LECLERCS CONJECTURE ON A CLUSTER STRUCTURE FOR TYPE A RICHARDSON VARIETIES KHRYSTYNA SERHIYENKO AND MELISSA SHERMAN-BENNETT

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LECLERC’S CONJECTURE ON A CLUSTER STRUCTURE FOR TYPE A

RICHARDSON VARIETIES

KHRYSTYNA SERHIYENKO AND MELISSA SHERMAN-BENNETT

Abstract. Leclerc [Lec16] constructed a conjectural cluster structure on Richardson varieties in

simply laced types using cluster categories. We show that in type A, his conjectural cluster structure

is in fact a cluster structure. We do this by comparing Leclerc’s construction with another cluster

structure on type A Richardson varieties due to Ingermanson [Ing19]. Ingermanson’s construction

uses the combinatorics of wiring diagrams and the Deodhar stratiﬁcation. Though the two cluster

structures are deﬁned very diﬀerently, we show that the quivers coincide and clusters are related

by the twist map for Richardson varieties, recently deﬁned by Galashin–Lam [GL22].

1. Introduction

In this article, we consider cluster structures on open Richardson varieties ˚

Rv,w in the complete

ﬂag variety F`n. For v≤w∈Sn, the open Richardson variety ˚

Rv,w is the intersection of the

Schubert cell Cwwith the opposite Schubert cell Cv, and is smooth, aﬃne, and irreducible. Open

Richardson varieties are related to the geometric interpretation of Kazhdan-Lusztig polynomials

[Deo85, KL79, KL80]; open Richardson varieties also arise in total positivity for F`n[Lus94, Rie99].

Special cases include (open) positroid varieties [KLS13, Pos06], which are Richardson varieties ˚

Rv,w

where whas a single descent.

A cluster structure on ˚

Rv,w is an identiﬁcation of the coordinate ring C[˚

Rv,w] with a cluster

algebra A(Σ). Cluster algebras were introduced by Fomin–Zelevinsky [FZ02] to provide an algebraic

and combinatorial framework for Lusztig’s dual canonical bases and total positivity [Lus90, Lus94].

Cluster algebras have appeared in a wide range of ﬁelds, including Teichm¨uller theory [FG06],

mirror symmetry [GHKK18], Poisson geometry [GSV10], symplectic geometry [STWZ19], knot

theory [FPST22], and scattering amplitudes in high energy physics [GGS+14]. One major direction

of research is to understand when varieties naturally ocurring in representation theory have a cluster

structure; examples of varieties with cluster structures include Grassmannians [Sco06], double

Bruhat cells in semisimple Lie groups [BFZ05], and unipotent cells in Kac-Moody groups [GLS11].

Cluster algebras are commutative rings with a distinguished set of generators called cluster

variables, which are grouped together into clusters. A cluster can be mutated into another cluster,

and any two clusters are related by a sequence of mutations. The information of all mutations of a

cluster is encoded in a quiver, i.e. a directed graph. A cluster and its quiver together form a seed.

Leclerc [Lec16] used categoriﬁcation to construct a conjectural cluster structure1for ˚

Rv,w. In

particular, he deﬁned a cluster category inside the module category of a preprojective algebra and

identiﬁed certain cluster tilting objects in this category. Each cluster tiliting object naturally gives

rise to a seed ΣLec

v,w= (Bv,w, QLec

v,w), where the cluster Bv,wis obtained via a cluster character

map and the quiver QLec

v,wrecords irreducible morphisms between indecomposable summands of the

cluster tilting module. In this contruction, it is relatively easy to obtain the cluster, but quite

KS and MSB were supported by the National Science Foundation under Award No. DMS-2054255 and Award

No. DMS-2103282 respectively. 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.

1Leclerc’s results and conjecture are in types ADE. We will deal only with the type A case in this paper.

arXiv:2210.13302v1 [math.CO] 24 Oct 2022

2 KHRYSTYNA SERHIYENKO AND MELISSA SHERMAN-BENNETT

diﬃcult to compute the quiver. Leclerc showed that the cluster algebra A(ΣLec

v,w) is a subring of

C[˚

Rv,w].

Leclerc conjectured that A(ΣLec

v,w) is equal to C[˚

Rv,w], and showed that equality holds in some

special cases. One of the obstacles in proving Leclerc’s conjecture is the diﬃculty in computing

the quiver QLec

v,win general. In [SSBW19], the authors together with L. Williams showed that for

certain positroid varieties, the quiver QLec

v,wcoincides with a plabic graph quiver. Later, Galashin–

Lam [GL19] extended this result to all positroid varieties, and used this to show Leclerc’s conjecture

in the positroid variety case.

Our main result is a proof of Leclerc’s conjecture in type A.

Theorem A. Let v≤wand let wbe a reduced expression for w. Then

C[˚

Rv,w] = A(ΣLec

v,w).

Moreover, the cluster algebra A(ΣLec

v,w)does not depend on the choice of w.

We prove Theorem A by comparing A(ΣLec

v,w) with another cluster structure on ˚

Rv,w, deﬁned

by Ingermanson [Ing19]. Ingermanson constructed a seed ΣIng

v,w= (Av,w, QIng

v,w) using the wiring

diagram of a unipeak expression for w. The cluster variables Av,ware particular factors of the

chamber minors of Marsh–Rietsch [MR04], and the quiver QIng

v,wcan be read oﬀ from the wiring

diagram. In her construction, determining the factorization of chamber minors into cluster variables

is quite involved, but once this has been done, it is easy to write down the quiver. Ingermanson

showed that the upper cluster algebra U(ΣIng

v,w) is equal to C[˚

Rv,w]; recent results of [GLSBS22]

imply that Ingermanson’s quiver is locally acyclic, and so A(ΣIng

v,w) = C[˚

Rv,w].

We show the following relationship between Ingermanson’s seed and Leclerc’s.

Theorem B. Let v≤wand let wbe a unipeak expression for w. Let τv,w be the right twist map

for ˚

Rv,w from [GL22]. Then

(Av,w, QIng

v,w)=(Bv,w◦τv,w, QLec

v,w).

We separately show that all of the seeds Leclerc deﬁnes are related by mutations. As a corollary

of these result, we show that the positive part of ˚

Rv,w deﬁned by Leclerc’s cluster structure agrees

with the totally positive part ˚

R>0,Lus

v,w deﬁned by Lusztig [Lus94].

Theorem C. Let v≤w. The subset

R>0,Lec

v,w := {F∈˚

Rv,w :all cluster variables in A(ΣLec

v,w)are positive on F}

coincides with ˚

R>0,Lus

v,w .

Theorem B has the eﬀect of simplifying the deﬁnitions of both Leclerc’s and Ingermanson’s seeds.

We obtain a much more straightforward method to factor chamber minors into Ingermanson’s

cluster variables, and an elementary method to draw Leclerc’s quiver from a wiring diagram. This

alternate description of Leclerc’s quiver is in the same vein as the descriptions in the positroid

variety case given by [SSBW19, GL19]. As such, we hope our results make cluster structures on

Richardson varieties more accesible.

Our results show that Leclerc’s and Ingermanson’s cluster structure on ˚

Rv,w are related by the

twist map for Richardsons, which generalizes the twist map for positroid varieties [MS17]. In the

positroid variety case, the twist map is conjectured to be a quasi-cluster transformation [Fra16];

i.e. the twist map is conjectured to be a sequence of mutations followed by rescaling by Laurent

monomials in frozens. We make the same conjecture in the Richardson variety case.

LECLERC’S CONJECTURE ON A CLUSTER STRUCTURE FOR TYPE A RICHARDSON VARIETIES 3

Conjecture 1.1. The twist map τv,w :˚

Rv,w →˚

Rv,w is a quasi-cluster transformation. As a result,

any cluster of A(ΣIng

v,w)is related to any cluster of A(ΣLec

v,w)by a sequence of mutations and rescaling

by Laurent monomials in frozens.

We brieﬂy discuss related work on Richardson variety cluster structures, which has been a very

active topic of late. In types ADE, M´enard [M´

22] constructed another cluster tilting object for

each reduced word wof w; his construction has the advantage that the quiver is constructed

algorithmically. Cao–Keller [CK22] recently showed that, if ΣM

v,wis the seed obtained from M´enard’s

cluster tilting object via the cluster character map, then U(ΣM

v,w) = C[˚

Rv,w] (again, in types

ADE). The relation between ΣM

v,wand ΣLec

v,wis as yet unknown; the quivers are conjectured to be

mutation-equivalent. In a separate direction, Casals–Gorsky–Gorsky–Le–Shen–Simental [CGG+22]

and Galashin–Lam–SB–Speyer [GLSBS22, GLSBS] have independently given cluster structures on

braid varieties in arbitrary type, which generalize Richardson varieties; it is not known how these

two cluster structures are related. Ingermanson’s construction is a special case of the construction

in [GLSBS22]; M´enard’s seeds are special cases of those in [CGG+22].

We begin with background on Richardson varieties and various combinatorial constructions in

Section 2. We review the constructions of ΣLec

v,wand ΣIng

v,win Section 3. In Section 4, we give

the relationship between Leclerc’s cluster variables and Ingermanson’s. In Section 5 we describe

Leclerc’s quiver using wiring diagrams, and in Section 6 we use this description to prove that

Leclerc’s quiver coincides with Ingermanson’s quiver. Finally, in Section 7, we complete the proofs

of Theorems A, B and C.

Acknowledgements: MSB thanks Pavel Galashin, Thomas Lam, and David Speyer for helpful

conversations related to Ingermanson’s construction.

2. Background

We use the following standard combinatorial notation: [n] := {1, . . . , n},[n]

his the set of

cardinality hsubsets of [n], w0is the longest permtuation of Sn,si∈Snis the transposition

exchanging iand i+ 1, for v, w ∈Sn,v≤wmeans vis less than win the Bruhat order and `(w)

denotes the length of w.

2.1. Background on Richardson varieties. Let G=SLn(C) and let B, B−⊂Gdenote the

Borel subgroups of upper and lower triangular matrices, respectively. Let N, N−denote the corre-

sponding unipotent subgroups of upper and lower unitriangular matrices, respectively. For g∈G,

let gidenote the ith column of g. We denote the minor of gon rows Rand columns Cby ∆R,C (g).

For w∈Sn, we choose a distinguished lift ˙wof wto G. The lift satisﬁes

˙wij =(±1 if i=w(j)

0 else

and the signs of entries are determined by the condition that ∆w[j],[j]( ˙w) = 1 for all j∈[n]. If the

particular lift of wto Gdoes not matter, we also write wfor the lift (e.g. we write BwB rather

than B˙wB).

We identify the ﬂag variety F`nwith the quotient G/B. Concretely, a matrix g∈Grepresents

the ﬂag V•= (V1⊂V2⊂ · · · ⊂ Vn=Cn) where Viis the span of g1, . . . , gi. The ﬂag variety has

two well-known decompositions into cells, the Schubert decomposition

G/B =G

w∈Sn

BwB/B =G

w∈Sn

4 KHRYSTYNA SERHIYENKO AND MELISSA SHERMAN-BENNETT

and the opposite Schubert decomposition

G/B =G

w∈Sn

B−wB/B =G

w∈Sn

Cw.

The stratum Cwis a Schubert cell and is isomorphic to C`(w). The stratum Cwis an opposite

Schubert cell and is isomorphic to C`(w0)−`(w). For a ﬁxed lift w, it is well-known that the projection

map G→G/B restricts to isomorphisms

(1) Nw ∩wN−

∼

−→ CwN−w∩wN−

∼

−→ Cw.

Or, more concretely, each coset in Cw(resp. Cw) has a unique representative matrix which diﬀers

from wonly in entries that lie both above and to the left (resp. both below and to the left) of a

nonzero entry of w(see e.g. [Ful97]).

We are concerned with the intersection of an opposite Schubert cell and a Schubert cell

Rv,w := Cv∩Cw

which is called an (open) Richardson variety. We usually drop the adjective “open.” The Richardson

variety ˚

Rv,w is nonempty if and only if v≤w, in which case it is a smooth irreducible aﬃne variety

of dimension `(w)−`(v) [Deo85].

Open Richardson varieties were studied in the context of Kazdhan-Lusztig polynomials [KL79];

the number of Fqpoints of ˚

Rv,w is exactly the R-polynomial indexed by (v, w) [Deo85], which

can be used to recursively compute Kazhdan-Lusztig polynomials. The Fq-point counts and more

generally the cohomology of ˚

Rv,w are also related to knot homology [GL20]. Real points of ˚

Rv,w,

and in particular positive points, feature in work of Lusztig and Rietsch [Lus94, Rie99] on total

positivity. Special cases of Richardson varieties include the (open) positroid varieties of [KLS13],

which are Richardson varieties ˚

Rv,w where whas a single descent. Richardson varieties themselves

are special cases of braid varieties (see e.g. [CGGS20]).

We identify ˚

Rv,w with two diﬀerent subsets of G, one for Ingermanson’s construction and one

for Leclerc’s. We will later use these identiﬁcations to deﬁne functions on ˚

Rv,w. Below, we use the

involutive automorphism g7→ gθof Gfrom [FZ99, (1.11)]; the (i, j) entry of gθis the minor of g

obtained by deleting the ith row and jth column. It is not hard to check that Bθ=B−,Nθ=N−,

and ˙vθis another lift of vto G.

Lemma 2.1. For v≤w, let

Nv,w := N∩˙wN−˙w−1∩B−vB ˙w−1and N0

v,w := N∩˙v−1N˙v∩˙v−1B−wB−.

Also, let D:Nv,w →Gbe the renormalization map sending g→gdg, where dgis the unique

diagonal matrix so that ∆v[j],w[j](gdg) = 1 for all j.

We have isomorphisms

α:D(Nv,w)→˚

Rv,w β:N0

v,w →˚

Rv,w

gdg7→ gdg˙wB g 7→ ( ˙vg)θB.

Proof. If g∈Nv,w, then g˙wis in B−vB. In particular, the minors ∆v[j],[j](g˙w)=∆v[j],w[j](g) are

nonzero. This implies the map Dis well-deﬁned. It is also an isomorphism onto its image.

The map αcan be written as a composition of two maps

D(Nv,w)D−1

−−−→Nv,w α0

−→ ˚

Rv,w

gdg7−→ g7−→ g˙wB

since gdg˙wB is equal to g˙wB.

Now, it follows easily from (1) that α0and βare both isomorphisms, noting in the ﬁrst case that

g˙wis in N˙w∩˙wN−∩B−vB and in the second that ( ˙vg)θis in N−˜v∩˜vN−∩BwB where ˜v= ˙vθ.

LECLERC’S CONJECTURE ON A CLUSTER STRUCTURE FOR TYPE A RICHARDSON VARIETIES 5

Remark 2.2. Leclerc identiﬁes the ﬂag variety with B−\Grather than G/B, and so considers the

variety

−˚

Rv,w := B−\(B−vB ∩B−wB−)

which is diﬀerent from, though isomorphic to, ˚

Rv,w. We ﬁx an isomorphism so that we can pullback

functions on −˚

Rv,w to functions on ˚

Rv,w. The isomorphism we choose is

Rv,w Θ

−→ (BvB−∩B−wB−)/B−

δ1

−→ ˙vN0

v,w

δ2

−→ −˚

Rv,w.

The map Θ : gB 7→ gθB−is induced by the involution g7→ gθon G; from [FZ99, Section 2], one

can see that Bθ=B−and that ˙wθis another lift of wso it is indeed an isomorphism. The maps

(δ1)−1and δ2are the natural projections from ˙vN0

v,w to G/B−and B−\Grespectively; these are

isomorphisms using the appropriate analogue of (1). The composition δ=δ2◦δ1is called the left

chiral map in [GL22, Deﬁnition 2.2].

2.2. Background on wiring diagrams and chamber minors. Before describing Ingermanson’s

and Leclerc’s seeds, we need some combinatorial background.

Given w∈Sn, a reduced expression for wis an expression w=sh1. . . sh`where `is as small as

possible. The number `is the length of w, denoted `(w). We use the notation

w(i):= sh1. . . shi−1and w(i):= sh`. . . shi=w−1w(i)

for preﬁxes of wand preﬁxes of w−1, setting w(1) =e.

As a shorthand, we write v≤wto indicate a pair of permutations v≤wand a choice of reduced

expression wfor w.

Deﬁnition 2.3. Let v≤w=sh1. . . sh`. A subexpression for vin wis an expression for vof the

form v=sv

h1. . . sv

h`where sv

hi∈ {e, shi}. As for w, we deﬁne

v(i):= sv

h1. . . sv

hi−1and v(i):= sv

h`. . . sv

hi.

The indices iwhere sv

hi6=eis the support of the subexpression. The subexpression is reduced if

the support has size `(v). The positive distinguished subexpression (PDS) for vin wis the reduced

subexpression whose support is lexicographically largest. If wis ﬁxed, we denote the PDS for vby

We denote the support of the PDS by J+

v, and call these the hollow crossings of w. The

complement of the support is J•

v; we call these the solid crossings of w. Note that |J•

v|=`(w)−

`(v) = dim ˚

Rv,w.

Example 2.4. Let w=s1s2s1s3s2s1and let v= 3214. Reduced subexpressions for vin winclude

ees1es2s1es2s1es2e s1s2eees1s1s2s1eee.

The ﬁrst subexpression has support {3,5,6}and is the PDS for vin w. So the hollow crossings

are J+

v={3,5,6}and the solid crossings are J•

v={1,2,4}.

Remark 2.5. Alternatively, the PDS for vcan be deﬁned using a greedy procedure, moving from

right to left. Set v(`+1) =v. If v(i+1) is already determined, then v(i)is equal to either v(i+1) or

v(i+1)shi, whichever is smaller. In the ﬁrst case, sv

hi=e; in the second, sv

hi=shi.

Remark 2.6. The notion of positive distinguished subexpressions (and more generally, distin-

guished subexpressions) is due to Deodhar [Deo85]. Our notation for the support and complement

of the support is inspired by [MR04], as is the terminology “solid” and “hollow” crossing. The +

in the superscript of J+

vis to indicate that J+

vrecords where the length of v(i)increases. The •in

the superscript of J•

vis to remind the reader that these are the solid crossings.

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LECLERC'SCONJECTUREONACLUSTERSTRUCTUREFORTYPEARICHARDSONVARIETIESKHRYSTYNASERHIYENKOANDMELISSASHERMAN-BENNETTAbstract.Leclerc[Lec16]constructedaconjecturalclusterstructureonRichardsonvarietiesinsimplylacedtypesusingclustercategories.WeshowthatintypeA,hisconjecturalclusterstructureisinfactaclusterstr...

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LECLERCS CONJECTURE ON A CLUSTER STRUCTURE FOR TYPE A RICHARDSON VARIETIES KHRYSTYNA SERHIYENKO AND MELISSA SHERMAN-BENNETT

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