MV polytopes and reduced double Bruhat cells

When $G$ is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of $G$. By fixing $w$ from the Weyl group, we can define MV polytopes whose highest vertex is labelled by $w$. We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by $w^{-1}$. To do this, we define a collection of generalized minor functions $\Delta_\gamma^\text{new}$ which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex $w$. We also describe the combinatorial structure of MV polytopes of highest vertex $w$. We explicitly describe the map from the Weyl group to the subset of elements bounded by $w$ in the Bruhat order which sends $u \mapsto v$ if the vertex labelled by $u$ coincides with the vertex labelled by $v$ for every MV polytope of highest vertex $w$. As a consequence of this map, we prove that these polytopes have vertices labelled by Weyl group elements less than $w$ in the Bruhat order.


Introduction
For G a complex reductive algebraic group, the irreducible representations are highest weight representations.To understand the tensor products of these irreducible representations, Lusztig defined a canonical basis for each V (ω i ), which behaves nicely with the decomposition of these tensor products into their irreducible subrepresentations [Lus90].In [MV07], Mirkovíc and Vilonen provide another basis using the geometric Satake correspondence, which relates the representation theory of the Langlands dual group G ∨ with the intersection homology of the affine Grassmanian, Gr.
Under this correspondence, the bases of the representations correspond to certain subvarieties of Gr, called Mirkovíc-Vilonen (MV) cycles.These MV cycles are the irreducible components of the intersection of infinite cells and as such, are difficult to understand as geometric objects.Anderson first conjectured that MV cycles could be analysed by studying their moment polytopes [And03] and in [Kam10], Kamnitzer gives a combinatorial description of MV cycles using these moment polytopes, called MV polytopes.Goncharov and Shen [GS15] take this one step further by explicitly showing that the set of MV polytopes are the tropical points of the unipotent subgroup of G.The benefit to this point of view is that the tropical Plücker relations come from the Plücker relations on N , which arise naturally by studying the transition maps of Lusztig's positive atlas [BZ97].
In this paper, for w ∈ W , we investigate a subset of MV polytopes called MV polytopes of highest vertex w, denoted by P w .These polytopes are MV polytopes whose vertex labelled by w is equal to the vertex labelled by w 0 .
The original motivation to study these polytopes was to develop a better understanding of affine MV polytopes, although these MV polytopes are also of interest due to their connection to preprojective algebra modules and MV cycles.In [BKT14], the authors define a class of preprojective algebra modules of interest, T w and in [Mén22], Ménard proves that P w is exactly set of MV polytopes associated to these modules.
For any MV polytope, there is a canonical labelling of the vertices by the Weyl group, so that the vertex data of P ∈ P w can be labelled (µ v ) v∈W .The main result of this paper is that the vertex data is only dependent on the Weyl group elements bounded by w.
This theorem is proven by explicitly describing the map W → {v ≤ w}.We would also like to realize P w as the non-negative tropical points on some subvariety of N such that the tropicalized generalized minors functions send a non-negative tropical point to the BZ data of an MV polytope of highest vertex w.The candidate for this subvariety is the reduced double Bruhat cell, L w −1 = N ∩ B − w −1 B − .On this subvariety, some of these generalized minor functions vanish.Instead, we redefine these minors ∆ new vωi to be the smallest weight γ such that ∆ γ,vωi = 0 on L w −1 .Consider the collection of tropical functions M γ = (∆ new γ • η −1 w −1 ) trop for γ ∈ Γ, where η w −1 is a necessary change of coordinates.
Using this new collection of tropical functions (M γ ) γ∈Γ , we obtain an identical result to the case of N .
This paper is organized as follows.In Section 3, we give a brief background of the theory of MV polytopes.In Section 4, we describe the Lusztig and vertex data of P w and prove Theorem A. In Section 5, we outline the theory which relation MV polytopes of highest vertex w to the tropical points of the reduced double Bruhat cells.

Notation
Let G be a semisimple, simply connected, complex group.Let T be a maximal torus of G.We define the weight and coweight lattice as X * = Hom(T, C × ) and X * = Hom(T, C × ) respectively.Let W = N G (T )/T be the Weyl group.
Fix B be a Borel subgroup of G such that T ⊂ B. Let N be the unipotent subgroup of B. Let I be the index set of the simple roots and denote α i as the simple root associated to the index i while α ∨ i is the simple coroot.Let ∆ be the set of roots and ∆ + the set of positive roots while ∆ ∨ is set of coroots and ∆ ∨ + the set of positive coroots.Let •, • : X * × X * → C be the pairing of the weight and the coweight lattice and set a ij = α ∨ i , α j .Denote by Q = N∆ the root lattice so Q + = N∆ + is the positive root cone.Similarly, let Q ∨ = N∆ ∨ be the coroot lattice and Q + = N∆ ∨ + the positive coroot cone.Let ω i be the fundamental weights, which form a basis of the weight lattice X * such that α ∨ i , ω j = δ i,j .Consider the space t R = X * ⊗R and t * R = X * ⊗R.Define a partial order on X * by µ ≤ λ ⇐⇒ λ−µ ∈ Q ∨ + and a partial order on X * by µ ≤ λ ⇐⇒ λ − µ ∈ Q + .Define the twisted partial order ≥ w on t R by β ≤ w α ⇐⇒ β − α, wω i ≥ 0 for all i ∈ I. Let s i be the simple reflection associated to the simple root α i , i.e. s i (α) = α − α ∨ i , α α i and set S = {s i : i ∈ I}.Then W is also the Coxeter group generated by S. W acts on the weight lattice by s i (β) = β − α ∨ i , β α i for β ∈ X * .Similarly, W acts on the coroots and the coweight lattice by s i (β) = β − β, α i α ∨ i for β ∈ X * .For w ∈ W , let ℓ(w) denote the length of w.We say the product Let ≤ denote the Bruhat order and let ≤ R , ≤ L denote the right and left weak Bruhat orders respectively.We will denote intervals in the strong Bruhat order by [v, w] = {x : v ≤ x ≤ w}.Similarly, the weak Bruhat intervals are [v, w]

MV polytopes
MV polytopes were originally defined by Anderson [And03] as the moment polytopes of certain subvarieties of the affine Grassmanian called MV cycles.In [Kam10], Kamnitzer gave a completely combinatorial description of MV polytopes using their hyperplane data.In particular, a GGMS polytope is an MV polytope exactly when the hyperplane data are a BZ datum.In this section, we review MV polytopes as combinatorial objects and outline their relation to preprojective algebra modules.We describe the crystal structure on the set of MV polytopes and define the Saito crystal reflection.
To define MV polytopes, we first consider GGMS polytopes.
We call (µ • ) the vertex data of the polytope.We can also define a GGMS polytope using the hyperplane data.The hyperplanes are indexed by weights of the form wω i .Define the set of chamber weights Γ = {wω i : w ∈ W, i ∈ I}.Let M • = (M γ ) γ∈Γ be a collection of integers that satisfy the edge inequalities for each w ∈ W and i ∈ I: where a ji = α ∨ j , α i .Then the polytope P (M • ) defined by the hyperplane data is By [Kam10, Proposition 2.2], these two definitions are equivalent in the following way.If P = P (µ • ), then P = P (M • ) where we set M wωi = µ w , w • ω i .If P = P (M • ) then P = P (µ • ) where we set From now on, for a GGMS polytope P , we will denote (µ • ) as the vertex data and (M • ) as the hyperplane data.
For w ∈ W and s i such that ℓ(s i w) > ℓ(w), there is an edge in P (µ • ) connecting µ w and µ wsi where and c = −M wωi − M wsiωi − j =i a ji M wωj .Note that from the edge inequalities (1) c ≥ 0. We call c the length of the edge from µ w to µ wsi .The next lemma follows directly from (2).
When a GGMS polytope is an MV polytope, the hyperplane data satisfy certain relations.First, we recall the tropical Plücker relations, which come from the tropicalization of the Plücker relations of [BZ97].
Definition 3.4.The collection (M γ ) γ∈Γ satisfies the tropical Plücker relations if for each w ∈ W and every i, j ∈ I such that i = j and s i , s j ∈ D R (w), then either a ij = 0 or the following holds: We define an MV polytope as GGMS polytope P whose hyperplane data (M • ) are a BZ datum.This definition is equivalent to the original definition of MV polytopes as the moment polytopes of MV cycles.
Theorem 3.6 ([Kam10, Theorem 3.1]).A GGMS polytope P (M • ) is an MV polytope if and only if it is the moment polytope of a stable MV cycle.
Denote by P the set of MV polytopes.For any P ∈ P, the polytope is determined by its vertex data (µ • ), which are a collection of points in Q ∨ , or its BZ data (M • ), which are a collection of integers.There is one more set of combinatorial data which determines P , closely related to the vertex data.
For a reduced word i = (i 1 , . . ., i m ) of w 0 , define the Weyl group elements Definition 3.7.Let P ∈ P with vertex data (µ • ) and BZ data (M • ).For a reduced word i = (i 1 , . . ., i m ) of w 0 , the Lusztig data of P with respect to i is defined by By the edge inequalities (1), n i k ≥ 0 for 1 ≤ k ≤ m.The Lusztig data corresponds to the lengths of the edges along the path determined by i above.Note that for any P ∈ P and any i, n i For convenience, we will call the path µ e , µ si 1 , . . ., µ si 1 •••si m−1 , µ w0 determined by a reduced word i of w 0 a minimal path from µ e to µ w0 in P .We will also use the shorthand Example 3.8.For the A 2 polytope in Figure 1, the reduced word i = (1, 2, 1) gives the Lusztig data n 121 • = (1, 2, 2), which are the lengths of the edges on the right side of the polytope.For i = (2, 1, 2), n 212 • = (3, 1, 2) which are the lengths of the edges on the left side of the polytope.
Any MV polytope is completely determined by its Lusztig data along one minimal path.Theorem 3.9 ([Kam10, Theorem 7.1]).Let i be any reduced word of w 0 .The Lusztig data with respect to i gives a bijection P → N m .

Crystal structure of P
The set of MV polytopes has a bicrystal structure and hence a reflection of the crystal will result in an action on the set of MV polytopes.First, we define a crystal structure as in [Kas95, Section 7.2].Definition 3.10.A crystal is a set B along with the maps wt : B → X * , ẽi : for each i ∈ I with the following axioms: A highest weight crystal has a unique element b 0 such that b 0 can be obtained by any element b ∈ B by applying a sequence of ẽi for different i ∈ I.
In particular, we are interested in the crystal B(∞).This is the highest weight crystal determined by the relations wt(b 0 ) = 0 and ε i (b) = max{n : ẽn ) is also a crystal.We call B(∞) a bicrystal with these two crystal structures where the weight functions agree and wt(b) ∈ −Q + for every b ∈ B(∞).In [Kam07], Kamnitzer defines the bicrystal structure on the set of MV polytopes and proves that this structure is isomorphic to the B(∞) bicrystal.
We can explicitly see how these operators act on the Lusztig data of a polytope.Suppose n i • (P ) is the Lusztig data with respect to i for a polytope P ∈ P. Then . The value of the crystal operators ε i can be easily determined by the Lusztig data.
. The Saito reflection is the map Define the * -Saito reflection as the map Note that by definition, The operators σ i , σ * i satisfy the same braid relations as the simple reflections s i thus for any w ∈ W , it is well defined to set σ w := σ i1 • • • σ im where i is a reduced word of w.Proof.Consider Pol(b) with vertex data (µ • ).For w ∈ W and j ∈ I such that ws j > w, there is a reduced word i such that This corollary allows us to write µ w (b) in a closed form.Note that the map σi has the property that wt As µ e = 0, it follows from Corollary 3.16 that Thus the vertex data (µ • (b)) of Pol(b) can be explicitly determined by the Saito reflection where Note that we shift by wt(b) so that µ e (b) = wt(b) − wt(b) = 0.

Preprojective algebra modules
We give a very brief background on preprojective algebra modules and the associated MV polytope.This section is needed to prove the generalized diagonal relations of Section 4.2.
In this section, we restrict to the case that G is a simply-laced algebraic group.First, we start with some general definitions.Definition 3.17.A quiver Q = (I, E, s, t) consists of a vertex set I, an arrow set E, a source map s : E → I and a target map t : E → I.We write the arrow α ∈ E as α : i → j, where i = s(α) and j = t(α). Define The path algebra of Q over C is the algebra CQ.Consider the ideal J generated by α∈E (αα * − α * α).
Definition 3.18.The preprojective of Q over C, denoted by Λ(Q), is the quotient of CQ by the ideal J.
We consider a few special Λ(Q)-modules.For i ∈ I, let S i be the 1-dimensional module concentrated at the vertex i, where all arrows act as zero.Let I i be the annihilator of S i .For any w ∈ W we define where i is a reduced word of w.Note that this is independent of the choice of i and thus is well-defined.
For a module M , the i-socle is the largest submodule of M which is isomorphic to S ⊕k i for some k ∈ N, while the i-head is the largest quotient of M which is isomorphic to Let G be a simply-laced complex algebraic group.Fix Q to be an orientation of the Dynkin diagram associated to the simple coroots of G and set Λ := Λ(Q).For M a Λ-module, we can define dimension vector By [BKT14, Theorem 5.4], for any w ∈ W we define the submodules M w ⊆ M as the image of the map I w ⊗ Λ Hom Λ (I w , M ) → M .By Remark 5.19 (i), Pol(M ) will have vertex data (µ w ) w∈W where µ w = dimM − dimM w .For certain modules M , M w and M siw are closely related.
Finally, a result of Crawley-Boevey tells us that we can switch the rolls of S i and M in the previous lemma.
Finally, we define the subset of Λ-modules T w .
Definition 3.21.Let T w to be the set of Λ-modules M such that M w = M .
By [BKT14, Remark 5.5 (ii)], this is the same category T w defined in [BKT14] and is also the category of modules C w −1 w0 defined in [Mén22, Definition 2.5].Note that for M ∈ T w , the vertex data of Pol(M ) will satisfy µ w = µ e .

Combinatorial data of MV polytopes of highest vertex w
In this section, we define a subset of MV polytopes, called MV polytopes of highest vertex w, and show that these polytopes only have vertices labelled by elements bounded by w in the Bruhat order.First, we introduce the definition of an MV polytope of highest vertex w.
Definition 4.1.Fix w ∈ W .Let P be an MV polytope with vertex data (µ • ).We say P is an MV polytope of highest vertex at most w if µ w = µ w0 .Denote by P w the set of MV polytopes of highest vertex w.
Remark 4.2.Recall in Section 3.2 we define the set of MV polytopes associated to T w as the set of Λmodules M such that M w = M .By [Mén22, Proposition 5.33], P ww0 is the set of MV polytopes associated to the modules in T w (under a reflection by w 0 and a shift to make µ e = 0).
Example 4.3.Consider MV polytopes associated to the group of type B 2 .For w = s 2 s 1 s 2 , P s2s1s2 is the set of polytopes such that µ s2s1s2 = µ w0 , see Figure 2 for an example.This condition will also imply that µ s1s2 = µ s1s2s1 .In Section 4.2 we will explore how the condition µ w = µ w0 affects the vertex data of a rank 2 polytope.
A reduced word i for w 0 gives a minimal path in the polytope of P beginning at µ e and ending at µ w0 .If this path passes through µ w , then the condition µ w = µ w0 forces the Lusztig data n i • (P ) to be zero in the coordinates after ℓ(w).More precisely, we can show that every vertex which appears after µ w in such a minimal path will necessarily be equal to µ w .Lemma 4.4.Let P be an MV polytope with vertex data Lemma 4.5.Fix w ∈ W and suppose But the sum of non-zero points in Q ∨ + is still a non-zero point in Q ∨ + and hence the only possible values of µ w0 − µ v and µ v − µ w are zero.Thus By the definition of the Lusztig data and its relation to the vertices (see (2)), this lemma allows us to characterize P w in terms of its Lusztig data with respect to certain reduced words.
Corollary 4.6.Fix w ∈ W .The following conditions are equivalent: (i) P ∈ P w , (ii) There exists a reduced word i of w 0 with (i 1 , . . ., i ℓ(w) ) a reduced word for w such that the Lusztig data n i (iii) For every reduced word i of w 0 with (i 1 , . . ., i ℓ(w) ) a reduced word for w, the Lusztig data n i Recall that an MV polytope P is determined by its BZ data (M γ ) γ∈Γ .We characterize the BZ data for P ∈ P w .
Lemma 4.7.The collection (M γ ) γ∈Γ is the BZ datum of an MV polytope with highest vertex w exactly when (i) (M γ ) γ∈Γ is the BZ datum of an MV polytope, (ii) There exists a reduced word j = (j 1 , . . ., j k ) of w −1 w 0 such that for ℓ = 0, . . ., k − 1, Proof.Consider P ∈ P with vertex data (µ • ) and BZ data (M • ).The only thing we need to show is that (ii) is equivalent to µ w = µ w0 .Suppose that µ w = µ w0 .For any reduced word Suppose j = (j 1 , . . ., j k ) is a reduced word for w −1 w 0 such that (ii) holds.As (4) is equivalent to µ ww j ℓ = µ ww j ℓ+1 and this holds for 0 = µ w0 and so P ∈ P w .Lemma 4.7 and Corollary 4.6 both only give information about the structure of the polytope P ∈ P w along the minimal paths from µ e to µ w0 that pass through the vertex µ w .To understand the whole structure of P , we need to understand the Lusztig data along any minimal path from µ e to µ w0 .
We will prove that for every P ∈ P w with vertex data (µ • ), µ v = µ vw for some well defined element v w .The proof is organized as follows.In Section 4.1, we define this element v w for v, w ∈ W .In Section 4.2, we outline the generalized diagonal relations and see how these relations completely determine the vertex data for rank 2 polytopes.In Section 4.3, we show that the Saito reflection acts on P w in a useful way and finally, in Section 4.4 we will show exactly where the Lusztig data are zero for an arbitrary reduced word of w 0 .

Intersections of Bruhat intervals
In this section, we will investigate the intersections of intervals in the Bruhat order with intersections in the weak Bruhat order.
First we recall some definitions and properties of Coxeter groups.Using the length, we can define the left descent set and right descent set respectively as Note that the left and right descent sets can be defined via the weak orders: We can relate the weak Bruhat orders to the length function in the following way.
For convenience, if u ≤ R w, we will say that u is an initial word of w, while we will say that u is a terminal word of w if u ≤ L w.We will also say for As W is finite, there is a unique longest element w 0 .This element has a special property for any reduced decomposition into two elements.Lemma 4.9.For any x, y ∈ W such that Proof.By the conditions of ℓ(x) + ℓ(y) = ℓ(w 0 ) and Suppose there exists s ∈ D R (x) ∪ D L (y).Then x • s • y is an element of length ℓ(w 0 ) + 1, which contradicts the maximality of w 0 .
Finally, Coxeter groups have three important properties that we will make use of multiple times throughout this section: Theorem 4.10 ([BB05, Proposition 2.2.7,Theorem 1.5.1,Theorem 3.3.1]).For W a Coxeter group, Lifting Property: Suppose u < w and s i ∈ D L (w)\D L (u).Then u ≤ s i w and s i u ≤ w.

Exchange Property
where s ij means that this term is deleted.
Word Property: Every two reduced words for w can be connected via a sequence of braid relations.
In Section 4.4, we will prove that for P ∈ P w with vertex data (µ v ) v∈W , P = conv{µ v : v ∈ W, v ≤ w}.The main result will be to explicitly describe the map W → [e, w] which arises by sending v → u if for every . Let w = s 1 s 2 s 3 and consider P ∈ P w with Lusztig data (1, 1, 1, 0, 0, 0) associated to the reduced word i = (1, 2, 3, 1, 2, 1).This polytope has the following form: Note that the vertices are indeed labelled by the set {v ∈ W : v ≤ w}.For v ∈ W larger than w, the relations on the vertices µ v are: Suppose for v ∈ W , u is the Weyl group element such that u ≤ w and µ v = µ u for every P ∈ P w .By examining the previous example, we expect two conditions on u: first, we expect that u ≤ R v; equivalently, this says there must be a minimal path from µ e to µ w0 in the polytope that passes through both the vertices µ u and µ v .Second, we expect that u is the longest element such that u ≤ w and u ≤ R v. First we prove that this element is well-defined.To do this, we will need a result of Björner and Wachs.
For x, y ∈ W , we will say z is a minimal upper bound for x and y if x, y ≤ z and for any Proof.As [e, v] R ∩ [e, w] is a finite set, there exists an element of longest length.Suppose there exists two distinct elements x, y of longest length.
Consider the set [x, w] ∩ [y, w].As this set is finite, there exists an element z (not necessarily unique) of minimal length.This element z has the property that for any z ′ ∈ W such that z ′ ≤ z, x ≤ z ′ and y ≤ z ′ , then ℓ(z) = ℓ(z ′ ) by the minimality of z and hence z = z ′ .We apply [BW88, Theorem 3.7] (see Theorem 4.12), so z ≤ R v as well.Thus z ∈ [e, v] R ∩ [e, w] but ℓ(z) > ℓ(x) = ℓ(y), which contradicts that x, y are of longest length.
, this lemma also holds for the left Bruhat order.This element is closely related to the Demazure product.For w ∈ W and s i ∈ S, let s i * w := max{w, s i w}, where the maximum is the element in the set of maximal length.The Demazure product can be defined recursively by Using the same proof technique as Proposition 6.4, we can relate the Demazure product to the weak orders.
Proof.If v = e, then e * w = w and clearly (5) holds.We proceed by induction.For v = e, there exists and ℓ(s i x) = ℓ(x) + 1.This implies ℓ(s i xw) = ℓ(s i x) + ℓ(w).Finally, by the Lifting Property, s i x ≤ v and (5) holds.A similar proof works for (6).
The maximal length element in the set {xw : x ≤ v and ℓ(xw) = ℓ(x) + ℓ(w)} must occur when ℓ(x) is of maximal length.Thus v * w = xw where x is the maximal length element such that x ≤ v and xw is reduced.By an identical argument, v * w = vy for y the maximal length element such that y ≤ w and vy is reduced.
It immediately follows that w ≤ L v * w and v ≤ R v * w by Proposition 4.8.To relate v w to the Demazure product, we first need the following lemma.
Lemma 4.18.For u, v ∈ W , the following conditions are equivalent: This lemma is applying the fact that for the longest element, w 0 = w • (w −1 w 0 ) is a reduced product for any w, so w −1 w 0 multiplied on the left with any terminal word of w will also be reduced.Proof.By Proposition 4.17, (w 0 v −1 ) * w = (w 0 v −1 ) • x where x is the maximal length element such that x ≤ w and (w 0 v −1 ) • x is reduced.By Lemma 4.18, w 0 v −1 • x is reduced if and only if x ≤ R v. Thus x is the maximal length element such that x ≤ w and x ≤ R v so by definition, x = v w .
We could alternatively take (vw 0 )((w 0 v −1 ) * w) as the definition of v w and the uniqueness of v w will be automatic as this product is well-defined.For our purposes, it is useful to use the Bruhat orders to define v w but this connection to the Demazure product simplifies some of the proofs.
As v w is an initial word of v,

Generalized diagonals
In this section we prove two technical lemmas, which state the generalized diagonal relations on MV polytopes.These relations are inspired by the diagonal relations in the rank 2 case, see the discussion at the end of [Kam10, Section 3] for more details.These inequalities are interesting because they relate vertices of the form µ w and µ sj w which are vertices that do not necessarily share a face of the polytope (see Figure 3).On the other had, the tropical Plücker relations only give relations amongst vertices with a shared face.
The first lemma requires the use of preprojective algebra modules, see Section 3.2 for more details.We recall a few definitions.For G a simply-laced complex algebraic group, let Λ be the preprojective algebra associated to the double quiver of an orientation of the Dynkin diagram of the coroots of g.For a Λ-module The following proof of the simply-laced case is due to Pierre Baumann.We thank him for allowing its inclusion in this text.
Lemma 4.21.Assume G is simply-laced.Let P be an MV polytope with vertex data (µ w ) w∈W .For every w ∈ W , s j ∈ D L (w), the inequality µ w − µ sj w , ω k ≤ 0 holds for every k = j.
As the tensor product is right exact, by applying the functor ⊗ Λ M sj w we get the long exact sequence Note that (Λ/I j ) ⊗ Λ M sj w = M siw /I j ; by definition, M sj w /I j = hd j (M sj w ) = S ⊕n j for some n ∈ N. Thus we have the resulting exact sequence with the dimension vectors dimM sj w + dim ker(φ) = dimM w + nα i .As dim ker(φ) ∈ Q ∨ + , the claim holds for M of this form.
Suppose M is a general Λ-module.Let N be the maximal extension of M by S j , i.e. take m ∈ N such that 0 → S ⊕m By the proof of [BGK12, Lemma 2], 0 → S ⊕m j → N sj w → M sj w → 0 is exact and thus dimN sj w = dimM sj w + mα ∨ j .Also, as the composition Finally, the difference between the dimension vector of M w and M sjw is as follows: Now, we implement the folding technique of to prove the general case.We will follow the notation used in [JS17].
Let G be a simply-laced algebraic group.Consider a bijection σ : I → I with a ij = a σ(i)σ(j) .This induces a Dynkin diagram automorphism on G by σ : G → G such that σ(x ±i (a)) = x ±σ(i) (a).Let G σ be the fixed point group on G and call the pair (G, G σ ) a symmetric pair.
Denote by g to be the Lie algebra of G σ .Let W be the Weyl Group of g, generated by simple reflections si .There is a group isomorphism Θ : W → W σ defined by where k i is the number of elements in the σ-orbit of i.Now, we will consider the σ-invariant MV polytopes of G. Denote P to be the set of MV polytopes for g.The diagram automorphism σ induces an action on P by σ(P ) := conv{σ −1 (µ σ(w) ) : w ∈ W }.
If σ(P ) = P , we call P σ-invariant.Denote the set of σ-invariant MV polytopes by P σ and let P be the set of MV polytopes for g.There is an identification between these two sets of polytopes.
Lemma 4.23.Assume G is non-simply-laced.Let P be an MV polytope with vertex data (µ w ) w∈W .For every w ∈ W , s j ∈ D L (w), the inequality µ w − µ sj w , ω k ≤ 0 holds for every k = j.
Proof.Let σ be a Dynkin diagram automorphism of G and let P σ be the set of σ-invariant MV polytopes of G. Let P be the set of MV polytopes associated to G σ , the fixed point group of σ.Recall by Theorem 4.22, Φ : P σ → P is a bijection, where P with vertex data (µ w ) w∈W is sent to P with vertex data (µ Θ(w) ) = (µ w ).
Let P ∈ P. Consider w ∈ W arbitrary.Let sj ∈ D L (w) and k ∈ I such that k = j.We want to show that µ w − µ sj w , ω k ≤ 0.
Since Φ is a bijection, there exists a P ∈ P σ such that µ w = µ Θ(w) .Then for the vertices of P , Note that s σ i depends on the number of elements in the σ-orbit of i, which can only equal 1, 2 or 3. We consider the case where there are 3 elements in the orbit.Then Since k ∈ I but {j, σ(j), σ 2 (j)} ∩ I = {j}, then k = σ(j) or σ 2 (j).Thus we can apply the simply-laced case to each term on the right side of the above equation, and hence µ Θ(w) − µ s σ j Θ(w) , ω k ≤ 0. As ω k is the restriction of ω k to the subspace h σ , then µ w − µ sj w , ω k = µ Θ(w) − µ s σ i Θ(w) , ω k ≤ 0. For the cases with 1 or 2 elements in the σ-orbit, we will simply have fewer terms on the right side of the above equation.
Recall we define * : I → I where i * is the index such that s i w 0 = w 0 s i * .For w = s i1 • • • s im , we define Lemma 4.24.Fix w ∈ W .For every P ∈ P w , and for every s j ∈ D L (w), µ sj w = µ w0s j * .
When G is of rank 2, then Lemma 4.24 completely determines the vertex data of a polytope in P w .To see this, consider w = s i1 s i2 . . .s im ∈ W .As G has two simple roots, there are only two simple reflections and so w is an alternating product of s 1 and s 2 .
The existence of only two simple roots means that MV polytopes are 2-dimensional polygons.The two simple reflections generate two distinct reduced words for w 0 : the alternating product s 1 s 2 s 1 . . . of length ℓ(w 0 ) and the alternating product s 2 s 1 s 2 . . . of length ℓ(w 0 ).These two reduced words give two minimals paths from µ e to µ w0 and correspond to the two sides of the polygon.
For P ∈ P w , µ w is on one side of the polygon and the vertex data for any vertex along this minimal path is described by Lemma 4.5, i.e. if v ≥ R w, then µ v = µ w , otherwise v ≤ R w and µ v can be distinct.For v ≤ R w and v ≥ R w, then µ v is necessarily on the minimal path from µ e to µ w0 which does not contain µ w , and hence either v ≤ R s i1 w or v ≥ R s i1 w.By Lemma 4.24, D L (w) = s i1 and so For the first case, we have already shown µ v = µ w .For the second case, as µ v is between the vertices µ si 1 w and µ w0s i * 1 , the equality µ si 1 w = µ w0s i * 1 forces µ v = µ si 1 w .It follows that on the side of the polygon which does not contain µ w , the highest vertex is labelled by µ si 1 w and the only possible distinct vertices are labelled by v ≤ w.Hence P = conv{µ v : v ∈ W, v ≤ w}.

Crystal action on P w
In this section, we will show that the Saito reflection behaves well with P w .First, we briefly recall the crystal structure of MV polytopes and the Saito reflection, see Section 3.1 for more details.

Lusztig and vertex data of P w
The goal of this section is to show that for any P ∈ P w , µ v = µ vw for every v ∈ W . First, we need to investigate where the zeros in the Lusztig data are located.Let i = (i 1 , . . ., i m ) be a tuple.Consider two subwords of i, a = (i a1 , . . ., i a k ) and b = (i b1 . . ., i b k ).We say the subword a comes after b in the reverse-lexicographical order if for some n, a n < b n and a j = b j for every j ≥ n.
Definition 4.28.Let i be a reduced word of w 0 .For w ∈ W , define the rightmost subword i w as the first subword in the reverse-lexicographical ordering that is a reduced word of w.
The next two lemmas will show that this rightmost word for w −1 w 0 will always start with a reduced word for v −1 w v. Lemma 4.29.Fix w ∈ W .Let i = (i 1 , . . ., i m ) be a reduced word for w 0 and let i w = (i j1 , . . ., i j ℓ(w) ).
For any terminal subword i ′ = (i k , i k+1 , . . ., i m ) of i, the subword of i indexed by the intersection {k, k + 1, . . ., m} ∩ {j 1 , . . ., j ℓ(w) } is a reduced word for the maximal length element in Proof.We proceed by induction on m + 1 − k, the length of the terminal subword i ′ .
Suppose k = m.If [e, w] L ∩[e, s im ] = {e}, then the maximal element in this set is s im .Then s im ∈ D R (w) and hence j ℓ(w) = m by definition.Thus the intersection {m} ∩ {j 1 , . . ., j ℓ(w) } = m and the reduced word (i m ) is a reduced word for s im .If [e, w] L ∩ [e, s im ] = {e}, then s im ∈ D R (w) and so {m} ∩ {j 1 , . . ., j ℓ(w) } = ∅ which is a reduced word for the maximal element.
Assume the hypothesis holds for the subword (i k+1 , . . ., i m ) and let y ′ be Weyl element given by the subword of i indexed by {k + 1, . . ., m} For i ′ = (i k , . . ., i m ), let y be the Weyl element given by the subword of i indexed by {k, k + 1, . . ., m} ∩ {j 1 , . . ., j ℓ(w) }.Then either y = y ′ or y = s i k • y ′ is a reduced product and so ℓ(y ′ ) ≤ ℓ(y).By definition of i w , y ≤ L w and hence y ∈ [e, w] L ∩ [e, s i k . . .s im ].
Note that by the definition of i w , the phrase "the subword of i indexed by" in the previous lemma can be replaced with "the terminal subword of i w indexed by".Lemma 4.30.Let v, w ∈ W .For every reduced word i = (i 1 , . . ., i m ) of w 0 such that v w = s i1 . . .s i ℓ(vw ) and Proof.Let i be a reduced word of w 0 as in the statement of the lemma.To show that the word i w −1 w0 begins with a reduced word for v −1 w v, we will show that The longest element in this intersection is the Demazure product (w 0 ww 0 )((w 0 w −1 w 0 ) * (w 0 v)) by Proposition 4.19.We want to show that this product is equal to w 0 w(v −1 w v).By Proposition 4.17, ((w 0 w −1 w 0 ) * (w 0 v)) = w 0 w −1 w 0 x where x is the maximal length element such that x ≤ w 0 v and (w 0 w −1 w 0 ) • x is reduced.Recall that a → w 0 aw 0 is an automorphism of the weak and strong Bruhat orders.Then x ≤ w 0 v ⇐⇒ w 0 xw 0 ≤ vw 0 .Also, by Lemma 4.18, Now, by applying Lemma 4.29, the terminal subword of i w −1 w0 indexed by the intersection of the indices of i w −1 w0 with {ℓ(v) + 1, . . ., m} is a reduced word of (v −1 w v) −1 w −1 w 0 .Thus this intersection is of length ℓ(w 0 ) + ℓ(v w ) − ℓ(w) − ℓ(v) and is equal to {j ℓ(v)+1 , j ℓ(v)+2 , . . ., j m+ℓ(vw)−ℓ(w) } for some indices ℓ ) is a reduced word of v −1 w v, then the word (i ℓ(vw)+1 , • • • , i ℓ(v) , i j ℓ(v)+1 , . . ., i j m−ℓ(w) ) is a reduced word for w −1 w 0 and must be the rightmost such word.
We will show that for any P ∈ P w , the Lusztig data of P with respect to the reduced word i will have zeros in the position of the subword i w −1 w0 .Proposition 4.31.Let i = (i 1 , . . ., i m ) by any reduced word of w 0 .For any w ∈ W and any P ∈ P w , the Lusztig data of P with respect to i will have zeros in the position of the subword i w −1 w0 .
When ℓ(w −1 w 0 ) = 1, then i w −1 w0 = i j for some j.If j = m, then (i 1 , . . ., i m−1 ) is a reduced word for w and by Lemma 4.5, (b) ∈ P w by Lemma 4.27 so the reduced word i ′ = (i * j+1 , . . ., i * m , i 1 , . . ., i j ) has i ′ w −1 w0 in the last position, hence n ij = 0 by above.Assume for ℓ(w −1 w 0 ) = k, the zeros of the Lusztig data n i • are in the position i w −1 w0 .Suppose w is such that ℓ(w −1 w 0 ) = k +1 and n i • is the Lusztig data with respect to i.If i j is the final coordinate of i w −1 w0 , then s ij+1 , . . ., s im ∈ D R (w −1 w 0 ) so we can apply Lemma 4.27 j − 1 times so that the Lusztig data with respect to (i * j+1 , . . ., i * m , i 1 , . . ., i j ) of the resulting polytope is (0, . . ., 0, n 1 , . . ., n j ).By the base case, n j = 0 and hence the Lusztig datum in the position of the final term of i w −1 w0 is zero.Now, apply σ * i * j so that the Lusztig data with respect to (i * j , i * j+1 , . . ., i * m , i 1 , . . ., i j−1 ) of the resulting polytope is (0, 0, . . ., 0, n 1 , . . ., n j−1 ).By Corollary 4.26, this polytope is in P s i * j w , where ℓ(s i * j w) = ℓ(w) − 1.Thus, by the induction assumption, the Lusztig data corresponding to the rest of the coordinates of i w −1 w0 will be zero.
For a word i, the zeros in the Lusztig data are given by the rightmost appearance of a word of w −1 w 0 .The location of these zeros in various reduced words of w 0 prove the vertex equalities in Example 4.11.

Reduced word Lusztig data
Equality of vertices Table 1: The zeros in the Lusztig data for A 3 MV polytopes Finally, we can prove that the Lusztig data will have zeros in the positions between µ vw and µ v for every Weyl group element v.
Remark 4.35.The description of P w given by Corollary 4.34 suggests a relationship between P w and extremal MV polytopes defined by [NS09].Naito and Sagaki prove that extremal MV polytopes can be explicitly described as P w•λ = conv{v • λ : v ≤ w}, where λ is a dominant coweight.Using the Lusztig data description of these extremal MV polytopes in that paper, we can see (up to a reflection by w 0 and a shift to make µ e = 0), these polytopes are in P w .
In [BJK22], Besson, Jeralds and Keirs study the weight polytopes of Demazure modules and prove they are extremal MV polytopes.These polytopes can be described in the following ways: where g(v) = v(v −1 * w).By Proposition 4.19 proved above, g(vw 0 ) = v w .Thus, under the identification µ v = g(vw 0 )λ − g(w 0 )λ, P w λ ∈ P w .Another related concept are the polytopes defined in [TW15].For w ∈ S n , the Bruhat interval polytope Q e,w is the polytope with vertex data (µ v ) v≤w given by µ v = (v(1), . . ., v(n)).By extending the vertex data to (µ v ) v∈W by µ v = µ vw , we see this is a polytope of highest vertex w.

The dual fan
A GGMS polytope can be characterized by its dual fan in relation to a standard fan, called the Weyl fan.To describe this relationship, first we define fans and dual fans of polytopes.
Let V be a real vector space and let V * be the dual space.A polyhedral cone in V is an finite intersection of closed linear half spaces.A fan F of V * is a collection of polyhedral cones with the following properties: (i) Every nonempty face of a cone in F is also a cone in F , (ii) The intersection of any two cones in F is a face of both, (iii) The union F = V * .
A fan F 1 is a coarsening of F 2 if every cone of F 1 is a union of cones in F 2 .
Define the Weyl fan W in t * R as the fan generated by the maximal cones For any convex polytope P ⊂ V , we can define the support function of P as ψ P : V * → R by ψ P (α) = min x∈P x, α .Define the dual fan N (P ) = {C * F : F is a face of P } in V * , where The dual fan is a useful tool to study the vertices and hyperplanes of P .Maximal cones of the dual fan correspond to vertices of the polytope.If N (P ) is a coarsening of W, then there is an surjection from W to the set of vertices of P ; in fact, this surjection determines the choice of labelling on the vertices µ w .
Additionally, the defining rays of the maximal cones of the dual fan correspond to the codimension 1 faces of P .This correspondence defines a surjection from the chamber weights Γ to the defining rays of the maximal cones of N (P ).As a result of this correspondence, the cones of the dual fan of P correspond with the vertices of P while the defining rays of the maximal cones of N (P ) will correspond with the codimension 1 faces of P .In the standard case, these codimension 1 faces are exactly the hyperplanes M γ for every γ ∈ Γ.When w = w 0 , some of the hyperplanes M γ of P w may have larger codimension and hence P w can have fewer than |Γ| codimension 1 faces.An interesting question would be to find exactly which chamber weights label these codimension 1 faces in P w .Question 4.40.What are the defining rays of the maximal cones of F w ?These rays will correspond to some subset of the chamber weights Γ Pw .This subset will give us the defining hyperplanes of P , i.e.P = {x : x, γ ≤ M γ , ∀γ ∈ Γ Pw }.
Example 4.41.For B 2 polytopes in P s1s2s1 , the hyperplanes labelled by s 2 s 1 ω 1 and s 1 s 2 s 1 ω 1 are not defining hyperplanes of these polytopes.See Figure 4 for the dual fan of such a polytope and see Figure 5 for the defining hyperplanes.Remark 4.42.The normal cone of the Bruhat interval polytope Q e,w was described by [Gae22] in type A as the equivalence class of all the linear extensions of the graphs Γ w (u) associated to the polytope Q e,w .Using both descriptions of the normal cone, we expect that the set of linear extensions of the graph Γ w (u) will be exactly the set of v ∈ W such that v w = u.

Tropical geometry
In this section, we outline the basic theory of tropical geometry to describe the correspondence between MV polytopes and the non-negative tropical points of the unipotent subgroup N ⊆ G. First, we recall the concepts of positive spaces and tropical points as in [FG09, Section 1].
Definition 5.1.Let X be an irreducible variety.A positive atlas on X is a collection of birational isomorphisms {α} α∈CX over Q where α : T → X and T is a split algebraic torus.These coordinate systems satisfy: (i) Each α is regular on the complement of a positive divisor in T and is given by a positive rational function, (ii) For any pair α, β of coordinate systems, β −1 • α is a positive birational isomorphism of T .
If X has a positive atlas, we call X a positive space.
On an algebraic torus T , define the tropical points as the cocharacters of T , i.e T (Z trop ) = X * (T ).Using a positive atlas, there is a unique way to define the Z-tropical points of the variety X.
Definition 5.2.The tropical points of a positive space X is defined as For a subtraction-free function F on T , we can tropicalize it to a function F trop on the tropical points.To see how a tropical function is related to the original function, consider the following example: We will call a function F on X positive if it can be written as a subtraction-free expression in the coordinates of a positive atlas of X.We will denote the tropical function by F trop .

MV polytopes as tropical points
For G a reductive complex algebraic group, let T be a torus of G, B a Borel subgroup containing T and N its unipotent subgroup.Consider the map x i : C → N with image in the Chevalley subgroup of α i by x i (a) = exp(aE i ).For the tuple i = (i 1 , . . ., i k ), we define x i (a 1 , . . ., a We will show that the variety N is a positive space.For a reduced word i of w 0 , define the Lusztig parameterization associated to i as the map x i : (C × ) m → N by (a 1 , . . ., a m ) → x i (a 1 , . . .a m ), where m = ℓ(w 0 ).This map is a birational isomorphism by [Lus94] and hence gives a coordinate system on N .In fact, the collection of the charts (x i ) form a positive atlas of N , called Lusztig's positive atlas [Lus94].Thus the tropical points of N are defined and N (Z trop ) ∼ = Z m .
In Section 3, we saw that the set of MV polytopes are in bijection with N m by fixing a reduced word i and considering the Lusztig data of P with respect to i.We would like to show that the set of MV polytopes of G are in bijection with the non-negative tropical points of N .First, we define a positive function to pick out the "non-negative" points.
Theorem 5.4 ([GS15, Theorem 5.4], [Kam10, Theorem 4.5]).For G a reductive algebraic group, there is a bijection between the non-negative tropical points N (Z trop ) ≥ and the set of MV polytopes P.
As Lusztig's positive atlas consists of x i for all reduced words, this bijection is independent of the reduced word used for the Lusztig data in the bijection P → N m .This bijection is also compatible with the hyperplane data in the sense that the following diagram commutes: As these ∆ γ • η functions satisfy the Plücker relations, a consequence of this diagram commuting is that the tropical Plücker relations are automatically satisfied by the tropical functions M γ .
For a generic MV polytope, the highest vertex is labelled by the longest element of the Weyl group, w 0 .In this thesis, we will consider MV polytopes where the highest vertex is labelled by w for some w ∈ W .We will prove that Theorem 5.4 is also true for this class of polytopes, where N is replaced by a subvariety of N .

Tropical geometry of reduced double Bruhat cells
We will define functions M γ on the tropical points of the reduced double Bruhat cell L w −1 that will send non-negative tropical points to the BZ data associated to MV polytopes of highest vertex w.These functions will come from the tropicalization of the generalized minors functions.
Recall we define the maps x i : C → N by x i (a) = exp(aE i ).We similarly define y i : C → N by y i (a) = exp(aF i ).For i ∈ I, we fix a representative of s i ∈ G by s i = y i (1)x i (−1)y i (1) and thus for any w ∈ W we can fix a representation for w ∈ G by w = s i1 • • • s im where i = (i 1 , . . ., i m ) is a reduced word for w.
Definition 5.5.For u, v ∈ W , the reduced double Bruhat cell is L u,v := N uN ∩ B − vB − .
In particular, we are interested in the reduced double Bruhat cell L w −1 := L e,w −1 = N ∩ B − w −1 B − .Following [GS15, Section 5], we have a positive structure on L w −1 described as follows.Let x i : C × → L w −1 be defined as in Section 5.For the reduced word i = (i 1 , . . ., i m ) of w −1 , define x i : (C × ) m → L w −1 by x i (a 1 , . . ., a m ) = x i1 (a 1 ) . . .x im (a m ).From the application of [FZ99, Theorem 1.2], this is a coordinate system on L w −1 .Consider the atlas given by the charts (x i ) where i runs over all reduced words of w −1 .This atlas gives a positive structure on L w −1 which we will still call Lusztig's positive atlas.As in the case of N , define the potential function χ(x i (a 1 , . . ., a m )) = m i=1 a i .
The potential χ is still independent of i and is positive on this atlas, so we can define the non-negative tropical points L w −1 (Z trop ) ≥ = {a ∈ L w −1 (Z trop ) : χ t (a) ≥ 0}.
To define the functions M γ on L w −1 (Z trop ), we need to introduce the generalized minors.We use the shorthand ∆ γ when δ = λ.

Corollary 4. 14 .
For every v, w ∈ W , the set [e, v] L ∩ [e, w] has a unique element of longest length.Lemma 4.13 ensures the following definition is well-defined.Definition 4.15.For any v, w ∈ W , denote v w to be the unique element of maximal length in [e, v] R ∩ [e, w].

Corollary 4 .
36 ([Kam10, Corollary A.4]).A GGMS polytope P is a polytope in Q ∨ whose dual fan N (P ) is a coarsening of the Weyl fan W.

Figure 4 :
Figure 4: The dual fan of a B 2 polytope of highest vertex s 1 s 2 s 1

Figure 5 :
Figure 5: The hyperplanes of a B 2 polytope of highest vertex s 1 s 2 s 1

χ
(x i (a 1 , . . ., a m )) = m k=1 a k .This function is subtraction-free in the Luzstig coordinates x i .In fact, χ is independent of i and thus positive on Lusztig's positive atlas.Hence we have a tropical function χ trop acting on N (Z trop ).Define the non-negative points as N (Z trop ) ≥ = {ℓ ∈ N (Z trop ) : χ trop (ℓ) ≥ 0}.
where ψ P (β) = min y∈P y, β .By [Kam10, Corollary A.4], (see Corollary 4.36) P is a coarsening of the Weyl fan W and the following corollary is immediate.Corollary 4.37.For any GGMS polytope P with vertex data (µ • ), C * v ⊆ C * µv ,P for every v ∈ W . Definition 4.38.Fix w ∈ W .Let F w be the fan of t * R defined by the maximal cones for v ∈ W Proposition 4.39.Let w ∈ W and suppose P is an MV polytope.P ∈ P w if and only if N (P ) is a coarsening of F w .Proof.Consider a polytope P ∈ P w with vertex data (µ • ) and v ∈ W arbitrary.For every β ∈ t * R , µ v , β = µ vw , β so by definition C * µv ,P = C * µv w ,P .By Corollary 4.37, it follows that C * v ⊆ C * µv w ,P for every v ∈ W , hence D * v ⊆ C * µv w ,P and N (P ) is a coarsening of F w .For the converse, consider an MV polytope P such that the dual fan is a coarsening of F w . .Thus for every β ∈ C * w0 , µ w , β = µ w0 , β .But this is only possible when µ w = µ w0 so P ∈ P w .
* v is indexed by v ∈ W such that v ≤ w.Clearly, F w is a coarsening of the Weyl fan.