Augmentations, Fillings, and Clusters

Abstract

We investigate positive braid Legendrian links via a Floer-theoretic approach and prove that their augmentation varieties are cluster K₂ (aka. $\mathcal{A}$-) varieties. Using the exact Lagrangian cobordisms of Legendrian links in Ekholm et al. (J. Eur. Math. Soc. 18(11):2627–2689, 2016), we prove that a large family of exact Lagrangian fillings of positive braid Legendrian links correspond to cluster seeds of their augmentation varieties. We solve the infinite-filling problem for positive braid Legendrian links; i.e., whenever a positive braid Legendrian link is not of type ADE, it admits infinitely many exact Lagrangian fillings up to Hamiltonian isotopy.

Appendices

Appendix A: Cluster varieties

We provide a rapid review on cluster varieties in the skew-symmetric cases. Below we set [n]₊:=max{0,n} for $n\in \mathbb{R}$.

1.1 A.1 Definitions

A quiver is a triple $Q=\left (I,I^{\mathrm {uf}}, \varepsilon \right )$, where I is a finite set, I^uf is a subset of I, and ε is an I×I skew-symmetric matrix whose entries ε_ij are integers when i∈I and j∈I^uf.

Let k∈I^uf. The mutation in the direction k produces a new quiver $\mu _{k}Q = \left (I',I^{\prime \,\mathrm {uf}},\varepsilon '\right )$ where I′=I, I^′ uf=I^uf, and

$$ \varepsilon '_{ij}=\left \{ \textstyle\begin{array}{l@{\quad}l} -\varepsilon _{ij} & \text{if $k\in \{i,j\}$}, \\ \varepsilon _{ij} & \text{if $\varepsilon _{ik} \varepsilon _{kj}< 0$, and $k\notin \{i,j\}$}, \\ \varepsilon _{ij}+ |\varepsilon _{ik}| \varepsilon _{kj} & \text{if $\varepsilon _{ik} \varepsilon _{kj}\geq 0$, and $k\notin \{i,j\}$}. \end{array}\displaystyle \right . $$

Two quivers are mutation equivalent if they are related by a sequence of mutations. Denote by |Q| the class of quivers that are mutation equivalent to Q.

Each Q induces a directed graph with vertex set I. For i,j∈I, the number of arrows from i to j is [ε_ij]₊. Vertices parametrized by i∈I−I^uf are called frozen vertices. In this paper, arrows among frozen vertices are allowed to be of half weight and will be illustrated by dashed arrows.

The unfrozen part of Q is the full subquiver Q^uf containing the unfrozen vertices.

A quiver Q is said to be acyclic if there is no directed cycle inside Q^uf.

A quiver Q is said to be connected if the underlying graph of Q^uf is connected.

A quiver Q is said to have full-rank if the submatrix $\varepsilon |_{I^{\mathrm {uf}}\times I}$ is of full-rank.

Connectedness and being full-rank are invariant under mutations and therefore descend to properties of mutation equivalence classes of quivers.

Definition A.1

A cluster K₂ variety 𝒜 is an affine variety together with a collection $\mathcal{C}$ of triples α=(Q_α,T_α,A_α), where

• $Q_{\alpha}=\left (I, I^{\mathrm {uf}}, \varepsilon \right )$ is a quiver;

• T_α is a split algebraic torus of rank |I| inside 𝒜;

• A_α={A_i;α}_i∈I is a coordinate system of T_α.

We require that

• For any unfrozen vertex k of the quiver Q_α, there is an $\alpha '=(Q_{\alpha '}, {\mathrm{T}}_{\alpha '}, \mathbf {A}_{\alpha '}) \in \mathcal{C}$, where Q_α′=μ_kQ_α, and the transition map between A_α′ and A_α is

  $$ A_{i; \alpha '}=\left \{ \textstyle\begin{array}{l@{\quad}l} A_{k; \alpha}^{-1}\left (\prod _{j}A_{j;\alpha}^{[-\varepsilon _{kj}]_{+}}+ \prod _{j}A_{j;\alpha}^{[\varepsilon _{kj}]_{+}}\right ) & \text{if $i=k$}, \\ A_{i;\alpha} & \text{if $i\neq k$}. \end{array}\displaystyle \right . $$

  We say that α′ is a mutation of α in the direction k and write α′=μ_kα.

• Every pair $\alpha , \alpha ' \in \mathcal{C}$ are related by a finite sequence of mutations.

• The complement of the union of T_α for all α is of codimension 2 in 𝒜.

Each α is called a cluster seed, T_α is called a cluster chart, A_α is called a cluster, and A_i;α is called a cluster K₂ coordinate or a cluster variable. Each A_i;α for i∈I−I^uf is invariant under mutations and is called a frozen variable. We will suppress the subscript α when the context is clear.

Remark A.2

The coordinate ring of a cluster chart T_α is a Laurent polynomial ring $\mathbb{L}_{\alpha}$ in the variables A_i;α. The intersection $\bigcap _{\alpha \in \mathcal{C}} \mathbb{L}_{\alpha}$ is an upper cluster algebra of [BFZ05]. A cluster K₂ variety 𝒜 is an affine variety whose coordinate ring is an upper cluster algebra. It is worth mentioning that our cluster K₂ varieties are different from the cluster $\mathcal{A}$ varieties in [FG09]. The latter is defined to be the union of the tori T_α for $\alpha \in \mathcal{C}$, and is not affine in general.

Each cluster seed α of 𝒜 encodes a 2-form on T_α:

$$ \Omega _{\alpha}:= \sum \varepsilon _{ij} \frac{{\mathrm{d}} A_{i;\alpha}}{A_{i;\alpha}} \wedge \frac{{\mathrm{d}} A_{j;\alpha}}{A_{j;\alpha}}. $$

(1.1)

By Corollary 6.5 of [FG09], this 2-form does not depend on the choice of cluster seeds and therefore defines a global 2-form Ω on 𝒜.

Borrowing ideas from mirror symmetry, Gross, Hacking, Keel, and Kontsevich interpreted the cluster structures in terms of wall-crossing structures called scattering diagrams [G+18]. In detail, associated to a quiver Q is a scattering diagram $\mathfrak{D}$. Inside $\mathfrak{D}$ is a simplicial fan consisting of cones called cluster chambers. The paper [G+18] establishes a one-to-one correspondence between the cluster seeds of 𝒜 and the cluster chambers of $\mathfrak{D}$. The mutation from α to μ_kα corresponds to crossing the sharing facet (a.k.a the wall) of their corresponding cluster chambers.

The following proposition is crucial for this paper.

Proposition A.3

Let Q be a quiver of full rank and let 𝒜 be its associated cluster K₂ variety over an algebraically closed field (of any characteristic). The cluster charts of distinct cluster seeds of 𝒜 do not coincide.

Remark A.4

Proposition A.3 may not hold when Q is not of full rank, e.g., if Q contains one vertex and no arrows, then its cluster variety has two cluster seeds but only one cluster chart.

Proof

Let 𝒜 be defined over an algebraically closed field of characteristic p. The characteristic 0 case follows by the same argument. Let α and α′ be two distinct cluster seeds of 𝒜. By Corollary 6.3 of [FZ07], the transition map between $\mathbf {A}_{\alpha '}=\{A_{i}'\}$ and A_α={A_i} takes the form

$$ A_{i}'= \left (F_{i}{\Big|}_{X_{k}=\prod _{l}A_{l}^{\varepsilon _{kl}}} \right )\prod _{j\in I} A_{j}^{g_{ij}}, $$

(1.2)

where g_ij are integers, and each F_i is a polynomial in the variables X_k for k∈I^uf. The matrix G=(g_ij) is called a g-matrix. The polynomials F_i are called F-polynomials.

By [G+18], each F_i is a generating function that counts broken lines in the scattering diagram associated to Q. For distinct α and α′, there is at least one wall between their corresponding chambers. In particular, there is an i∈I^uf such that F_i≠1. By [LS15] and [G+18], we have

• all coefficients of F_i are positive integers;

• the constant term of F_i is 1;

• the coefficient of the highest term of F_i is 1.

Here the highest term of F_i is the monomial $\prod _{j} X_{j}^{a_{j}}$ such that for any other term $\prod _{j} X_{j}^{b_{j}}$ in F_i, we have a_j≥b_j for all j. The above last two properties are equivalent due to [FZ07, Prop.5.3].

By the above discussion, there exists an i∈I^uf such that the polynomial F_i has at least two terms even after reducing to a polynomial with coefficients in the finite field $\mathbb{F}_{p}$. The quiver Q is of full rank. The substitution $X_{k}=\prod _{l}A_{l}^{\varepsilon _{kl}}$ gives rise to an injective homomorphism from the polynomial ring $\mathbb{F}_{p}[X_{i}]_{i\in I^{\mathrm {uf}}}$ to the Laurent polynomial ring $\mathbb{F}_{p}[A_{j}^{\pm 1}]_{j\in I}$. Therefore $A'_{i}$ is not a Laurent monomial of A_j for j∈I. On the other hand, biregular isomorphisms between algebraic tori over an algebraically closed field are of monomial coordinate transformations. Thus T_α≠T_α′. □

1.2 A.2 Cluster ensembles

Following [FG09], cluster Poisson varieties are the cluster dual of cluster K₂ varieties. Each cluster Poisson variety 𝒳 is equipped with a collection of torus charts with coordinate systems $\mathbf {X}_{\alpha} =\{{X}_{i;\alpha}^{\pm 1}\}_{i\in I}$. The transition map between $\mathbf {X}_{\alpha '}=\mathbf {X}_{\mu _{k}\alpha}$ and X_α is

$$ X_{i;\alpha '}=\left \{ \textstyle\begin{array}{l@{\quad}l} X_{k;\alpha}^{-1} & \text{if $i=k$,} \\ X_{i; \alpha} X_{k; \alpha}^{\left [\varepsilon _{ik}\right ]_{+}} \left (1+X_{k;\alpha}\right )^{-\varepsilon _{ik}} & \text{if $i\neq k$.} \end{array}\displaystyle \right . $$

The coordinates X_i;α are called cluster Poisson coordinates.

Let 𝒜 and 𝒳 be a pair of cluster varieties associated to a mutation equivalence class |Q|. There is a natural one-to-one correspondence between the cluster seeds of 𝒜 and the cluster seeds of 𝒳. Each pair of corresponding cluster seeds is called a cluster seed of (𝒜,𝒳). Following [FG09, §1.2], there is a canonical map p:𝒜→𝒳 such that⁴

$$ p^{*}\left (X_{i;\alpha}\right )=\prod _{j} A_{j;\alpha}^{ \varepsilon _{ij}} $$

for every cluster seed of (𝒜,𝒳). The triple (𝒜,𝒳,p) is called a cluster ensemble.

Definition A.5

Suppose σ:I′→I is an injective map such that

• (1) $\sigma |_{I^{\prime \,\mathrm {uf}}}:I^{\prime \,\mathrm {uf}}\rightarrow I^{\mathrm {uf}}$ is a bijection,

• (2) $\varepsilon '_{ij}=\varepsilon _{\sigma (i)\sigma (j)}$ for all i,j∈I′.

Then σ induces a morphism of algebraic tori σ:α′→α and σ:χ→χ′, which are extended to morphisms of cluster varieties σ:𝒜′→𝒜 and σ:𝒳→𝒳′, called cluster morphisms. If σ is bijective, then the induced cluster morphisms are called cluster isomorphisms.

Example A.6

Consider the inclusion of the unfrozen part $Q^{\mathrm {uf}}=\left (I^{\mathrm {uf}},I^{\mathrm {uf}},\varepsilon |_{I^{\mathrm {uf}}\times I^{\mathrm {uf}}} \right )$ into Q. This inclusion induces cluster morphisms 𝒜^uf→𝒜 and 𝒳→𝒳^uf. More properties about these cluster morphisms can be found in [She14, §3].

Definition A.7

A cluster automorphism is a cluster isomorphism from a cluster variety to itself. Cluster automorphisms form a group $\mathcal{G}$ called the cluster modular group.

Fix an initial cluster seed of (𝒜,𝒳). Every cluster automorphism maps the initial seed to another cluster seed. We can express the obtained new cluster coordinates in terms of the initial ones as in (A.2). As a consequence, one may assign the c-matrix, g-matrix, and F-polynomials of [FZ07] to each cluster automorphism with respect to a fixed initial seed.

Proposition A.8

A cluster automorphism σ is the identity map on 𝒜 if and only if it is the identity map on 𝒳.

Proof

The separation formula of Fomin-Zelevinsky [FZ07] implies that σ is the identity map on 𝒜 (resp. 𝒳) if and only if its g-matrix G (resp. c-matrix C) is the identity matrix with respect to one (equivalently any) initial seed. The proposition then follows from the tropical duality theorem [NZ12, Theorem 1.2], which says that C⁻¹=G^t. □

Definition A.9

[GS18]

A cluster Donaldson-Thomas transformation on a cluster variety is a cluster automorphism whose c-matrix is equal to minus identity.

For any cluster ensemble, its cluster Donaldson-Thomas transformation, if exists, is a unique central element in the cluster modular group.

1.3 A.3 Quasi-cluster morphisms

Define $N:=\bigoplus _{i\in I}\mathbb{Z}e_{i}$ for a quiver $Q=\left (I,I^{\mathrm {uf}},\varepsilon \right )$, and let N^uf its the sub-lattice spanned by e_i for i∈I^uf. The exchange matrix ε equips N with a skew-symmetric form $\{\cdot , \cdot \}:N\times N\rightarrow \mathbb{Q}$ such that $\left \{e_{i},e_{j}\right \}=\varepsilon _{ij}$. Let M be the dual lattice of N.

One should think of N as the character lattice of a cluster chart χ and think of M as the character lattice of the cluster chart α dual to χ. For n∈N and m∈M we denote the corresponding character functions as Xⁿ and A^m. In particular, $X^{e_{i}}$ are precisely the cluster Poisson coordinates X_i, and the map p:𝒜→𝒳 is induced by the linear map p^∗:N→M,n↦{n,⋅}. We will use this set-up to define quasi-cluster morphisms. More detailed discussions can be found in [Fra16, GS19, SW19].

Definition A.10

Let N and N′ be the lattices associated to Q and Q′ respectively. Suppose σ:N′→N is an injective linear map such that

• (1) $\sigma |_{N^{\prime \,\mathrm {uf}}}$ is an isomoprhism onto N^uf;

• (3) for any i∈I^′ uf, we have $\sigma \left (e'_{i}\right )=e_{j}$ for some j∈I^uf,

• (3) σ preserves the skew-symmetric forms.

Then σ induces a morphism of algebraic tori σ:χ→χ′ which extends to a morphism σ:𝒳→𝒳′. On the dual side, σ induces a linear map σ:M→M′, which defines a morphism of algebraic tori σ:α′→α and extends to a morphism σ:𝒜′→𝒜. We call the induced morphisms σ:𝒳→𝒳′ and σ:𝒜′→𝒜 quasi-cluster morphisms.

A quasi-cluster isomorphism is a quasi-cluster morphism where σ:N′→N is an isomorphism. A quasi-cluster automorphism is a quasi-cluster isomorphism from a cluster variety to itself. Quasi-cluster automorphisms form a group $\mathcal{QG}$ called the quasi-cluster modular group.

The cluster modular group $\mathcal{G}$ is a subgroup of the quasi-cluster modular group $\mathcal{QG}$. There is a natural map $\mathcal{QG}\rightarrow \mathcal{G}^{\mathrm {uf}}$ where $\mathcal{G}^{\mathrm {uf}}$ denotes the cluster modular group for the unfrozen part.

Remark A.11

Quasi-cluster automorphisms are also known as (quasi-)cluster transformations.

The restriction of quasi-cluster morphisms to cluster charts commute with cluster mutations. Consequently, we have the following theorem.

Theorem A.12

Let $\mathcal{V}$ and $\mathcal{W}$ be two cluster varieties of the same type (either K₂ or Poisson). If $\sigma :\mathcal{V}\rightarrow \mathcal{W}$ is a quasi-cluster morphism, then there is a one-to-one correspondence between their cluster seeds, and σ restricts to a morphism between the corresponding cluster charts.

Below we construct two types of quasi-cluster morphisms that are crucial for us.

Changing a frozen vertex

Recall the lattice N associated with a quiver $Q=\left (I,I^{\mathrm {uf}},\varepsilon \right )$. Let k be a frozen vertex of Q. Let $\left (\delta _{j}\right )_{j\in I}$ is an |I|-tuple of integers. We consider a lattice $N'=\bigoplus _{i\in I} \mathbb{Z}e_{i}'$ and define a linear map σ:N′→N such that

$$ \sigma \left (e'_{i}\right ):=\left \{ \textstyle\begin{array}{l@{\quad}l} e_{i} & \text{if $i\neq k$}, \\ \sum _{j\in I} \delta _{j}e_{j} & \text{if $i=k$}. \end{array}\displaystyle \right . $$

The exchange matrix ε equips N with a skew-symmetric form {⋅,⋅}, whose pull-back through σ induces a skew-symmetric form {⋅,⋅}′ on N′. Let ε′ be an I×I matrix such that

$$ \varepsilon '_{ij}=\left \{ e'_{i}, e'_{j}\right \}':=\left \{ \sigma (e'_{i}), \sigma (e'_{j})\right \}. $$

Let 𝒜′ and 𝒳′ be the cluster varieties associated with the quiver $Q'=\left (I, I^{\mathrm {uf}}, \varepsilon '\right )$. Note that σ satisfies the conditions (1) and (2) of Definition A.10. Therefore it defines quasi-cluster morphisms

$$ \sigma :\mathscr{A}'\rightarrow \mathscr{A} \quad \quad \text{and} \quad \quad \sigma :\mathscr{X}\rightarrow \mathscr{X}'. $$

Let α (resp. α′) be the K₂ cluster chart associated with the quiver Q (resp. Q′). Let χ (resp. χ′) be the Poisson cluster chart associated with the quiver Q (resp. Q′). Then the pull-back maps of σ can be written in terms of these cluster charts as

$$ \sigma ^{*}\left (A_{i}\right )=\left \{ \textstyle\begin{array}{l@{\quad}l} A'_{i}A^{\prime \,\delta _{i}}_{k} & \text{if $i\neq k$}, \\ A'_{k} & \text{if $i=k$}. \end{array}\displaystyle \right . \quad \quad \text{and} \quad \quad \sigma ^{*}\left (X'_{i} \right )=\left \{ \textstyle\begin{array}{l@{\quad}l} X_{i} & \text{if $i\neq k$}, \\ \prod _{j} X_{j}^{\delta _{j}} & \text{if $i=k$}. \end{array}\displaystyle \right . $$

(1.3)

Proposition A.13

With the above set-up, the following statements are true.

• (1) If δ_k=1, then the quasi-cluster morphisms σ are quasi-cluster isomorphisms.

• (2) If ∑_jε_ijδ_j=0 for every i∈I^uf, then there is no arrow between the vertex k and the unfrozen part of Q′.

Proof

(1) is obvious. For (2), it suffices to note that for i∈I^uf,

$$ \varepsilon _{ik}'=\left \{e'_{i},e'_{k}\right \}'=\left \{\sigma \left (e'_{i}\right ),\sigma \left (e'_{k}\right )\right \} = \bigg\{ e_{i}, \sum _{j} \delta _{j}e_{j}\bigg\} = \sum _{j} \delta _{j}\varepsilon _{ij} = 0. $$

Hence, there is no arrow between the vertex k and the unfrozen part of Q′. □

Merging frozen vertices

Let t₁ and t₂ be frozen vertices in a quiver $Q=\left (I,I^{\mathrm {uf}},\varepsilon \right )$. Define the quiver $Q'=\left (I',I^{\prime \,\mathrm {uf}},\varepsilon '\right )$, where $I':=(I\setminus \left \{t_{1},t_{2}\right \})\sqcup \{t\}$, I^′ uf:=I^uf, and

$$ \varepsilon '_{ij}:=\left \{ \textstyle\begin{array}{l@{\quad}l} \varepsilon _{ij} & \text{if $i,j\neq t$}, \\ \varepsilon _{t_{1}j}+\varepsilon _{t_{2}j} & \text{if $i=t$}, \\ \varepsilon _{it_{1}}+\varepsilon _{it_{2}} & \text{if $j=t$}. \end{array}\displaystyle \right . $$

We say that Q′ is obtained from Q by merging the frozen vertices t₁ and t₂ into a single frozen vertex t. Let N and N′ be the lattices associated with the quivers Q and Q′ respectively. There is an injective linear map

$$\begin{aligned} \sigma :N'&\rightarrow N \\ e'_{i} & \mapsto e_{i} \quad \text{for $i\neq t$}, \\ e'_{t} & \mapsto e_{t_{1}}+e_{t_{2}}. \end{aligned}$$

Note that σ satisfies the conditions in Definition A.10. It defines quasi-cluster morphisms

$$ \sigma :\mathscr{A}'\rightarrow \mathscr{A} \quad \text{and} \quad \sigma :\mathscr{X}\rightarrow \mathscr{X}'. $$

The next proposition is direct consequence of the construction of σ.

Proposition A.14

The quasi-cluster morphism σ:𝒜′→𝒜 embeds 𝒜′ as a subvariety of 𝒜 determined by the locus $\left \{A_{i}=A_{j}\right \}$.

Appendix B: Double Bott-Samelson cells

2.1 B.4 Definition and basic properties

Double Bott-Samelson (BS) cells, introduced in [SW19], are moduli spaces of flags with prescribed relative positions encoded by positive braids. In this section we briefly recall their definition and basic properties following loc. cit. Theorem 2.12 establishes natural isomorphisms between the augmentation varieties of positive braid closures and the double BS cells associated with SL_n.

Let B_± be a pair of opposite Borel subgroups of a Kac-Moody group G and let $\mathsf {U}_{\pm}:=\left [\mathsf {B}_{\pm},\mathsf {B}_{\pm}\right ]$ be the maximal unipotent subgroups. There are flag varieties $\mathcal{B}_{+}:=\mathsf {G}/\mathsf {B}_{+}$ and $\mathcal{B}_{-}:=\mathsf {B}_{-}\backslash \mathsf {G}$. By replacing B_± with U_± we define decorated flag varieties $\mathcal{A}_{+}:=\mathsf {G}/\mathsf {U}_{+}$ and $\mathcal{A}_{-}:=\mathsf {U}_{-}\backslash \mathsf {G}$. There are natural projections $\pi :\mathcal{A}_{\pm}\rightarrow \mathcal{B}_{\pm}$. If π(A)=B then we say that A is a decoration over B.

We denote elements in $\mathcal{B}_{+}$ as Bⁱ and elements in $\mathcal{B}_{-}$ as B_i. The same convention is applied to $\mathcal{A}_{\pm}$ with the letter B replaced by A.

Let T:=B₊∩B₋ and let $\mathsf {W}:=\left .N(\mathsf {T})\right /\mathsf {T}$ be the Weyl group. Consider the Bruhat decompositions and Birkhoff decomposition

$$ \mathsf {G}=\bigsqcup _{w\in \mathsf {W}}\mathsf {B}_{+} w\mathsf {B}_{+}=\bigsqcup _{w\in \mathsf {W}}\mathsf {B}_{-}w \mathsf {B}_{-}=\bigsqcup _{w\in \mathsf {W}}\mathsf {B}_{-}w\mathsf {B}_{+}. $$

We adopt the convention of writing

We often omit w in the notation when it is the identity. Moreover, when decorated flags are involved, the notations only concern the underlying flags; for example, means .

For a positive braid word $\beta =s_{i_{1}}\dots s_{i_{m}}$, a chain will be abbreviated as . By Theorem 2.18 of [SW19], the chains of flags associated to different words of the positive braid [β] have a natural one-to-one correspondence. In this sense, the chain does not depend on the word β chosen.

Definition B.1

Let β and γ be positive braids. The half decorated double BS cell $\mathrm {Conf}_{\beta}^{\gamma}(\mathcal{C})$, viewed as a $\mathbb{Z}$-scheme, is a moduli space parametrizes G-orbits of the chains of flags

If one forgets to choose a decoration A_m over B_m, then the resulting space is denoted by $\mathrm {Conf}_{\beta}^{\gamma}(\mathcal{B})$. Denote by π the forgetful map from ${\mathrm{Conf}}_{\beta}^{\gamma}(\mathcal{C})$ to $\mathrm {Conf}_{\beta}^{\gamma}(\mathcal{B})$.

Remark B.2

This version of double BS cells is slightly different from those in [SW19]: first, the two chains of flags swap places with the $\mathcal{B}_{+}$-chain at the bottom and the $\mathcal{B}_{-}$-chain at the top now; second, there is only one decoration A_m over B_m and the flag B⁰ is no longer decorated.

For , there is a unique flag B₋₁ such that . It then follows from that . This construction gives rise to the following reflection maps.

Definition B.3

The left reflection map $l_{i}: \mathrm {Conf}^{\gamma}_{s_{i}\beta}(\mathcal{C})\rightarrow \mathrm {Conf}^{s_{i} \gamma}_{\beta}(\mathcal{C})$ is an isomorphism mapping

Its inverse map $l^{i}:\mathrm {Conf}^{s_{i}\gamma}_{\beta}(\mathcal{C}) \rightarrow \mathrm {Conf}^{ \gamma}_{s_{i}\beta}(\mathcal{C})$ is defined by an analogous process.

Let φ_i:SL₂→G be the group homomorphism associated to the simple root α_i. Define

$$\begin{aligned} &e_{i}(q) :=\varphi _{i} \begin{pmatrix} 1 & q \\ 0 & 1 \end{pmatrix} , \quad e_{-i}(q) :=\varphi _{i} \begin{pmatrix} 1 & 0 \\ q & 1 \end{pmatrix} , \\ & \overline{s}_{i} :=\varphi _{i} \begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix} , \quad \overline {\overline {s}}_{i} :=\varphi _{i} \begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix} . \end{aligned}$$

Consider a reduced expression $w=s_{i_{1}}\ldots s_{i_{n}}$ in W. Let

$$ \overline{w}=\overline{s}_{i_{1}}\ldots \overline{s}_{i_{n}}, \hspace{8mm}\overline {\overline {w}}=\overline {\overline {s}}_{i_{1}}\ldots \overline {\overline {s}}_{i_{n}}. $$

The elements $\overline{w}$ and $\overline {\overline {w}}$ in G do not depend on the reduced expression chosen. We set

$$ R_{i}(q) := e_{i}(q)\overline{s}_{i} = \varphi _{i} \begin{pmatrix} q & -1 \\ 1 & 0 \end{pmatrix} . $$

(2.1)

Lemma B.4

Fix a flag B^j. The space of flags B^k such that is isomorphic to $\mathbb{A}^{1}$. In particular, if B^j=B₊, then B^k=R_i(q)B₊ for some unique $q\in \mathbb{A}^{1}$.

Proof

It suffices to prove the lemma for B^j=B₊. Let $\mathsf {U}_{i}:=\left \{e_{i}(t)\ \middle | \ t\in \mathbb{A}^{1}\right \}$ be the 1-dimensional unipotent subgroup associated to the simple root α_i and let Q_i:=B₊∩s_iB₊s_i. By [Kum02, Lemma 6.1.3], we know that B₊=U_iQ_i. Therefore

$$ \left .\mathsf {B}_{+} s_{i} \mathsf {B}_{+}\right /\mathsf {B}_{+}=\left .\mathsf {U}_{i}\mathsf {Q}_{i} s_{i} \mathsf {B}_{+}\right /\mathsf {B}_{+}=\left .\mathsf {U}_{i}s_{i}\mathsf {Q}_{i}\mathsf {B}_{+}\right /\mathsf {B}_{+}= \left .\mathsf {U}_{i}s_{i}\mathsf {B}_{+}\right /\mathsf {B}_{+}. $$

Hence B^k=R_i(q)B₊ for some unique $q\in \mathbb{A}^{1}$. □

We prove an important property of the double BS cells, following [SW19, §2.4].

Proposition B.5

The space $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})$ is the non-vanishing locus of a polynomial in $\mathbb{A}^{l(\beta )+l(\gamma )}$. Consequently, it is a smooth affine variety.

Proof

It suffices to prove the lemma for $\mathrm {Conf}^{e}_{\beta}(\mathcal{C})$; the general case will follow by using the reflections to shift the top γ to the bottom. Suppose β is of length l. Every point of $\mathrm {Conf}^{e}_{\beta}(\mathcal{C})$ admits a unique representative as follows

Using Lemma B.4 recursively, we obtain parameters $(q_{1},\dots ,q_{l}) \in \mathbb{A}^{l}$ such that

$$ \mathsf {B}^{k}= R_{i_{1}}(q_{1})\cdots R_{i_{k}}(q_{k}) \mathsf {B}_{+}, \quad \quad k=1,\ldots ,l. $$

By definition, we require that the rightmost pair (U₋,B^l) is in general position.

Let ω_i be the ith fundamental weight. The ith principal minor $\Delta _{i}: \mathsf {G}\rightarrow \mathbb{A}$ is a regular function uniquely determined by the following two conditions: (1) Δ_i(u₋gu₊)=Δ_i(g), where u_±∈U_±; (2) $\Delta _{i}(h)=h^{\omega _{i}}$ for h∈T. When G=SL_n, the function Δ_i coincides with (2.7). Note that g∈B⁻B⁺ if and only if Δ_i(g)≠0 for all i. Therefore the pair (U₋,B^l) is in general position if and only if

$$ f(q_{1},\ldots ,q_{l}):=\prod _{1\leq i \leq {\mathrm{rk}}\mathsf {G}} \Delta _{i} \left (R_{i_{1}}(q_{1})\cdots R_{i_{l}}(q_{l})\right )\neq 0. $$

 □

2.2 B.5 Cluster structures on double Bott-Samelson cells

A pair of positive braids (β,γ) can be regarded as a single braid in the product Br×Br. We shall prove that every word of (β,γ) gives rise to a cluster seed of $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})$. First, each word determines a labeling of arrows and a triangulation on the configuration diagram. Then we require that every pair of flags that are connected by a diagonal in the triangulation are in general position. The subspace of $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})$ that satisfy these general position conditions is an algebraic torus. The algebraic tori obtained from all words of (β,γ) form a subset of the atlas of cluster charts.

In detail, let t be a word of (β,γ). We label the arrows and draw the triangulation on the configuration diagram according to t as shown in Example B.8. On top of the triangulation, we draw rank(G) many parallel lines, each of which represents a simple root of G. Triangles in the triangulation are either upward pointing or downward pointing (as shown below), and depending on the orientation and the labeling of the base, each triangle places a node at one of the lines, cutting it into segments called strings. The segments from such cutting become the vertices of the quiver Q_t, and the arrows in Q_t are drawn according to the pictures below, where the dashed arrows between different levels i≠j are weighted by weights that are related to Cartan numbers (see [SW19] for more details). In particular, in the simply-laced cases (which include SL_n), the dashed arrows all have weight 1/2. In the end, we delete the left most vertices (together with all incident arrows) and freeze the right most vertices on each level.

To define the cluster K₂ coordinates, we first need to decorate the flags. Two decorated flags (resp. ) are said to be compatible if $x^{-1}y\in \mathsf {U}_{+}\overline{w}\mathsf {U}_{+}$ (resp. $xy^{-1}\in \mathsf {U}_{-}\overline {\overline {w}}\mathsf {U}_{-}$). Two decorated flags is called a pinning if xy∈U₋U₊.

Lemma B.6

Given (resp. or ), for every decoration A over B, there exists a unique decoration A′ over B′, such that are compatible (resp. are compatible or is a pinning).

Using the above lemma, we can begin with the decoration A_m over B_m and induce decorations one-by-one over the rest flags following the C-shape path illustrated by the dashed circles below

The next proposition presents a standard representative for every point in $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})$.

Proposition B.7

Every point in $\mathrm {Conf}_{\beta}^{\gamma}(\mathcal{C})$ admits a unique representative in the following form:

where

$$ x_{k} = R_{i_{1}}\left (q_{1}\right )R_{i_{2}}\left (q_{2}\right ) \dots R_{i_{k}}\left (q_{k}\right ), \qquad y_{k} = R_{j_{k}}\left (p_{k} \right )\dots R_{j_{2}}\left (p_{2}\right ) R_{j_{1}}\left (p_{1} \right ). $$

This gives an open embedding $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})\hookrightarrow \mathbb{A}^{m}_{p_{1}, \dots , p_{m}}\times \mathbb{A}^{l}_{q_{1},\dots , q_{l}}$.

Proof

Let us first verify that adjacent decorated flags along the top chain and the bottom chain are compatible. Let x₀=y₀=e. Note that

$$ \mathsf {U}_{+}x_{k-1}^{-1}x_{k}\mathsf {U}_{+}=\mathsf {U}_{+}e_{i_{k}}\left (q_{k}\right ) \overline{s}_{i_{k}}\mathsf {U}_{+}=\mathsf {U}_{+}\overline{s}_{i_{k}}\mathsf {U}_{+}, $$

$$ \mathsf {U}_{-}y_{k-1}y_{k}^{-1}\mathsf {U}_{-}=\mathsf {U}_{-}\left (e_{j_{k}}\left (p_{k} \right )\overline{s}_{j_{k}}\right )^{-1}\mathsf {U}_{-}=\mathsf {U}_{-}e_{-j_{k}} \left (-p_{k}\right )\overline {\overline {s}}_{j_{k}}\mathsf {U}_{-}=\mathsf {U}_{-} \overline {\overline {s}}_{j_{k}}\mathsf {U}_{-}. $$

Since a compatible decoration on one end of any adjacent pair of flags along either of the horizontal chains can be uniquely determined by the decoration on the other end of the pair, the existence of uniqueness of such representative automatically follows from the fact that G acts freely and transitively on the space of pinnings. □

Now for a fixed word t of (β,γ), we get a quiver Q_t with vertices corresponding to strings, which necessarily cross certain diagonals (possibly more than one) in the triangulation.

The cluster K₂ coordinate associated to the string a is defined to be the ith principal minor of xy:

$$ A_{a}=\Delta _{i}(yx). $$

The function A_a is independent of the choice of diagonals if a crosses more than one diagonals.

Example B.8

Let G=SL₃, β=s₂s₁s₂s₁, and γ=s₂s₁. For Br×Br, we use negative numbers for letters in the second factor. The word t=(2,−2,1,2,−1,1) for (β,γ) gives rise to the following triangulation, string diagram, and quiver

Remark B.9

In [SW19] a cluster K₂ structure is constructed on the decorated double BS cell $\mathrm {Conf}^{\gamma}_{\beta}\left (\mathcal{A}_{\mathrm {sc}}\right )$ for a simply-connected group G, which has frozen vertices on both sides of the quiver. The cluster K₂ structure on $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})$ is essentially obtained from that of $\mathrm {Conf}^{\gamma}_{\beta}\left (\mathcal{A}_{\mathrm {sc}}\right )$ by setting all the frozen variables on the left to be 1 due to the pinning condition on .

The next Proposition provides an interpretation of left reflections in terms of standard representatives in Proposition B.7. It implies that the left reflections are cluster transformations.

Proposition B.10

The left reflection $\mathrm {Conf}^{\gamma}_{s_{i}\beta}(\mathcal{C})\rightarrow \mathrm {Conf}^{s_{i} \gamma}_{\beta}(\mathcal{C})$ can be expressed in terms of standard representatives as

$$ \longmapsto $$

Proof

The left reflection does the following.

To restore to the standard representative, we need to act on the resulting configuration by $\left (R_{i}(q)\right )^{-1}$. Note that under the such action, $x\mathsf {U}_{+}\mapsto \left (R_{i}(q)\right )^{-1}x\mathsf {U}_{+}$ and U₋y↦U₋yR_i(q). It is not hard to see that such action will give the standard configuration as claimed in the proposition. □

2.3 B.6 An open embedding

In this section we construct an open embedding $\psi :\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})\times \mathbb{G}_{m} \hookrightarrow \mathrm {Conf}^{\gamma}_{s_{i}\beta}(\mathcal{C})$ whose image is the localization (freezing) at a cluster variable of the latter.

Recall from Lemma B.4 that the moduli space of B⁻¹ that fits into the triangle in the picture on the left below is parametrized by the multiplicative group scheme $\mathbb{G}_{m}$. Note that the base change of $\mathbb{G}_{m}$ to any field is isomorphic to as affine schemes over .

On the other hand, consider a standard representative and let us temporarily forget about the decorations on the pinning and the bottom chain, as shown in the picture on the right above. By gluing these two figures along the pinning , we end up with a point in $\mathrm {Conf}^{\gamma}_{s_{i}\beta}(\mathcal{C})$, which defines a morphism

$$ \psi :\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})\times \mathbb{G}_{m} \rightarrow \mathrm {Conf}^{\gamma}_{s_{i}\beta}(\mathcal{C}), $$

(2.3)

It is easy to see that ψ is an open embedding.

Proposition B.11

The image of ψ in $\mathrm {Conf}^{\gamma}_{s_{i}\beta}(\mathcal{C})$ is the distinguished open subset corresponding to the localization (freezing) at the leftmost cluster variable A_c in the picture below

Proof

There is a unique representative of (B.4) such that B₀=B₋, B⁻¹=B₊, and B⁰=R_i(d)B₊. The principal minors of R_i(d) are

$$ \Delta _{k}(R_{i}(d))=\left \{ \textstyle\begin{array}{l@{\quad}l} d & \text{if } i=k; \\ 1 & \text{if } i\neq k. \end{array}\displaystyle \right . $$

Hence, the left cluster variable A_c=d. By definition, (B.4) is in the image of ψ when B₀ and B⁰ are in general position, or equivalently when d≠0. In other words, the image of ψ is precisely the non-vanishing locus of the cluster variable A_c. In cluster theory, localization of a cluster K₂ variety at a cluster variable A_c is again a cluster K₂ variety, which can be obtained by freezing the vertex c. Therefore the image of ψ is also a cluster K₂ variety. □

Now we make $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})\times \mathbb{G}_{m}$ into a cluster K₂ variety by adding an extra frozen variable d corresponding to the $\mathbb{G}_{m}$ factor. There should not be no arrows connecting c and the unfrozen variables of $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})$ because the extra $\mathbb{G}_{m}$ factor will not affect their mutations. However, there is freedom of adding arrows connecting c and the frozen variables of $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})$. The next proposition shows that these arrows can be uniquely determined by requiring ψ to be a quasi-cluster isomorphism onto its image.

Proposition B.12

The space $\mathrm {Conf}_{\beta}^{\gamma}(\mathcal{C})\times \mathbb{G}_{m}$ can be equipped with a unique cluster K₂ structure which extends the cluster K₂ structure on $\mathrm {Conf}_{\beta}^{\gamma}(\mathcal{C})$ by adding one extra frozen vertex c and possibly arrows between c and the original frozen part, such that ψ becomes a quasi-cluster isomorphism onto its image.

Proof

Suppose we start with a standard representative in the image of ψ as follows.

From the last proposition we know that x₀U₊=R_i(d)U₊ for some non-zero d.

To obtain the preimage of this representative under ψ, we need to delete the flag U₊ at the lower left corner and re-scale the decorations along the bottom chain as follows.

Here h_k∈T are such that (U₋,x₀h₀U₊) is a pinning and (x_k−1h_k−1U₊,x_kh_kU₊) are compatible.

Set $\lambda _{0}^{\vee }= -\alpha _{i}^{\vee}$. Define co-characters $\lambda ^{\vee}_{k}$ of T for 1≤k≤l by the recursive relation

$$ \lambda ^{\vee}_{k}:=s_{i_{k}}\left (\lambda ^{\vee}_{k-1}\right ). $$

Note that x₀=R_i(d). An easy calculation shows that x₀h₀∈U₋U₊ if and only if $h_{0}= d^{\lambda _{0}^{\vee}}$. Since (x_k−1U₊,x_kU₊) is a compatible pair, by definition we get $x_{k-1}^{-1}x_{k}\in \mathsf {U}_{+}\overline{s}_{i_{k}}\mathsf {U}_{+}$. Therefore,

$$ \mathsf {U}_{+}\left (x_{k-1}h_{k-1}\right )^{-1}\cdot x_{k}h_{k}\mathsf {U}_{+} = \mathsf {U}_{+}\overline{s}_{i_{k}}\cdot s_{i_{k}}(h_{k-1}^{-1})h_{k}\mathsf {U}_{+}. $$

The pair (x_k−1h_k−1U₊,x_kh_kU₊) is compatible if and only if $h_{k}=s_{i_{k}}(h_{k-1})$. By induction we get $h_{k}= d^{\lambda _{k}^{\vee}}$ for 0≤k≤l.

Next we investigate the pull-back of cluster K₂ coordinates of $\mathrm {Conf}^{\gamma}_{s_{i}\beta}(\mathcal{C})$ under ψ. Fix a word t for (β,γ) and consider the word (i,t) for $\left (s_{i}\beta ,\gamma \right )$. Let Q_t be the quiver associated to t, and Q_i,t the quiver associated to (i,t) with the leftmost vertex c frozen.

Recall that

$$ \psi ^{*}\left (A_{c}\right )=\Delta _{i}(x_{0})=\Delta _{i}(R_{i}(d))=d. $$

We define d to be the cluster variable $A'_{c}$ for the new frozen vertex c.

For any other string (vertex) a associated to (i,t) as the left picture below, there is a corresponding string (vertex) a associated to t as the right right below.

Let $\delta _{a}:=-\left \langle \lambda ^{\vee}_{k},\omega _{h}\right \rangle \in \mathbb{Z}$; then

$$ \psi ^{*}\left (A_{a}\right )=\Delta _{h}\left (y_{j}x_{k}\right )= \Delta _{h}\left (y_{j}x_{k}h_{k}\right )d^{- \left \langle \lambda _{k}^{\vee},\omega _{h}\right \rangle } =A'_{a}A^{\prime \,\delta _{a}}_{c}. $$

In addition we define $\delta _{c}:=-\langle \lambda _{0}^{\vee}, \omega _{i}\rangle =1$. Let I denote the vertices of Q_i,t and let ε_ij be the exchange matrix encoded by Q_i,t. The set I′=I−{c} consists of vertices of Q_t and I^uf consists of unfrozen vertices of Q_t. We claim that for any a∈I^uf, we have

$$ \sum _{b\in I}\varepsilon _{ab}\delta _{b}=0. $$

(2.5)

To see this, recall that there is projection map

$$ p:~~\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})\times \mathbb{G}_{m} \longrightarrow \mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C}) \stackrel{\pi}{\longrightarrow} \mathrm {Conf}^{\gamma}_{\beta}(\mathcal{B}) $$

As in [SW19, §3], $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{B})$ is equipped with the cluster Poission variables $\{X_{a}'\}_{a\in I^{\mathrm {uf}}}$ such that

$$ p^{*}\left (X'_{a}\right )=\prod _{b\in I'} A^{\prime \,\varepsilon _{ab}}_{b}. $$

(2.6)

Consider the composition

$$ p':= \mathrm {Conf}^{\gamma}_{s_{i}\beta}(\mathcal{C}) \stackrel{\pi}{\longrightarrow} \mathrm {Conf}^{\gamma}_{s_{i}\beta}( \mathcal{B}) \longrightarrow \mathrm {Conf}^{\gamma}_{\beta}(\mathcal{B}) $$

Here the second map is rational, obtained by forgetting the flag B⁻¹. Note that B⁻¹ only changes the decorations on the other flags. Therefore we have p=p′∘ψ. Therefore for a∈I^uf we have

$$\begin{aligned} p^{*}\left (X'_{a}\right )&=\psi ^{*}\circ p^{\prime \,*}\left (X'_{a}\right )= \psi ^{*}\left (\prod _{b\in I}A^{\varepsilon _{ab}}_{b}\right )=A^{\prime \,\varepsilon _{ac}}_{c} \prod _{b\in I'}A^{\prime \,\varepsilon _{ab}\delta _{b}}_{c}A^{\prime \,\varepsilon _{ab}}_{b}\\ &= A_{c}^{\prime \,\sum _{b\in I}\varepsilon _{ab}\delta _{b}}\cdot \prod _{b \in I'} A^{\prime \,\varepsilon _{ab}}_{b}. \end{aligned}$$

Comparing it with (B.6), we arrive at the identity (B.5).

Note that identity (B.5) satisfies the assumptions stated in Proposition A.13. Therefore we know that there is a unique way to extend the quiver of $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})$ so that ψ becomes a quasi-cluster isomorphism onto its image. □

Example B.13

We continue from Example B.8. Consider the map $\phi _{1}:\mathrm {Conf}_{\beta}^{\gamma}(\mathcal{C})\times \mathbb{G}_{m} \rightarrow \mathrm {Conf}_{s_{1}\beta}^{\gamma}(\mathcal{C})$. Let $d=A'_{c}$ be the coordinate for the $\mathbb{G}_{m}$ factor. Then in the preimage,

The change of decorations gives rise the the pull-backs $\phi _{1}^{*}\left (A_{c}\right )=A'_{c}$ and $\phi _{1}^{*}\left (A_{a}\right )=A'_{a}d^{\delta _{a}}=A'_{a}A^{\prime \,\delta _{a}}_{c}$ for a≠c. The integers δ_a assigned to the vertices a are as follows.

Using these δ_a we conclude that the cluster structure on $\mathrm {Conf}^{\gamma}_{\beta}(\mathcal{C})\times \mathbb{G}_{m}$ is given by the following quiver, where the right most vertex is the extra frozen vertex c.