cluster ensembles using methods in mirror symmetry, and proved that the theory of cluster algebras also can be used to obtain toric degenerations of projective varieties. Let U − w ⊆ G be the unipotent cell associated with w ∈ W , which is naturally regarded as an open subvariety of X(w). Berenstein-Fomin-Zelevinsky [4] gave an upper cluster algebra structure on the coordinate ring C[U − w ]. When G is simply-laced, the first named author and Oya [21] constructed a family {∆(X(w), L λ , v s )} s∈S of Newton-Okounkov bodies parametrized by the set of seeds for C[U − w ] such that • this family contains ∆ i (λ) and ∆ i (λ) for all i ∈ R(w) up to unimodular transformations, • the Newton-Okounkov bodies ∆(X(w), L λ , v s ), s ∈ S, are all rational convex polytopes, • the Newton-Okounkov bodies ∆(X(w), L λ , v s ), s ∈ S, are all related by tropicalized cluster mutations. If w = w 0 and λ = 2ρ, then the Newton-Okounkov body ∆(G/B, L 2ρ , v s ) contains exactly one lattice point a s in its interior, and the dual ∆(G/B, L 2ρ , v s ) ∨ is a lattice polytope (see Theorem 3.3 and Corollary 3.13). Realizing tropicalized cluster mutations as combinatorial mutations in M R , we obtain the following.
Theorem 1 (Theorem 4.9). If G is simply-laced, then the following hold.
(1) For fixed w ∈ W and λ ∈ P + , the Newton-Okounkov bodies ∆(X(w), L λ , v s ), s ∈ S, are all related by combinatorial mutations in M R up to unimodular transformations. (2) For w = w 0 and λ = 2ρ, the translated polytopes ∆(G/B, L 2ρ , v s ) − a s , s ∈ S, are all related by combinatorial mutations in M R up to unimodular transformations. In particular, the dual polytopes ∆(G/B, L 2ρ , v s ) ∨ , s ∈ S, are all related by combinatorial mutations in N R up to unimodular transformations.
In order to relate FFLV polytopes with these Newton-Okounkov bodies, we use Ardila-Bliem-Salazar's transfer map [3] between the Gelfand-Tsetlin polytope GT (λ) and the FFLV polytope F F LV (λ) in type A n , where λ ∈ P + . Note that the Gelfand-Tsetlin polytope GT (λ) is unimodularly equivalent to the string polytope ∆ i (λ) associated with specific i ∈ R(w 0 ) (see Example 3.6). We realize their transfer map as a composition of combinatorial mutations in M R (see Theorem 5.5). Combining this with Theorem 1, we obtain the following in type A n .
are all related by combinatorial mutations in N R up to unimodular transformations.
Note that Ardila-Bliem-Salazar [3] gave such a transfer map also in type C n . Since their transfer map in type C n can be also described as a composition of combinatorial mutations in M R , we obtain the following.
(1) For fixed λ ∈ P + , the Gelfand-Tsetlin polytope GT Cn (λ) and the FFLV polytope F F LV Cn (λ) are related by combinatorial mutations in M R up to translations by integer vectors. (2) The dual polytopes GT Cn (2ρ) ∨ and F F LV Cn (2ρ) ∨ are related by combinatorial mutations in N R .
In order to relate string polytopes and Nakashima-Zelevinsky polytopes by combinatorial mutations, we use Newton-Okounkov bodies of flag varieties arising from cluster structures. In Sect. 2.1, we recall the definitions of higher rank valuations and Newton-Okounkov bodies. We also review their basic properties. In Sect. 2.2, we define valuations using cluster structures, following [21].
2.1. Basic definitions on Newton-Okounkov bodies. We first recall the definition of Newton-Okounkov bodies, following [23,28,29,30]. Let R be a C-algebra without nonzero zero-divisors, and m ∈ Z >0 . We fix a total order ≤ on Z m respecting the addition.
Note that we need to fix a total order on Z m whenever we consider a valuation. For a ∈ Z m and a valuation v on R with values in Z m , we define a C-subspace R a ⊆ R by Then the leaf above a ∈ Z m is defined to be the quotient space Example 2.2. Fix a total order ≤ on Z m respecting the addition, and let C(z 1 , . . . , z m ) be the field of rational functions in m variables. The total order ≤ on Z m induces a total order (denoted by the same symbol ≤) on the set of Laurent monomials in z 1 , . . . , z m as follows: where c ∈ C × , and the summand "(higher terms)" stands for a linear combination of Laurent monomials bigger than z a1 1 · · · z am m with respect to ≤.

Then this map v low
≤ is a well-defined valuation with 1-dimensional leaves with respect to the total order ≤. We call v low ≤ the lowest term valuation with respect to ≤. Definition 2.3 (see [28,Sect. 1.2] and [30, Definition 1.10]). Let X be an irreducible normal projective variety over C, L a line bundle on X generated by global sections, and m := dim C (X). Take a valuation v : C(X) \ {0} → Z m with 1-dimensional leaves, and fix a nonzero section τ ∈ H 0 (X, L). We define a subset S(X, this is called the Newton-Okounkov body of (X, L) associated with (v, τ ).
The definition of valuations implies that S(X, L, v, τ ) is a semigroup. Hence it follows that C(X, L, v, τ ) is a closed convex cone, and that ∆(X, L, v, τ ) is a convex set. In addition, we deduce by [30,Theorem 2.30] that ∆(X, L, v, τ ) is a convex body, i.e., a compact convex set. If L is ample, then it follows from [30, Corollary 3.2] that the real dimension of ∆(X, L, v, τ ) equals m; this is not necessarily the case when L is not ample. By definition, we have Since S(X, L, v, τ ) is a semigroup, the definition of Newton-Okounkov bodies implies that Remark 2.4. If we take another nonzero section τ ∈ H 0 (X, L), then it follows that which implies that the Newton-Okounkov body ∆(X, L, v, τ ) does not essentially depend on the choice of τ . Hence it is also denoted simply by ∆(X, L, v).

2.2.
Cluster algebras and valuations. The first named author and Oya [21] constructed valuations using the theory of cluster algebras. In this subsection, we review this construction. We first recall the definition of (upper) cluster algebras of geometric type, following [4,17]. Note that we use the notation in [15,22]. Fix a finite set J and a subset J uf ⊆ J. We write J fr := J \ J uf . Let F := C(z j | j ∈ J) be the field of rational functions in |J| variables. Then a seed s = (A, ε) of F is a pair of • a J-tuple A = (A j ) j∈J of elements of F, and • ε = (ε i,j ) i∈J uf ,j∈J ∈ Mat J uf ×J (Z) such that (i) A forms a free generating set of F, and Then µ k (s) is again a seed of F, and we have µ k µ k (s) = s. Two seeds s and s are said to be mutation equivalent if there exists a sequence (k 1 , k 2 , . . . , k ) in J uf such that Let T be the |J uf |-regular tree whose edges are labeled by J uf so that the |J uf |-edges emanating from each vertex receive different labels. If t, t ∈ T are joined by an edge labeled by k ∈ J uf , i,j ) i∈J uf ,j∈J . Definition 2.6 (see [4, Definitions 1.6 and 1.11]). We set which is called an upper cluster algebra of geometric type. The (ordinary) cluster algebra A (S) of geometric type is defined to be the C-subalgebra of F generated by We usually fix t 0 ∈ T, and construct a cluster pattern S = {s t } t∈T from a seed s t0 . In this case, s t0 is called the initial seed.
for all t ∈ T; this property is called the Laurent phenomenon. In particular, A (S) is included in the upper cluster algebra U (S).
In the rest of this subsection, we assume that ( †) the exchange matrix ε t0 is of full rank for some t 0 ∈ T, which implies that ε t is of full rank for all t ∈ T; see [4, Lemma 3.2].
Definition 2.8 ([40, Definition 3.1.1]). Let S = {s t = (A t , ε t )} t∈T be a cluster pattern, and fix t ∈ T. For a, a ∈ Z J , we write a εt a if and only if a = a + vε t for some v ∈ Z J uf ≥0 , where elements of Z J (resp., Z J uf ≥0 ) are regarded as 1 × J (resp., 1 × J uf ) matrices. This εt defines a partial order on Z J , called the dominance order associated with ε t . Definition 2.9 ([21, Definition 3.8]). Let S = {s t = (A t , ε t )} t∈T be a cluster pattern, and op εt the opposite order of εt . We fix a total order ≤ t on Z J which refines op εt . It induces a total order (denoted by the same symbol ≤ t ) on the set of Laurent monomials in Following [17,15], we define a tropicalized cluster mutation as follows: for t k -t , for j ∈ J. As we review in Sect. 3.2, the tropicalized cluster mutation µ T k can be used to connect Newton-Okounkov bodies associated with v t and v t .

Case of flag and Schubert varieties
In this section, we restrict ourselves to the case of flag varieties and Schubert varieties. In Sect. 3.1, we review fundamental properties of these varieties, and recall basic facts on their Newton-Okounkov bodies. In Sect. 3.2, we review results of [21], which connect string polytopes and Nakashima-Zelevinsky polytopes by tropicalized cluster mutations.
3.1. String polytopes and Nakashima-Zelevinsky polytopes. Let G be a connected, simplyconnected semisimple algebraic group over C, and g its Lie algebra. Choose a Borel subgroup B ⊆ G and a maximal torus H ⊆ B. Then the full flag variety is defined to be a quotient space G/B, which is a nonsingular projective variety. Denote by h ⊆ g the Lie algebra of H, by h * := Hom C (h, C) its dual space, and by ·, · : h * × h → C the canonical pairing. Let P ⊆ h * be the weight lattice for g, P + ⊆ P the set of dominant integral weights, {α i | i ∈ I} ⊆ P the set of simple roots, {h i | i ∈ I} ⊆ h the set of simple coroots, and the Cartan matrix. Denote by N G (H) the normalizer of H in G, and by W := N G (H)/H the Weyl group of g. The Weyl group W is generated by the set {s i | i ∈ I} of simple reflections. We call i = (i 1 , . . . , i m ) ∈ I m a reduced word for w ∈ W if w = s i1 · · · s im and if m is minimum in such expressions of w. In this case, the length m is called the length of w, which is denoted by (w). Let R(w) be the set of reduced words for w. The Schubert variety X(w) is a normal projective variety of complex dimension (w) (see, for instance, [26, Sects. II.13.3, II.14.15]). If w is the longest element w 0 in W , then the Schubert variety X(w 0 ) coincides with the full flag variety G/B. For λ ∈ P + , we define a line bundle L λ on G/B by where B acts on G × C from the right as follows: for g ∈ G, c ∈ C, and b ∈ B. By restricting this bundle, we obtain a line bundle on X(w), which we denote by the same symbol L λ . By [ where ρ ∈ P + denotes the half sum of the positive roots. For λ ∈ P + , let V (λ) be the irreducible highest weight G-module over C with highest weight λ. We fix a highest weight vector v λ of V (λ). The Demazure module V w (λ) corresponding to w ∈ W is defined to be the B-submodule of V (λ) given by We fix a lowest weight vector τ λ ∈ H 0 (G/B, L λ ). By restricting this section, we obtain a section in H 0 (X(w), L λ ), which we denote by the same symbol τ λ . Let ∆ i (λ) (resp., ∆ i (λ)) denote the string polytope (resp., the Nakashima-Zelevinsky polytope) associated with i ∈ R(w) and λ ∈ P + ; see [ Kaveh [28] proved that the string polytope ∆ i (λ) is identical to the Newton-Okounkov body ∆(X(w), L λ , v high i , τ λ ) of (X(w), L λ ) associated with a highest term valuation v high i . Using a different kind of highest term valuationṽ high i , the first named author and Naito [19] showed that the Nakashima-Zelevinsky polytope ∆ i (λ) can be realized as a Newton-Okounkov body ∆(X(w), L λ ,ṽ high i , τ λ ). Afterward, the first named author and Oya [20] proved that the Newton-Okounkov body ∆(X(w), L λ , v high i , τ λ ) (resp., ∆(X(w), L λ ,ṽ high i , τ λ )) is also identical to the one associated with a valuation given by counting the orders of zeros/poles along a specific sequence of Schubert varieties. This description leads to the realization of ∆ i (λ) (resp., ∆ i (λ)) in [21] as a Newton-Okounkov body arising from a cluster structure, which is reviewed in the next subsection.
In the context of mirror symmetry, when G is of type A n , Rusinko [41,Theorem 7] proved that the polar dual of the (properly translated) string polytope ∆ i (2ρ) is a lattice polytope for all i ∈ R(w 0 ). Using Hibi's criterion [24] on the integrality of the vertices of the dual polytopes, Steinert [43] generalized this result to all Lie types as follows. Theorem 3.3 (see [43,Sects. 4,6]). Take a valuation v : C(G/B) \ {0} → Z dim C (G/B) with 1dimensional leaves, and fix a nonzero section τ ∈ H 0 (G/B, L 2ρ ). If the semigroup S(G/B, L 2ρ , v, τ 2ρ ) is finitely generated and saturated, then the Newton-Okounkov body ∆(G/B, L 2ρ , v, τ 2ρ ) contains exactly one lattice point in its interior. In addition, the dual ∆(G/B, L 2ρ , v, τ 2ρ ) ∨ in the sense of Sect. 1 is a lattice polytope.
Remark 3.4. In the paper [43], the algebraic group G is assumed to be simple. However, the proof of Theorem 3.3 can also be applied to the case that G is semisimple. Let X n be the Lie type of G, and define i Xn ∈ R(w 0 ) as follows.
• If G is of type A n , then ) ∈ I n(n−1) .
• If G is of type E 8 , then ) ∈ Z n(n−1) .
Example 3.6 ([34, Corollary 5]). Let G = SL n+1 (C), and λ ∈ P + . We consider the reduced word i An ∈ R(w 0 ) in Example 3.5. Then the string polytope ∆ i An (λ) is unimodularly equivalent to the Gelfand-Tsetlin polytope GT (λ) which is defined to be the set of (a 1 , a 1 , a 2 , a 1 , a 2 , a satisfying the following conditions: where λ ≥k := k≤ ≤n λ, h for 1 ≤ k ≤ n, and the notation Then the string polytope ∆ i Cn (λ) is unimodularly equivalent to the Gelfand-Tsetlin polytope GT Cn (λ) of type C n which is defined to be the set of (a 1 , a 2 , a 3 , a 2 , a n−1 , a (1) n , . . . , a (n) 1 2n−1 ) ∈ R n 2 satisfying the following conditions as in Example 3.6: where λ ≥k := k≤ ≤n λ, h for 1 ≤ k ≤ n.

Newton-Okounkov bodies of Schubert varieties arising from cluster structures.
In this subsection, we assume that G is simply-laced. Let B − ⊆ G denote the Borel subgroup opposite to B, and U − the unipotent radical of B − . We regard U − as an affine open subvariety of G/B by the following open embedding: For w ∈ W , we set Define a J uf × J-integer matrix ε i = (ε s,t ) s∈J uf ,t∈J by ε s,t := For s ∈ J, we set D(s, i) := D w ≤s is , is .
Let S = {s t = (A t , ε t )} t∈T be the cluster pattern whose initial seed is given as s t0 = ((A s;t0 ) s∈J , ε i ).   . If G is simply-laced, then the following hold for all w ∈ W , λ ∈ P + , and t ∈ T.
(1) The Newton-Okounkov body ∆(X(w), L λ , v st , τ λ ) is independent of the choice of a refinement of the opposite dominance order op εt .

Combinatorial mutations on Newton-Okounkov bodies
In this section, we recall the notion of combinatorial mutations for lattice polytopes which was developed by Akhtar-Coates-Galkin-Kasprzyk in [1]. There are two kinds of combinatorial mutations: one is the operation in N R -side and the other one is in M R -side. Our main interest is the operation in M R -side (see Definition 4.3) and this is originally defined as a "dual version" of the operation in N R -side. See Proposition 4.4.

4.1.
Basic definitions on combinatorial mutations. We first introduce combinatorial mutations for lattice polytopes in N R . Let P ⊆ N R be a lattice polytope, and take w ∈ M . For h ∈ Z, write H w,h := {v ∈ N R | w, v = h}, and P w,h := P ∩ H w,h .
We use the notation w ⊥ instead of H w,0 . Let V (P ) ⊆ N denote the set of vertices of P . For each subset A ⊆ N R , we set A + ∅ = ∅ + A = ∅. . Let w ∈ M be a primitive vector, and take a lattice polytope F which sits in w ⊥ . Suppose that for every negative integer h, there exists a possibly-empty lattice polytope G h ⊆ N R such that the inclusions hold. Then we define the lattice polytope mut w (P, F ) as follows: Note that G h and P w,h + hF are empty except for finitely many h's. We call the lattice polytope mut w (P, F ) (or the operation mut w (−, F )) the combinatorial mutation in N R of P with respect to w and F . When (4.1) is satisfied, we say that mut w (P, F ) is well-defined.
It is proved in [1, Proposition 1] that mut w (P, F ) is independent of the choice of {G h } h .
Remark 4.2. In [25], the definition of combinatorial mutations in N R has been extended to rational convex polytopes and unbounded polyhedra. See [25,Sect. 2] for more details.
Next, we introduce another operation, which is a piecewise-linear transformation on M R .

Definition 4.3 ([1, Sect. 3]
; see also [25,Definition 3.1]). Let w ∈ M be a primitive vector, and take a lattice polytope F which sits in w ⊥ . We define a map ϕ w,F : We call the piecewise-linear map ϕ w,F a combinatorial mutation in M R .
Therefore, we see that mut w (P, F ) * = ϕ w,F (P * ) as in Proposition 4.4; see also We now introduce the notion of combinatorial mutation equivalence.
Definition 4.7 (see [25,Definition 3.5]). Two lattice polytopes P and P in N R are said to be combinatorially mutation equivalent in N R if there exists a sequence ((w 1 , F 1 ), . . . , (w , F )), where w i ∈ M is primitive and F i ⊆ w ⊥ i is a lattice polytope, such that P = mut w ((· · · mut w2 (mut w1 (P, F 1 ), F 2 ) · · · ), F ). Similarly, two rational convex polytopes Q and Q in M R are said to be combinatorially mutation equivalent in M R if there exists a sequence ((w 1 , F 1 ), . . . , (w , F )), where w i ∈ M is primitive and F i ⊆ w ⊥ i is a lattice polytope, such that Q = ϕ w ,F (· · · (ϕ w1,F1 (Q)) · · · ) and the image of each of the intermediate steps is always a rational convex polytope.

Tropicalized cluster mutations as combinatorial mutations.
Our main interest is the map ϕ w,F in Definition 4.3. We first prove the following.
Proposition 4.8. For k ∈ J uf , the tropicalized cluster mutation µ T k : R J → R J can be described as a composition of a combinatorial mutation in M R and f ∈ GL J (Z).
Proof. Recall that µ T k : R J → R J , (g j ) j∈J → (g j ) j∈J , is defined by For j ∈ J, we write the j-th unit vector of R J as e j ∈ R J . Define u k = (u k,j ) j ∈ Z J by u k,j := min{ε (t) k,j , 0} (j = k), 2 (j = k).
Let f : R J → R J be a linear map defined by the matrix (f i,j ) i,j∈J whose i-th row is e i if i = k and e k − u k if i = k, where f acts on g ∈ R J from the right, that is, we regard g as a row vector. Then we notice that f ∈ GL J (Z). Let us write w := 1 k is the greatest common divisor of the absolute values of the nonzero entries of ε (t) k . We set F := conv(0, −c (t) k e k ), where 0 denotes the origin of R J . Notice that w is primitive, and F ⊆ w ⊥ since the k-th entry of ε (t) k is 0, which follows from the skew-symmetrizability of ε • .
Our goal is to show that µ T k = f • ϕ w,F as maps. For g = (g j ) j∈J ∈ R J , the direct computation shows that Moreover, we see the following: which coincides with µ T k (g) in the case g k ≤ 0. Similar to this, we obtain the following: which coincides with µ T k (g) in the case g k ≥ 0. This proves the proposition. Recall from Corollary 3.13 that a t denotes the unique interior lattice point of ∆(G/B, L 2ρ , v st , τ 2ρ ). As an application of Theorem 3.11 and Proposition 4.8, let us prove the following. Theorem 4.9. If G is simply-laced, then the following hold.
for j ∈ J. In particular, a t is independent of the choice of t ∈ T, and fixed under the tropicalized cluster mutations.
Proof. Since g is isomorphic to a direct sum of simply-laced simple Lie algebras as a Lie algebra, there exists i g ∈ R(w 0 ) which is a concatenation of reduced words i Xn defined in Example 3.5.
Combining the computation of a Xn in Example 3.5 with Theorem 3.11 (4), we deduce the assertion for s t0 = s ig . We proceed by induction on the distance from t 0 in T. Take t, t ∈ T and k ∈ J uf such that t k -t . We assume that the assertion holds for t. By definition, the tropicalized cluster mutation µ T k is given by a unimodular transformation on each of the half spaces {(g j ) j∈J ∈ R J | g k ≥ 0} and {(g j ) j∈J ∈ R J | g k ≤ 0}. In addition, µ T k is identity on the boundary hyperplane {(g j ) j∈J ∈ R J | g k = 0} which includes the interior lattice point a t of ∆(G/B, L 2ρ , v st , τ 2ρ ). From these, we deduce that µ T k (a t ) is an interior lattice point of µ T k (∆(G/B, L 2ρ , v st , τ 2ρ )) = ∆(G/B, L 2ρ , v s t , τ 2ρ ). This implies the assertion for t , which proves the proposition. By Theorem 3.11 (4) and Proposition 4.10, we can compute the unique interior lattice point of the string polytope ∆ i (2ρ) as follows.
Corollary 4.11. Let i = (i 1 , . . . , i m ) ∈ R(w 0 ). If G is simply-laced, then the unique interior lattice point a i = (a j ) j∈J of the string polytope ∆ i (2ρ) is given by Proof of Theorem 4.9 (2). We write ∆ := ∆(G/B, L 2ρ , v st , τ 2ρ ), and set ∆ := µ T k (∆). Let us consider the unique interior lattice point a t = (a j ) j∈J of ∆. Recall from Proposition 4.8 that µ T k = f • ϕ w,F for specific w ∈ M , F ⊆ w ⊥ , and f ∈ GL J (Z). Since we have a j = 0 for all j ∈ J uf by Proposition 4.10, the definitions of ϕ w,F and f imply that ϕ w,F (a t ) = f (a t ) = a t .
In addition, we have which implies by Propositions 4.4, 4.5 that here, we note that (∆ ∨ ) * = ∆ − a t . Hence it follows that In general, for Q ⊆ M R containing the origin in its interior and γ ∈ GL(M R ), it follows from the definition of the polar dual that the equality γ(Q) * = t γ −1 (Q * ) holds, where t γ ∈ GL(N R ) denotes the dual map of γ. Indeed, we have Notice that if γ is unimodular, then so is t γ.
Hence we conclude that This implies the required assertion since t f is a unimodular transformation.

Relation with FFLV polytopes
FFLV polytopes were introduced by Feigin-Fourier-Littelmann [11,12] and Vinberg [44] to study PBW-filtrations of V (λ). Kiritchenko [31] proved that the FFLV polytope F F LV (λ) of type A n coincides with the Newton-Okounkov body of (G/B, L λ ) associated with a valuation given by counting the orders of zeros/poles along a specific sequence of translated Schubert varieties. Feigin-Fourier-Littelmann [13] realized the FFLV polytopes of types A n and C n as Newton-Okounkov bodies of (G/B, L λ ) using a different kind of valuation. Ardila-Bliem-Salazar [3] gave an explicit bijective piecewise-affine map from the Gelfand-Tsetlin polytope GT (λ) of type A n (resp., type C n ) to the FFLV polytope F F LV (λ) of type A n (resp., type C n ) by generalizing Stanley's transfer map [42] to marked poset polytopes. In this section, we relate Ardila-Bliem-Salazar's transfer map with combinatorial mutations. 5.1. Marked poset polytopes. In this subsection, we recall the definition of Ardila-Bliem-Salazar's marked order polytopes and marked chain polytopes together with their transfer map [3].
First, we recall what a marked poset is. Let Π be a poset equipped with a partial order ≺, and A ⊆ Π a subset of Π containing all minimal elements and maximal elements in Π. Take a vector λ = (λ a ) a∈A ∈ R A , called a marking, such that λ a ≤ λ b whenever a ≺ b in Π. We call the triple ( Π, A, λ) a marked poset.
In [3, Theorem 3.4], a piecewise-affine bijection φ from O( Π, A, λ) to C( Π, A, λ) was constructed, which is called a transfer map. The piecewise-affine map φ : R Π\A → R Π\A , (x p ) p → (x p ) p , is defined as follows: for p ∈ Π \ A, where for p, q ∈ Π, q p means that p covers q, that is, q ≺ p and there is no q ∈ Π \ {p, q} with q ≺ q ≺ p. Now, we recall a key notion which we will use in the proof of Theorem 5.3, called marked chain-order polytopes, introduced in [8] and developed in [9]. We remark that the original notion of marked chain-order polytopes is more general, but we restrict it for our purpose. Take a marked poset ( Π, A, λ) and fix Π ⊆ Π \ A. We define O Π ( Π, A, λ) as follows: where for c ∈ Π \ Π , we set We can directly check that O ∅ ( Π, A, λ) = O( Π, A, λ) and O Π\A ( Π, A, λ) = C( Π, A, λ). By taking Π with ∅ Π Π \ A, we obtain an "intermediate polytope" between a marked order polytope and a marked chain polytope. It is proved in [9,Proposition 2.4] that if λ ∈ Z A , then O Π ( Π, A, λ) is a lattice polytope for every Π ⊆ Π \ A. Define a map φ Π : R Π\A → R Π\A , (x p ) p → (x p ) p , by for p ∈ Π \ A. Notice that φ Π\A = φ and φ ∅ = id. In [9, Theorem 2.1], it is proved that the map φ Π gives a piecewise-affine bijection from O( Π, A, λ) to O Π ( Π, A, λ).

5.2.
Combinatorial mutation equivalence of marked poset polytopes. The second named author proved in [25,Theorem 4.1] that the transfer map between ordinary poset polytopes can be described as a composition of combinatorial mutations in M R . We can generalize this result to marked poset polytopes under some conditions. We say that a poset Π is pure if every maximal chain in Π has the same length. When Π is pure, all chains starting from a minimal element in Π and ending at p have the same length for each p ∈ Π. We denote by r(p) the length of such chains.
Let ( Π, A, λ) be a marked poset with λ ∈ Z A . Assume that Π is pure, and that λ satisfies λ a = λ b for all a, b ∈ A with r(a) = r(b). Then there exists u = (u p ) p∈ Π\A ∈ O( Π, A, λ) ∩ Z Π\A such that u p = u p for all p, p ∈ Π \ A with r(p) = r(p ), and u p = λ a for all p ∈ Π \ A and a ∈ A with r(p) = r(a). (5.2) Let λ r denote the marking given by (λ r ) a = r(a) for a ∈ A. Then it is proved in [10, Corollary 23] that for a pure poset Π, O( Π, A, λ r ) (resp., C( Π, A, λ r )) contains a unique interior lattice point. Indeed, the unique interior lattice point (r p ) p∈ Π\A ∈ O( Π, A, λ r ) is given by r p = r(p) for all p ∈ Π \ A, while the unique interior lattice point (r p ) p∈ Π\A ∈ C( Π, A, λ r ) is given by r p = 1 for all p ∈ Π \ A. We notice that (r p ) p satisfies (5.2) and φ((r p ) p ) = (r p ) p . Write Namely, O( Π, A, λ r ) (resp., C( Π, A, λ r )) contains the origin as the unique interior lattice point.
We regard polytopes appearing below as ones living in M R .
Theorem 5.3. Let Π be a pure poset.
Example 5.4. Let us consider the marked poset in Figure 5.1. We regard that the marked poset polytopes live in R 3 . In this case, the transfer map φ : R 3 → R 3 is given as follows: φ(x, y, z) = (min{x − 1, x − z}, min{y − z, y − 2}, z) Hence, even if we apply any translation to the marked order polytope, the transfer map never becomes piecewise-linear. This implies that the transfer map φ cannot be described as a composition of combinatorial mutations in M R .

5.3.
Type A case. Let G = SL n+1 (C), and λ ∈ P + . We write λ ≥k := k≤ ≤n λ, h for 1 ≤ k ≤ n. Let O λ (resp., C λ ) denote the marked order (resp., chain) polytope associated with a marked poset whose Hasse diagram is given in Figure 5  By the definition, the marked order polytope O λ coincides with the Gelfand-Tsetlin polytope GT (λ) (see Example 3.6), and the marked chain polytope C λ coincides with the FFLV polytope F F LV (λ) (see [11, equation ( Since the associated marked poset satisfies the assumption in Theorem 5.3, the following theorem is an immediate consequence of Theorem 5.3.
Theorem 5.5. The following hold.
Let O λ (resp., C λ ) denote the marked order (resp., chain) polytope associated with a marked poset whose Hasse diagram is given in Figure 5.3.
. . . By the definition, the marked order polytope O λ coincides with the Gelfand-Tsetlin polytope GT Cn (λ) of type C n (see Example 3.7), and the marked chain polytope C λ coincides with the FFLV polytope F F LV Cn (λ) of type C n (see [12, equation (1.2)]).
Similar to Theorem 5.5, the following theorem holds.
Theorem 5.6. The following hold.