BRAID VARIETY CLUSTER STRUCTURES I 3D PLABIC GRAPHS PAVEL GALASHIN THOMAS LAM MELISSA SHERMAN-BENNETT AND DAVID SPEYER Abstract. We introduce 3-dimensional generalizations of Postnikovs plabic graphs and

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BRAID VARIETY CLUSTER STRUCTURES, I: 3D PLABIC GRAPHS

PAVEL GALASHIN, THOMAS LAM, MELISSA SHERMAN-BENNETT, AND DAVID SPEYER

Abstract. We introduce 3-dimensional generalizations of Postnikov’s plabic graphs and

use them to establish cluster structures for type Abraid varieties. Our results include

known cluster structures on open positroid varieties and double Bruhat cells, and establish

new cluster structures for type Aopen Richardson varieties.

Contents

1. Introduction 1

2. Open Richardson varieties 3

3. Double braid quivers 8

4. Invariance under moves 19

5. Cluster algebras associated to 3D plabic graphs 27

6. Double braid varieties 30

7. Deodhar geometry and seeds 37

8. Moves preserve the cluster algebra 41

9. Proof of Theorem 7.12 47

10. Applications and relations to previous work 48

References 54

1. Introduction

Braid varieties ◦

Ru,β are smooth, aﬃne, complex algebraic varieties associated to a permu-

tation uand a braid word β, that is, a word representing an element of the positive braid

monoid. The purpose of this work is to construct a cluster algebra structure [FZ02] on the

coordinate ring of a braid variety.

Theorem 1.1. The coordinate ring C[◦

Ru,β]of a braid variety is a cluster algebra.

Date: May 31, 2024.

2020 Mathematics Subject Classiﬁcation. Primary: 13F60. Secondary: 14M15, 05E99.

Key words and phrases. Plabic graph, cluster algebra, open Richardson variety, conjugate surface, local

acyclicity, Deodhar hypersurface.

P.G. was supported by an Alfred P. Sloan Research Fellowship and by the National Science Foundation

under Grants No. DMS-1954121 and No. DMS-2046915. T.L. was supported by Grant No. DMS-1953852

from the National Science Foundation. M.S.B. was supported by the National Science Foundation under

Award No. DMS-2103282. D.E.S was supported by Grants No. DMS-1854225 and No. DMS-1855135. Any

opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the authors

and do not necessarily reﬂect the views of the National Science Foundation.

arXiv:2210.04778v3 [math.CO] 29 May 2024

2 PAVEL GALASHIN, THOMAS LAM, MELISSA SHERMAN-BENNETT, AND DAVID SPEYER

The braid varieties we consider were studied, rather recently, in [Mel19, CGGS20], gen-

eralizing varieties considered previously in [Deo85, MR04, WY07]. Theorem 1.1 resolves

conjectures of [Lec16, CGGS21].

Certain special cases of braid varieties, namely open Richardson varieties and open positroid

varieties, have played a key role in the combinatorial geometry of ﬂag varieties and Grassman-

nians. Open Richardson varieties ◦

Ru,w are subvarieties of the variety SLn/B+of complete

ﬂags in Cn, arising in the study of total positivity and Poisson geometry [Deo85, Lus98,

MR04, Pos06, BGY06, KLS13]. An explicit cluster structure for open Richardson varieties

was conjectured by Leclerc [Lec16]. M´enard [M´e22] gave an alternative conjectural cluster

structure, which was proved by Cao and Keller [CK22] to be an upper cluster structure;

M´enard’s and Leclerc’s cluster structures are expected to coincide. Another upper cluster

structure was constructed by Ingermanson [Ing19]. The cluster structure of Theorem 1.1, in

the case of open Richardson varieties, agrees with that of Ingermanson. It is related to the

cluster structure of Leclerc [Lec16] by the twist automorphism [GL22b, SSB22].

We also prove that the cluster varieties in Theorem 1.1 are locally acyclic [Mul13]; in

the case of open Richardson varieties we use this to establish a variant of a conjecture of

Lam and Speyer [LS16]. In particular, the cohomology of braid varieties satisﬁes the curious

Lefschetz phenomenon (Theorem 10.1).

Theorem 1.1 generalizes the (type A) results of [FZ99, GY20] on double Bruhat cells

and [SW21] on double Bott–Samelson cells, and also the results of Ingermanson [Ing19] and

Cao and Keller [CK22], who found upper cluster structures on open Richardson varieties

(see also [M´e22]). Furthermore, Theorem 1.1 generalizes the main result of [GL19] (see

also [Sco06, MS16, SSBW19]), where the same statement was proved for open positroid

varieties [KLS13], which are special cases of open Richardson varieties. Positroid varieties

are parametrized by plabic graphs [Pos06], whose planar dual quivers describe the cluster

algebra structure on the associated open positroid varieties.

Ever since the completion of [GL19], it has been our hope that constructing a cluster struc-

ture for open Richardson varieties would lead to a meaningful generalization of Postnikov’s

plabic graphs; indeed, discovering such a generalization turned out to be a crucial step in

our proof of Theorem 1.1. We associate a 3D plabic graph to each pair (u, β) consisting of

a permutation uand a (double) braid word β, and use the combinatorics of this graph to

construct our cluster structure. The reader is invited to look forward at the examples in

Figures 1–6. Code for computing 3D plabic graphs and the associated seeds is available at

[Gal23].

An important geometric ingredient in our approach is the study of the Deodhar geometry

of braid varieties, originally used by Deodhar [Deo85] in the ﬂag variety setting. We deﬁne an

open Deodhar torus Tu,β ⊂◦

Ru,β, and our cluster variables are interpreted as characters of Tu,β

that have certain orders of vanishing along the Deodhar hypersurfaces in the complement of

the Deodhar torus. We expect this geometric approach to have applications to other settings

where cluster structures are expected to make an appearance.

Our work has a number of applications. Our cluster structure implies, via [LS16], a curious

Lefschetz phenomenon for the cohomology of braid varieties. Our approach is closely related

to the combinatorics of braid Richardson links that are associated to a braid variety; in

particular, we relate certain quiver point counts to the HOMFLY polynomial of these links.

We learned at the ﬁnal stages of completing this manuscript that a cluster structure for

braid varieties (of arbitrary Lie type) was independently announced in a recent preprint [CGG+22].

BRAID VARIETY CLUSTER STRUCTURES, I: 3D PLABIC GRAPHS 3

It would be interesting to investigate the relation between our 3D plabic graphs and the ap-

proach of [CGG+22].

Our construction in other Lie types. The ﬁrst three authors have continued this line

of research to encompass braid varieties of arbitrary Lie type [GLSB23], again using the

Deodhar decomposition as the main geometric tool. The cluster structure that we construct

in type Ais much more explicit and combinatorial than the ones that we construct in

other Lie types. In particular, our type Astructure has clear connections to the plabic

graphs of [Pos06]. In all Lie types, there is a family of “chamber minors” (generalizing those

appearing in [FZ99, MR04]) which are monomials in the cluster variables. In type A, the

matrix transforming cluster variables to chamber minors is a {0,1}matrix; see Section 7.3

and Proposition 7.7. However, in other Lie types, this matrix can have entries larger than 1,

and the combinatorics of these matrix entries is complicated; see Remark 7.8 and [GLSB23,

Section 7].

On the other hand, there are arguments which are simpler and clearer when presented in

all Lie types. We have chosen to place those arguments in [GLSB23], and removed longer

computations which establish them only in type Afrom this paper.

Overview. In Section 2, we give a synopsis of our main results in the setting of open

Richardson varieties. In the rest of the paper we work in the setting of braid varieties. We

deﬁne 3D plabic graphs and the associated quivers e

Qu,β in Section 3. Next, we develop the

combinatorics of 3D plabic graphs and show that the quivers e

Qu,β are invariant under braid

moves on the word β, naturally extending square moves from Postnikov’s plabic graphs to

3D plabic graphs; see Section 4. We discuss cluster algebras associated to 3D plabic graphs

in Section 5 and show that they are locally acyclic in the sense of [Mul13]. In Sections 6

and 7, we study the Deodhar geometry of ◦

Ru,β and construct a seed in C(◦

Ru,β) for each 3D

plabic graph. We then show that the seeds are related by mutation in Section 8. Finally,

in Section 9, we prove Theorem 7.12 by induction on the length of β. We conclude with

some applications of our approach in Section 10, and explain how our results and deﬁnitions

specialize to those in [Pos06, FZ99, Ing19].

Acknowledgments. T.L. and D.E.S. thank our students Ray Karpman and Gracie Inger-

manson for helping us understand the relationship between Deodhar’s positive subexpres-

sions and Postnikov’s combinatorics and for the other ideas discussed in Section 2.6. M.S.B.

thanks Daping Weng for illuminating conversations on [SW21]. P.G. thanks Terrence George

for discussions regarding [GK13]. We thank Lauren Williams for her comments on the ﬁrst

version of this manuscript. We also appreciate many conversations with Allen Knutson about

Richardson and Bott–Samelson varieties, and Deodhar tori. We thank Roger Casals, Eugene

Gorsky, and Anton Mellit for conversations related to this project. Finally, we thank the

authors of [CGG+22] for sharing their exciting results with us.

2. Open Richardson varieties

In this section, we give a more detailed explanation of Theorem 1.1 in the case of open

Richardson varieties.

4 PAVEL GALASHIN, THOMAS LAM, MELISSA SHERMAN-BENNETT, AND DAVID SPEYER

Figure 1. A 3D plabic graph Gu,w.

Figure 2. Propagation rules (right to left) for the relative cycles in Gu,w.

Figure 3. Applying propagation rules to ﬁnd one relative cycle Ccof Gu,w.

2.1. Open Richardson varieties. Let G= SLn, and let B+,B−be the opposite Borel

subgroups of upper and lower triangular matrices, respectively. For two permutations u, w ∈

Snsuch that u≤win the Bruhat order, the open Richardson variety ◦

Ru,w is deﬁned as

◦

Ru,w := (B−uB+∩B+wB+)/B+.

To each pair u≤wand to each reduced word wfor wwe associate an ice quiver e

Qu,w

(Section 2.3). Let A(e

Qu,w) be the associated cluster algebra; see Section 5.1 for background.

Theorem 2.1. For all u≤win Sn, we have an isomorphism

C[◦

Ru,w]∼

=A(e

Qu,w).

Moreover, the cluster algebra A(e

Qu,w)is locally acyclic and really full rank.

The cluster algebra terminology in Theorem 2.1 will be introduced in Section 5.1. We

now describe the 3D plabic graph Gu,w, the quiver e

Qu,w, and the associated cluster algebra

A(e

Qu,w).

BRAID VARIETY CLUSTER STRUCTURES, I: 3D PLABIC GRAPHS 5

2.2. 3D plabic graphs. Let w= (i1, i2, . . . , im) be a reduced word for w. Consider the

unique rightmost subexpression ufor uinside w, and let Ju,w⊂[m] := {1,2, . . . , m}be the

set of indices not used in u. The 3D plabic graph Gu,wis obtained from the wiring diagram

for wby replacing all crossings in [m]\Ju,wby overcrossings and replacing each crossing

c∈Ju,wby a black-white bridge edge bc; see Figure 1. We place a marked point on each of

the nleftmost boundary vertices of Gu,w, and denote by Mthe set of these marked points.

The number of bridges in Gu,wis |Ju,w|=ℓ(w)−ℓ(u), which is the dimension of ◦

Ru,w.

To each index c∈Ju,wwe will associate an (oriented) relative cycle Ccin Gu,w, which by

deﬁnition is either a cycle in Gu,wor a union of oriented paths in Gu,wwith endpoints in M.

Each relative cycle Ccwill naturally bound a disk Dc.1For instance, in Figure 3, the

vertical sections of Dcare shown in wavy pink lines. We indicate the relative position of Dc

in R3with respect to the edges of Gu,wby over/under-crossings. We will compute Dc, and

therefore its boundary Cc, starting from the bridge bcand proceeding to the left using the

propagation rules in Figure 2. We choose the counterclockwise orientation of Cc, so that as

one traverses Cc, the disk Dcis to the left. See Section 3.4 for a description of relative cycles

in the case of double braid varieties.

2.3. The quiver. Aquiver Qis a directed graph without directed cycles of length 1 and 2.

An ice quiver e

Qis a quiver whose vertex set e

V=V(e

Q) is partitioned into frozen and mutable

vertices: e

V=Vfro ⊔Vmut. The arrows between pairs of frozen vertices are automatically

omitted.

The procedure in Section 2.2 yields a bicolored graph Gu,wdecorated with a family

(Cc)c∈Ju,wof relative cycles. To this data, we associate an ice quiver e

Qu,w. Our construction

will rely on the results of [FG09, GK13]. The vertex set V(e

Qu,w) := Ju,wis in bijection with

the set of relative cycles. If a relative cycle Ccis actually a cycle in Gu,wthen cis a mutable

vertex of e

Qu,w; otherwise, if Ccis a union of paths with endpoints in M,cis a frozen vertex

of e

Qu,w.

To compute the arrows of e

Qu,w, we consider Gu,was a ribbon graph, with counterclockwise

half-edge orientations around white vertices and clockwise half-edge orientations around

black vertices. Let Su,wbe the surface with boundary obtained by replacing every edge of

Gu,wby a thin ribbon and gluing the ribbons together according to the local orientations at

the vertices of Gu,w. See Figure 7(d) for an example of Su,w. The nmarked points of Gu,w

give rise to nmarked points on ∂Su,w, the set of which is also denoted by M.

We view each relative cycle Ccas an element of the relative homology Λu,w:= H1(Su,w,M).

It turns out that each mutable relative cycle can be also viewed as an element of the dual

lattice Λ∗

u,w; see Section 3.3. The (signed) number of arrows between two vertices c, d ∈Ju,w

in e

Qu,w, where dis mutable, is deﬁned to be the (signed) intersection number ⟨Cc, Cd⟩of

the relative cycles Ccand Cd. These intersection numbers can be computed explicitly using

simple pictorial rules; see Algorithm 3.9. Alternatively, one can compute the quiver e

Qu,wby

summing up half-arrow contributions; see Section 3.7. One can check that in the case of our

running example (Figure 1), both descriptions yield the quiver shown in Figure 4. See also

Example 3.11.

1More precisely, when Ccis a cycle, ∂Dc=Cc, and when Ccis a union of paths with marked endpoints,

∂Dcis the union of Cctogether with several straight line segments connecting pairs of marked points.

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BRAIDVARIETYCLUSTERSTRUCTURES,I:3DPLABICGRAPHSPAVELGALASHIN,THOMASLAM,MELISSASHERMAN-BENNETT,ANDDAVIDSPEYERAbstract.Weintroduce3-dimensionalgeneralizationsofPostnikov’splabicgraphsandusethemtoestablishclusterstructuresfortypeAbraidvarieties.Ourresultsincludeknownclusterstructuresonopenpositroidvarie...

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BRAID VARIETY CLUSTER STRUCTURES I 3D PLABIC GRAPHS PAVEL GALASHIN THOMAS LAM MELISSA SHERMAN-BENNETT AND DAVID SPEYER Abstract. We introduce 3-dimensional generalizations of Postnikovs plabic graphs and

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