Canonical basis linearity regions arising from quiver representations

a

asis of(2001)of the

nd thearising

sighesteight

Journal of Algebra 270 (2003) 696–727

www.elsevier.com/locate/jalgebr

Canonical basis linearity regions arisingfrom quiver representations

Robert Marsha,∗ and Markus Reinekeb

a Department of Mathematics and Computer Science, University of Leicester, University Road,Leicester LE1 7RH, UK

b Bergische Universität Wuppertal, Gaußstraße 20, D-42097 Wuppertal, Germany

Received 29 October 2002

Communicated by Peter Littelmann

Abstract

In this paper we show that there is a link between the combinatorics of the canonical ba quantized enveloping algebra and the monomial bases of the second author [Math. Z. 237639] arising from representations of quivers. We prove that some reparametrization functionscanonical basis, arising from the link between Lusztig’s approach to the canonical basis astring parametrization of the canonical basis, are given on a large cone by linear functionsfrom these monomial bases for a quantized enveloping algebra. 2003 Elsevier Inc. All rights reserved.

Keywords:Quantum group; Lie algebra; Canonical basis; Parametrization functions; Monomial basis;Representations of quivers; Degenerations; Piecewise-linear functions

1. Introduction

Let U = Uq(g) be the quantum group associated to a semisimple Lie algebrag ofrank n. The negative partU− of U has a canonical basisB with favourable propertie(see Kashiwara [8] and Lusztig [14, Section 14.4.6]). For example, via action on hweight vectors it gives rise to bases for all the finite-dimensional irreducible highest wU -modules.

* Corresponding author.E-mail addresses:[email protected] (R. Marsh), [email protected] (M. Reineke).

0021-8693/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2003.04.002

R. Marsh, M. Reineke / Journal of Algebra 270 (2003) 696–727 697

tworessionr

ion

s ofof thevef the

ftion

fr

nythat

-linear

steins oftermssztig

We consider the reparametrization functions of the canonical basis arising fromdifferent types of parametrization, each dependent on the choice of a reduced expfor the longest elementw0 in the Weyl groupW of g. Let s1, s2, . . . , sn be the Coxetegenerators ofW , and let i = (i1, i2, . . . , iN) be such an expression, so thatw0 =si1si2 · · · siN andN is the number of positive roots ofg. The first parametrization ofBarises from the correspondence betweenB and a basis of PBW type associated toi [14,Section 14.4.6]. We denote it byϕi :B→ NN . The second is the string parametrizat(see [5, Section 2], [9], and [19, Section 2]) which we denote byψi :B→ Xst(i), whereXst(i) ⊆ NN is known as thestring conecorresponding toi. If i andj are two reducedexpressions forw0 then the reparametrization functionSj

i is the composition

Sji = ϕjψ

−1i :Xst(i)→NN .

These reparametrization functions are useful in relating the two different typeparametrizations, and in [6] were shown to have applications to the descriptiontensor product multiplicities of simpleg-modules in an approach involving totally positivarieties. They were also shown to be closely connected with the description ocanonical basis in [7,18].

We also consider the monomial basis ofU− as constructed in [22]. LetΓ be a quiverwith underlying graph∆, where∆ is the Dynkin diagram ofg. Let i(Γ )= (i1, i2, . . . , iN )

be a reduced expression forw0 adapted toΓ in the sense of [11]. LetkΓ denote the pathalgebra ofΓ (wherek is an arbitrary field). Any other quiverQ of type∆ defines, in anatural way, a directed partition (see Definition 4.2) of the Auslander–Reiten quiver okΓ .Using the representation theory ofkΓ , the second author [22] has associated a funcDΓQ :NN →NN to the pair(Γ,Q), such that{

FD1(c)i1

FD2(c)i2

· · ·FDN(c)iN

: c ∈NN}

is a monomial basis ofU− (whereDΓQ(c)= (D1(c),D2(c), . . . ,DN(c))).

Let i(Q) be a reduced expression forw0 adapted toQ. Our main result is that iΓ is the linearly oriented quiver in typeAn (see Fig. 2), andQ is an arbitrary quiveof typeAn, thenDΓ

Q is invertible, and its inverse,EΓQ , coincides withSi(Γ )

i(Q) on a largecone (which we call thedegeneration cone) which contains the Lusztig coneLst(i(Q))

of [15] associated toi(Q). We conjecture that this result should also hold for achoice of quiverΓ (and indeed, for any simply-laced simple Lie algebra), and alsothe degeneration cone should be an entire region of linearity for the piecewisefunctionSi(Γ )

i(Q) .

A consequence of our result is thatSΓi(Q) is linear on the Lusztig coneLst(i(Q). Thiswas already known (for an alternating orientation onΓ ) in typesA1, A2, A3, andA4 (fortypeA4 see [7, 6.1]).

We remark that our approach to these functions differs from that of Berenand Zelevinsky [6] in that we are particularly concerned with linearity propertiethe reparametrization functions. Our result has an interesting interpretation inof the canonical basis, which we explain in Section 11. We also remark that Lu

698 R. Marsh, M. Reineke / Journal of Algebra 270 (2003) 696–727

cell ofn thesenta-gions[7]),

f quiv-

of theducedall thegtativen

f

ch

cones are used to describe regular functions on a reduced real double Bruhatthe corresponding algebraic group [26], they have links with primitive elements idual canonical basis (this can be seen using [5]) and therefore with the repretion theory of affine Hecke algebras [10], they are known to correspond to reof linearity of the Lusztig reparametrization functions and tight monomials (seeand they can be described using the homological algebra of representations oers [2].

The structure of the paper is as follows. In Section 2 we recall the parametrizationscanonical basis we will need, and in Section 3 we recall the connection between reexpressions for the longest word in the Weyl group and quivers. In Section 4 we recmonomial basis and corresponding linear functionsDΓ

Q defined in [22], as well as showinthat they are invertible. In the remaining sections, we construct the following commudiagram (whereΓ is the linearly oriented quiver in typeAn andk is a reduced expressioadapted toΓ ):

Xst(i(Q)

) Ski(Q)

NN

Cst(Q)

EΓQ

CPBW(Q)DΓQ

Lst(i(Q)

) EΓQ

LPBW(Q).DΓQ

The Lusztig conesLst(i(Q)) will be recalled in Section 5, and the conesCPBW(Q) willbe defined in Section 6. The degeneration coneCst(Q) is defined to beDΓ

Q(CPBW(Q)).In Section 7 we define the conesLPBW(Q), and in Section 8 we show thatLPBW(Q) iscontained inCPBW(Q). In Section 9 we show thatEΓ

Q(Lst(Q))= LPBW(Q) and finally, inSection 10 we show thatEΓ

Q andSki(Q) coincide onCst(Q).

2. Parametrizations of the canonical basis

Let g be the simple Lie algebra overC of typeAn. Let R denote the set of roots og andR+ ⊆ R the positive roots. LetU be the quantized enveloping algebra ofg. ThenU is a Q(v)-algebra generated by the elementsEi , Fi , Kµ, i ∈ {1,2, . . . , n}, µ ∈ P , theweight lattice ofg. LetU+ be the subalgebra generated by theEi andU− the subalgebragenerated by theFi .

Let W be the Weyl group ofg with Coxeter generatorss1, s2, . . . sn. It has a uniqueelementw0 of maximal length. For each reduced expressioni for w0 there are twoparametrizations of the canonical basisB for U−. The first arises from Lusztig’s approa


wara’s

the

cedse4]. Let

a-

oor the

to the canonical basis [14, Section 14.4.6], and the second arises from Kashiapproach [8].

2.1. Lusztig’s approach

There is anQ-algebra automorphism ofU which takes eachEi to Fi , Fi to Ei , Kµ

to K−µ, andv to v−1. We use this automorphism to transfer Lusztig’s definition ofcanonical basis in [11, Section 3] toU−.

Let Ti , i = 1,2, . . . , n, be the automorphism ofU as in [13, Section 1.3] given by

Ti(Ej )=−FjKj if i = j,

Ej if |i − j |> 1,−EiEj + v−1EjEi if |i − j | = 1,

Ti(Fj )=−K

−1j Ej if i = j,

Fj if |i − j |> 1,−FjFi + vFiFj if |i − j | = 1,

Ti(Kµ)=Kµ−〈µ,αi 〉hi , for µ ∈ P,

where theαi are the simple roots and thehi are the simple coroots ofg.For eachi, let ri be the automorphism ofU which fixesEj and Fj for j = i or

|i−j |> 1 and fixesKµ for all µ, and which takesEj to−Ej andFj to−Fj if |i−j | = 1.Let T ′′i,−1 = Tiri be the automorphism ofU as in [14, Section 37.1.3]. Letc ∈NN , whereN = &(w0), andi be a reduced expression forw0. Let

F ci := F

(c1)i1

T ′′i1,−1

(F(c2)i2

) · · ·T ′′i1,−1T′′i2,−1 · · ·T ′′iN−1,−1

(F(cN)iN

).

DefineBi = {F ci : c ∈NN }. ThenBi is the basis of PBW-type corresponding to the redu

expressioni. Note that, if the reduced expressioni is adapted to a quiver in the senof [11], then this basis can also be constructed using the Hall algebra approach of [2¯ be theQ-algebra automorphism fromU toU takingEi toEi , Fi to Fi , Kµ toK−µ, andv to v−1. Lusztig proves the following result in [11, Sections 2.3, 3.2].

Theorem 2.1 (Lusztig).TheZ[v]-spanL of Bi is independent ofi. Let π :L→ L/vLbe the natural projection. The imageπ(Bi) is also independent ofi; we denote it byB.The restriction ofπ to L ∩ L is an isomorphism ofZ-modulesπ1 :L ∩ L→ L/vL. AlsoB = π−1

1 (B) is a Q(v)-basis ofU−, which is thecanonical basisofU−.

Lusztig’s theorem provides us with a parametrization ofB , dependent oni. If b ∈ B,we writeφi(b)= c, wherec ∈NN satisfiesb ≡ F c

i modvL. Note thatφi is a bijection.For reduced expressionsj andj ′ for w0, Lusztig defines in [11, Section 2.6] a rep

rametrization functionRj ′j = φj ′φ

−1j :NN → NN . This function was shown by Lusztig t

be piecewise linear and its regions of linearity were shown to have significance fcanonical basis, in the sense that elementsb of the canonical basis withφj (b) in the same


thegions

t

cides

t on

triza-

as

region of linearity ofRj ′j often have similar form. For example, this can be seen from

explicit descriptions of the canonical basis of typeA2 andA3, as computed by Lusztiin [11] and by Xi in [25], respectively. More evidence for the importance of these regis their connection [5] with the multiplicativity properties of dual canonical bases.

2.2. The string parametrization

Let Ei and Fi be the Kashiwara operators onU− as defined in [8, Section 3.5]. LeA⊆Q(v) be the subring of elements regular atv = 0, and letL′ be theA-lattice spannedby arbitrary productsFj1Fj2 · · · Fjm ·1 inU−. We denote the set of all such elements byS.The following results were proved by Kashiwara in [8].

Theorem 2.2 (Kashiwara).

(i) Let π ′ :L′ → L′/vL′ be the natural projection, and letB ′ = π ′(S). ThenB ′ is aQ-basis ofL′/vL′ (the crystal basis).

(ii) Furthermore,Ei and Fi each preserveL′ and thus act onL′/vL′ . They satisfyEi(B

′) ⊆ B ′ ∪ {0} and Fi(B′) ⊆ B ′. Also for b, b′ ∈ B ′ we haveFib = b′, if and

only if Eib′ = b.

(iii) For eachb ∈B ′, there is a unique elementb ∈ L′ ∩L′ such thatπ ′(b)= b. The set ofelements{b: b ∈B ′} forms a basis ofU−, theglobal crystal basisofU−.

It was shown by Lusztig [12, 2.3] that the global crystal basis of Kashiwara coinwith the canonical basis ofU−.

There is a parametrization ofB arising from Kashiwara’s approach, again dependena reduced expressioni for w0. Let i = (i1, i2, . . . , iN) andb ∈B . Let a1 be maximal suchthatEa1

i1b �≡ 0 modvL′; let a2 be maximal such thatEa2

i2Ea1i1b �≡ 0 modvL′, and so on, so

thataN is maximal such that

EaNiNEaN−1iN−1· · · Ea2

i2Ea1i1b �≡ 0 modvL′.

Let a = (a1, a2, . . . , aN). We write ψi(b) = a. This is the crystal string ofb (see [5,Section 2], [9], and [19, Section 2]). It is known thatψi(b) uniquely determinesb ∈ B

(see [19, Section 2.5]). We haveb ≡ Fa1i1Fa2i2· · · F aN

iN· 1 modvL′. The image ofψi is

a cone which appears in [5], known as thestring conecorresponding toi.We next consider a function which links the string parametrization with the parame

tion arising from Lusztig’s approach. For reduced expressionsi andj for w0, consider themaps

Xst(i)ψ−1

i

Bφj

NN .

We defineSji = φjψ

−1i :Xst(i)→ NN , a reparametrization function. This function h

appeared in, for example, [6].


the

enotedch

s forns.f

e

ng

ssion

.square

beredthe

or

,

r the

tein–

Remark on notation. We shall use the following notation convention. A coneC which isto be regarded as a subset ofXst, i.e., to be thought of as a set of strings, will be givensubscript “st” to denote this, thusCst. A coneC which is to be regarded as a subset ofNN ,and is to be regarded as a set of PBW parameters for the canonical basis, will be dwith the subscript “PBW,” thusCPBW. In each case, it will be clear from the context whireduced expression forw0 is being used.

3. Reduced expressions compatible with quivers

In this section we shall recall from [3] an explicit description of reduced expressionw0 compatible with quivers in typeAn, which we shall need in studying such expressioGiven two such reduced expressionsi and i′, we write i ∼ i′ if there is a sequence ocommutations (of the formsisj = sj si with |i − j |> 1) which, when applied toi, give i′.This is an equivalence relation on the set of reduced expressions forw0, and the equivalencclasses are called commutation classes.

Let Q = (Q0,Q1) be a quiver, i.e., a finite oriented graph with set of verticesQ0 andset of arrowsQ1. We assumeQ to be of typeAn, i.e., that the unoriented graph underlyiQ is the Dynkin diagram of typeAn. If Q is a quiver andi is a sink inQ (i.e., all thearrows incident withi have i as target), we denote bysi(Q) the quiver with all of thearrows incident withi reversed. Ifi is a reduced expression forw0, we say (followingLusztig [11]) thati is compatible withQ if i1 is a sink inQ, i2 is a sink insi1(Q), i3 is asink insi2si1(Q), . . ., iN is a sink insiN−1siN−2 · · · si1(Q). It is known that ifQ is any quiverof typeAn, then there is always at least one reduced expressioni compatible withQ, andthat the set of reduced expressions compatible withQ is the commutation class ofi.

Berenstein, Fomin, and Zelevinsky give a nice description of a reduced exprecompatible with a given quiverQ in type An in [3, Section 4.4.3]. Suppose thatQ issuch a quiver. Number the edges ofQ from 1 to n − 1, starting from the left-hand endBerenstein, Fomin, and Zelevinsky construct an arrangement as follows. Consider ain the plane, with horizontal and vertical sides. We will drawn+ 1 ‘pseudo-lines’ in thissquare. Putn+ 1 points onto the left-hand edge of the square, equally spaced, num1 to n+ 1 from top to bottom, so that 1 andn+ 1 are at the corners. Do the same forright-hand edge, but number the points from bottom to top. Lineh will join point h on theleft with pointh on the right. Forh= 1, n+ 1, Lineh will be a diagonal of the square. Fh ∈ [2, n], Lineh will be a union of two line segments of slopesπ/4 and−π/4. There aretherefore precisely two possibilities for Lineh. If edgeh − 1 in Q is oriented to the leftthe left segment has positive slope, while the right one has negative slope; if edgeh− 1 isoriented to the right, it goes the other way round.

Example. Berenstein, Fomin, and Zelevinsky give the following example. ConsidecaseA5, with Q given by the quiver with vertices 1, 2, 3, 4, 5 and arrows 1→ 2,2← 3, 3→ 4, and 4← 5. (We shall denote such a quiver byRLRL, where anR(respectivelyL) denotes an edge oriented to the right (respectively left).) The BerensFomin–Zelevinsky arrangement for this quiver is shown in Fig. 1.


square,st

e

sistingimple

inesnting,ch isin the

ction

gest

ve

Fig. 1. The Berenstein, Fomin, and Zelevinsky diagram for the quiverRLRL.

If i is a reduced expression forw0 in typeAn, then thechamber diagramfor i is givenby a set of pseudolines numbered from 1 ton. Two sets of points numbered 1 ton arearranged on the vertical lines of a square as in the above picture. Underneath thethe simple reflections ini are written from left to right. Theith pseudoline then linkthe point markedi on the left with the point markedi on the right, in such a way, thaimmediately above the simple reflectionij from i pseudolinesij and ij + 1 cross. See[3, 1.4] for details. Berenstein, Fomin, and Zelevinsky prove the following result.

Proposition 3.1 (Berenstein, Fomin, and Zelevinsky).A reduced expressioni for w0is compatible with the quiverQ if and only if its chamber diagram is isotopic to tharrangement defined above corresponding toQ.

Proof. See [3, Section 4.4.3].✷Since each simple reflection appearing in such a reduced expressioni corresponds to

a crossing of two pseudo-lines in the diagram, each pseudo-line in the diagram (conof a part of positive slope and a part of negative slope) gives rise to some of the sreflections appearing ini; each simple reflection appears twice in this way as two lcrossing correspond to a simple reflection. We can remove this duplication by coufor each line, only the simple reflections which arise during the part of the line whiof positive slope. In this way, each edge of the quiver, which corresponds to a linediagram, gives rise to some of the simple reflections in the reduced expressioni; we includealso the line from the bottom left of the diagram to the top right. Each simple reflearises for a unique edge of the quiver (or comes from the extra line).

Number the edges ofQ from 1 ton− 1, starting from the left. Suppose that the edl1, l2, . . . , la all point to the left, and that edgesr1, r2, . . . , rb all point to the right, and thaevery edge is one of these, wherel1 < l2 < · · · < la andr1 < r2 < · · · < rb. Form ∈ N,denote by(m↘ 1) the sequencem,m− 1, . . . ,2,1. Then it is easy to see, that the aboconstruction shows that the reduced expression


at

his ise

f

lt

om-

phsleetails.

echponding

i(Q)= (l1↘ 1)(l2↘ 1) · · · (la↘ 1)(n↘ 1)(n↘ n+ 1− rb)

× (n↘ n+ 1− rb−1) · · · (n↘ n+ 1− r1)

is compatible withQ; it follows that the reduced expressions compatible withQ areprecisely those commutation equivalent toi(Q).

4. Monomial bases arising from representations of quivers

Let Γ = (Γ0,Γ1) be a quiver. We assumeΓ to be of Dynkin type, which means ththe unoriented graph∆ underlyingΓ is a disjoint union of Dynkin diagrams of typeA,D, E. Let k be an arbitrary field. Then we can form the path algebrakΓ of Γ over k,which has the paths inΓ as ak-basis (including an ‘empty’ path for each vertexi ∈ Γ0),and multiplication of paths given by concatenation if possible, and zero otherwise. Ta finite dimensionalk-algebra sinceΓ , being of Dynkin type, has no oriented cycles. Wform the category modkΓ of finite-dimensional representations ofkΓ . The isoclasses osimple objectsSi in modkΓ correspond bijectively to the verticesi ∈ Γ0 of Γ . Let NΓ

be the free abelian semigroup spanned by elementsαi for i ∈ Γ0; it can be identified withthe positive root lattice. For a representationM ∈modkΓ , we denote bydi for i ∈ Γ0 theJordan–Hölder multiplicity of the simpleSi in M. This allows us to define a map dimfromthe set of isoclasses in modkΓ to NΓ by dim(M)=∑

i∈Γ0diαi . The fundamental resu

in the theory of Dynkin quivers is

Theorem 4.1 (Gabriel).The mapdim induces a bijection between isoclasses of indecposable objectsXα in modkΓ and the positive rootsα ∈ R+ ⊂NΓ of type∆.

The Auslander–Reiten quiver of the algebrakΓ is defined as the oriented grahaving the isoclasses of indecomposable representations ofkΓ as vertices, and arrowcorresponding to irreducible maps (i.e., morphisms in modkΓ between indecomposabobjects which cannot be factored into a composition of non-split maps); see [1] for dWe can construct the Auslander–Reiten quiver of our quiverΓ in the following way(see [1]). LetΓ op be the quiverΓ with the orientation of all arrows reversed. LetZΓ op

be the quiver with verticesZ × {1,2, . . . , n}. Whenever there is an arrowi→ j in Γ op,we draw one arrow(z, i)→ (z, j) and one arrow(z, j)→ (z + 1, i) for eachz ∈ Z.DefineA(Γ ) to be the full subquiver ofZΓ op consisting of all vertices(z, i) such that1 � z� (h+ai−bi)/2 where, for eachi ∈ {1,2, . . . , n}, ai (respectivelybi ), is the numberof arrows in the unoriented path inΓ from i toσ(i) that are directed towardsi (respectivelyσ(i)). Here,σ is the unique permutation of the vertices ofΓ such thatw0(αi)=−ασ(i),andh is the Coxeter number. ThenA(Γ ) is the Auslander–Reiten quiver ofΓ . (We willnot need the Auslander–Reiten translate here.)

Example. Let Γ be the linearly oriented quiver in typeA5 (see Fig. 2). Then thAuslander–Reiten quiver ofΓ is given in Fig. 3. The vertices in this diagram, whicorrespond to the isoclasses of indecomposable modules, are labelled by the correspositive roots (so, for example, 123 meansα1+ α2+ α3).


g

ath)d,

u-

nt

o

Fig. 2. Linearly oriented quiver of typeA5.

Fig. 3. The Auslander–Reiten quiver of the linearly oriented quiver of typeA5.

The following definition was first introduced in [22].

Definition 4.2. An ordered partitionR+ = I1 ∪ · · · ∪ Is of R+ into disjoint subsetsIk iscalleddirectedif

(i) Ext1kQ(Xα,Xβ)= 0 for all α,β in the same partIk ,

(ii) Ext1kQ(Xα,Xβ)= 0 and HomkQ(Xβ,Xα)= 0 if α ∈ Ik , β ∈ Il , where 1� k < l � s.

The existence of (several!) directed partitions of a given quiverΓ can be seen usinAuslander–Reiten theory (see [1]): if HomkQ(U,V ) �= 0 (respectively Ext1kQ(V,U) �= 0)for indecomposablesU,V ∈modkΓ , then there exists a path (respectively a proper pfrom [U ] to [V ] in the Auslander–Reiten quiver ofkΓ . But since this graph is directewe can enumerate the isoclasses of indecomposables in modkΓ as [U1], . . . , [Uν], suchthat HomkQ(Up,Uq)= 0 for p > q , and Ext1kQ(Up,Uq)= 0 for p � q . Define rootsαp

by [Up] = [Xαp ]. By definition, the partitionR+ = {α1} ∪ · · · ∪ {αν} is directed. All otherdirected partitions can be constructed by coarsening such a partition.

Fix a directed partitionR+ = I1 ∪ · · · ∪ Is from now on. We will associate to it a seqencei = (i1, i2, . . . , it ), as well as a functionD from the set of isoclasses in modkΓ to Nt .Enumerate the verticesΓ0 of Γ asΓ0= {1 . . .n} such that the existence of an arrowi→ j

in Γ impliesi < j . Denote the multiplicity of the simple rootαs in a rootα by [α : αs ]. Inparticular, writes ∈ α to indicate that the simple rootαs appears with non-zero coefficiein α, that is,[α : αs ] �= 0. For eachp = 1, . . . , s, write the subset ofΓ0 consisting of allverticesi such thati ∈ α for some rootα ∈ Ip as{ip1 , . . . , iptp }, increasing with respect tthe above defined ordering onΓ0. Then the sequencei is defined as

i = i11 . . . i1t i

21 . . . i

2t . . . i

s . . . ist .
1 2 1 s


d,

e

senonicalmely,ts have

lledpe

.,to this

r isine are

We remark that, in general, the length of the sequencei is notN (but in the cases we neewe shall see that the length isN ). Given an isoclass[M] in modkΓ , we can writeM asM =⊕

α∈R+ Xcαα , using Gabriel’s theorem and Krull–Schmidt. LetD(M) be the tuple

a = (a1

1, . . . a1t1, a2

1, . . . a2t2, . . . , as1, . . . , a

sts

),

given by

ap

j =∑

α∈Ip, ipj ∈α

[α : αipj

]cα. (1)

This defines a functionD from the set of isoclasses in modkΓ to Nt , which is obviouslyadditive, i.e.,D(M ⊕ N) = D(M) +D(N). Identifying the set of isoclasses in modkΓwith NR+ via [⊕α X

cαα ] �→∑

α cαα, we thus get a linear functionD :NR+ → Nt . Themain result of this paper is that, for suitable choices ofΓ and the directed partition, thfunctionD coincides with one of the reparametrization functions(S

ji )−1 on an explicitly

described region.The original use of the functionD lies in the following theorem (see [22]).

Theorem 4.3 (Reineke).Writing i = (i1, i2, . . . , it ) andD = (D1, . . . ,Dt ), the set{FD1(c)i1

. . .FDt (c)it

∈U−: c ∈NR+}

is a basis forU−.

Moreover, these monomial bases forU− have good properties with respect to bachange: the base change coefficients to a PBW basis (respectively to the cabasis) form upper unitriangular matrices with respect to a certain ordering (narelated to the degeneration ordering on quiver representations), and these coefficienrepresentation theoretic (respectively geometric) interpretations (see [22,23]).

The functionD is not invertible in general, but it is for special directed partitions, caregular in [22]. We now introduce a certain class of directed partitions for quivers of tyA,and compute the inverse ofD in these special cases.

We know from Section 3 that the reduced expression

k = (1,2,1,3,2,1, . . . , n, n− 1, . . . ,1)

is adapted to the linearly oriented quiverΓk , with arrows pointing to the left (see, e.gFig. 2). This is one of the most regular reduced expressions. We shall now specialisecase.

Remark 4.4. We conjecture that our results hold, with similar proofs, if this quivereplaced by an arbitrary Dynkin quiver of typeAn, but the Kashiwara operators arethis general case more difficult to describe combinatorially. In the special case w


on ofametri-d

zed,n

o on,

ected

ofg

c11 c22 c33 c44 c55

c12 c23 c34 c45

c13 c24 c35

c14 c25

c15

Fig. 4. Array of elements ofN15.

considering, it is possible to handle this combinatorics. See [20] for a descriptithe combinatorics of Kashiwara operators on canonical basis elements using parzations of the canonical basis arising from PBW-bases ofU− corresponding to reduceexpressions compatible with arbitrary quivers.

A reduced expressioni for w0 defines an ordering on the setΦ+ of positive roots ofthe root system associated toW . We writeαj = si1si2 · · · sij−1(αij ) for j = 1,2, . . . ,N .ThenΦ+ = {α1, α2, . . .αN }. Forc = (c1, c2, . . . , cN ) ∈ ZN , write cαj = cj . If α = αij :=αi + αi+1+ · · · + αj with i < j , we also writecij for cαij .

Letα1, α2, . . . , αN be the ordering induced onΦ+ byk. We can write an element ofNN

asc = (cij ), where 1� i � j � n, using the above. The canonical basis is parametrivia the Lusztig parametrizationφk :B → NN . We can writec as an array based othe Auslander–Reiten quiver forΓk ; see Fig. 4. We writecij in place of the modulecorresponding to the positive rootαi + αi+1 + · · ·αj , with c11, c22, . . . , cnn on the firstrow, c12, c23, . . . , cn−1,n along the second, interspersing the first row elements, and suntil c1,n on the last row. For example, whenn = 5, see Fig. 4. Suppose thatQ is anyquiver of typeAn. We can define a directed partition ofA(Γk) in the following way. Fixz ∈ Z, and setz1 = z. Let v1 := (z1,1) be a vertex ofZΓk . For i = 2, . . . , n, define avertexvi = (zi , i) of ZΓk inductively, as follows. Ifi − 1→ i is an arrow inQ, then letvi be the head of the unique arrow with sourcevi−1. If i − 1← i is an arrow inQ, thenlet vi be the source of the unique arrow with headvi−1. Let Sz = (v1, v2, . . . , vn)⊆ ZΓk .Then it is clear thatZΓk =

⋃z∈Z Sz. It follows thatA(Γk)=

⋃z∈Z(Sz ∩A(Γk)); note that

this decomposition must be finite. It is clear from the construction, that this is a dirpartition ofA(Γk). LetTz = Sz∩A(Γk). We call eachSz asliceof ZΓk . If Tz is non-empty,we call it a slice ofA(Γk) (note that for this to happen, we must havez� 1).

Example. Consider the caseA5, with Q = RLRL. The corresponding decompositionA(Γk) into slices is given in Fig. 5. Each vertex ofA(Γk) is denoted by a number, indicatinthe number of the slice it lies in.

1 2 3 4 5

1 2 3 4

2 3 4

2 3

3

Fig. 5. Slice structure ofA(Γk ) corresponding to the quiverRLRL.


tedn

that

ted

rite

Let Q be an arbitrary quiver (of typeAn). Note that we can identify(i, j), for1 � i � j � n, with the indecomposable module with dimension vectorαi + · · · + αj ,so each such pair lies in a corresponding sliceTz. If v = (z, a) ∈ A(Γk), let i(v), j (v) besuch that the dimension vector of the module at vertexv is αi(v) + · · · + αj(v).

Recall the functionD :ZR+ → Zt associated in Section 4 to a quiver and a direcpartition. Consider now our special case whereΓ = Γk , and where the directed partitiois associated to an arbitrary quiverQ of typeA as above. In this case, it is easy to seethe sequencei is given by

i(Q)= (l1↘ 1)(l2↘ 1) · · · (la↘ 1)(n↘ 1)(n↘ n+ 1− rb)

· (n↘ n+ 1− rb−1) · · · (n↘ n+ 1− r1),

a reduced expression for the longest word in the Weyl group compatible withQ. Here,each of the bracketed parts ofi(Q) arises from a part of the directed partition associatoQ.

In this special case, we can easily prove the invertibility ofDΓkQ .

Lemma 4.5. Let DΓkQ be the function associated to the directed partition ofA(Γk)

corresponding to an arbitrary quiverQ of typeA. Then the functionDΓkQ is an invertible

map fromNN to NN with inverse functionEΓkQ = (D

ΓkQ )−1. Moreover, the componentscα

of c = (cα)α = EΓkQ (a) for somea ∈ NN and α ∈ R+ are of the formcα = ak − al or

cα = ak.

Proof. Recall from Section 4 that we can write

i = (i11 . . . i

1t1. . . is1 . . . i

sts

)and thatDΓk

Q (c) is the tuplea = (apj ) given by

apj =

∑α∈Ip, ipj ∈α

cα.

In our case, it is easily seen from the definition of the directed partition that we can w

Ip ={αp

1 , . . . , αp

t ′p},

such that the rootαpu is of lengthu, andαpu = αp

u−1 + αju for some simple rootαju (weformally setαp0 = 0). It follows thatαpu = αj1 + · · · + αju , which in particular impliest ′p = tp . We also see that {

ip, . . . , i

pt

}= {j1, . . . , jtp

}.
1 p


y

a,

asisd

gs

l]

nfiniterlyingReitento the

It follows that DΓkQ :NN → NN , as claimed. Denote byσ the permutation defined b

jσ(u) = ipu . Then we can computeapu as

apu =

∑α∈Ip, ipj ∈α

cα = cαpσu + cαpσu+1+ · · · + cαptp

.

It follows that

cαpu ={ap

σ−1u− a

p

σ−1(u+1)if u �= tp,

ap

σ−1uotherwise.

This proves the claimed properties ofDΓkQ . ✷

Remark 4.6. It follows from the proof of Lemma 4.5 that, in the situation of the lemmwe have thatDΓk

Q (c) is the tuple(aj )j=1,2,...,N such that if the rootαj lies in Ip , then

aj =∑

α∈Ip, ij∈αcα.

5. The Lusztig cones

Lusztig [15] introduced certain regions which, in low rank, give rise to canonical belements of a particularly simple form. TheLusztig conecorresponding to a reduceexpressioni for w0 is defined to be the set of pointsa ∈ NN satisfying the followinginequalities.

For every pairs, s′ ∈ [1,N] with s < s′, is = is ′ = i, andip �= i whenevers < p < s′,we have (∑

p

ap

)− as − as ′ � 0, (2)

where the sum is over allp with s < p < s′ such thatip is joined toi by an edge in theDynkin diagram. We shall denote this cone byLst(i) (as we shall regard it as a set of strinof B in directioni). We shall shorten the notationLst(i(Q)) toLst(Q).

It was shown by Lusztig [15] that, in typeAn, if a ∈ Lst(i) then the monomiaF(a1)i1

F(a2)i2· · ·F (aN)

iNlies in the canonical basisB, providedn= 1,2,3. The first author [17

showed that this remains true ifn = 4, but it is false forn � 5 by [17,21]. The Lusztigcones have been studied in the papers [7,16,18] in typeA for every reduced expressioi for the longest word, and have also been studied by Bedard in [2] for arbitrary(simply-laced) type for reduced expressions compatible with a quiver whose undegraph is the Dynkin diagram. Bedard describes these vectors using the Auslander–quiver of the quiver and homological algebra, showing they are closely connectedrepresentation theory of the quiver.


s

e

,ts. For

te

s

eacht

6. The degeneration cones

We define the coneCPBW(Q)⊆NN corresponding to a quiverQ to be the set of pointc= (cij ) satisfying the inequalities (C1) and (C2) below. We define thedegeneration cone

corresponding toQ to be the coneCst(Q)=DΓkQ (CPBW(Q)).

Define acomponentof Q to be a maximal full subgraphX of Q subject to thecondition that all of the arrows ofX point in the same direction. CallX a left (respectivelyright) component ofQ if its arrows all point to the left (respectively right). IfX is acomponent ofQ (left or right), letSz(X) (respectivelyTz(X)) denote the part of the slicSz (respectivelyTz) corresponding toX.

Example. Consider the example withQ=RLRL given above. ThenQ has 4 componentseach containing one edge. Two are right components, and two are left componeneach componentX, we indicate the subsets of slices,Tz(X), in A(Γk); see Fig. 6. Ineach case, the numbersz denote elements of the setsTz(X), and the empty circles denoelements not in any subsetTz(X).

Firstly, let X be a left component ofQ, and supposeTz, Tz+1 are consecutive slicein A(Γk) such thatSz(X) = Tz(X) and Sz+1(X) = Tz+1(X) (i.e., the subsetsSz(X)andSz+1(X) of ZΓk are contained entirely insideA(Γk)). We can orderTz(X) linearlyby the second entry of the vertices appearing in it (recall from Section 4 thatvertex is regarded as a pair(r, i), wherer ∈ N and i is a vertex ofΓk , i.e., an elemenof {1,2, . . . , n}). Write Tz(X) = {x1, x2, . . . xk}. Similarly, we can writeTz+1(X) =

Fig. 6. The subsetsTz(X) of A(Γk) corresponding to components of the quiverRLRL.


sring

s

) and

of the

nit;

f

sn it;

{y1, y2, . . . , yk}. For t = 1, . . . , k, let it = i(xt ), and letjt = j (xt ). Note thati(yt)= it + 1andj (yt )= jt + 1. Then inequalities (C1) are

k∑r=a

cir ,jr �k∑

r=acir+1,jr+1 for a = 1, . . . , k. (C1)

Secondly, letX be a right component ofQ, and supposeTz, Tz+1 are consecutive slicein A(Γk). We can orderTz(X) linearly by the second component of the vertices appeain it; write Tz(X) = {x1, x2, . . . xk}. Similarly, we can writeTz+1(X) = {y1, y2, . . . , yl},where k � l, sinceX is a right component. Fort = 1, . . . , k, let it = i(xt), and letjt = j (xt ). Note thati(yt )= it + 1 andj (yt )= jt + 1 for t = 1, . . . , l. Then inequalities(C2) are

cir ,jr � cir+1,jr+1 for r = 1, . . . , l − 1. (C2)

The coneCPBW(Q) is defined to be the set of(cij ) ∈NN satisfying all of the inequalitie(C1) and (C2).

Example. In our running example (see the start of this section), the inequalities (C1(C2) are given as follows:

Component 1:c11 � c22 � c33 � c44 � c55.Component 2:c13 � c24 � c35 andc13+ c23 � c24+ c34 � c35+ c45.Components 3 and 4 give rise to no inequalities in this example.

7. The PBW-version of the Lusztig cones

We now define a coneLPBW(Q). Eventually, we shall see thatSki(Q)(Lst(Q)) =

LPBW(Q) (see Theorem 10.10), so this cone can be regarded as a PBW-versionLusztig coneLst(Q).

Firstly, letX be a left component ofQ, and supposeTz, Tz+1 are consecutive slices iA(Γk). We can orderTz(X) linearly by the second entry of the vertices appearing inwrite Tz(X) = {x1, x2, . . . xk}. Similarly, we can writeTz+1(X) = {y1, y2, . . . , yl}, wherek � l (sinceX is a left component). Fort = 1, . . . , k, let it = i(xt), and letjt = j (xt ).Note thati(yt )= it + 1 andj (yt )= jt + 1 for 1� t � k. Then the defining inequalities oLPBW(Q) are

(L1) If Sz(X)= Tz(X) andSz+1(X)= Tz+1(X) then∑k

r=1 cir ,jr �∑k

r=1 cir+1,jr+1, and(L2) cir ,jr � cir+1,jr+1 for r = 2, . . . , k − 1.

Secondly, letX be a right component ofQ, and supposeTz, Tz+1 are consecutive slicein A(Γk). We can orderTz(X) linearly by the second entry of the vertices appearing iwrite Tz(X)= (x1, x2, . . . , xk). Similarly, we can writeTz+1(X)= (y1, y2, . . . , yl), wherek � l (sinceX is a right component). Fort = 1, . . . , l, let it = i(xt ), and letjt = j (xt ).


f

fore,

Note thati(yt)= it + 1 andj (yt )= jt + 1 for 1� t � l. Then the defining inequalities oLPBW(Q) are

(L3) If Sz(X)= Tz(X) andSz+1(X)= Tz+1(X) then∑k

r=1 cir ,jr �∑k

r=1 cir+1,jr+1, and(L4) cir ,jr � cir+1,jr+1 for r = 2, . . . , l − 1.

Finally, if X is a left component and is the leftmost component ofQ, then we have

(L5) ci1,j1 � ci1+1,j1+1,

and ifX is a right component and is the leftmost component ofQ, then we have

(L6) ci1,j1 � ci1+1,j1+1.

8. The relationship between the Lusztig cones and the degeneration cones in thePBW-parametrization

We show in this section, that, for an arbitrary quiverQ of typeAn, we haveLPBW(Q)⊆CPBW(Q). Firstly, we show that certain inequalities hold inLPBW.

Lemma 8.1. Supposec ∈ LPBW(Q). Then the following inequalities hold:If X is a left component ofQ, then(with the notation above),

(I) ci1,j1 � ci1+1,j1+1, and(II) cik,jk � cik+1,jk+1.

If X is a right component, then

(III) ci1,j1 � ci1+1,j1+1, and(IV) cik,jk � cik+1,jk+1.

The inequalities(III) and(IV) only hold if they make sense—i.e., ifk = l (which is the casewhenSz(X)= Tz(X) andSz+1(X)= Tz+1(X)).

Proof. Let c ∈ LPBW(Q). Suppose first that the leftmost componentX1 of Q is a leftcomponent. The inequality (I) forX1 is the defining inequality (L5) ofLPBW(Q). SupposethatSz(X)= Tz(X) andSz+1(X)= Tz+1(X). Then we have, from (L1) and (L2), that

(a)∑k

r=1 cir ,jr �∑k

r=1 cir+1,jr+1, and(b) cir ,jr � cir+1,jr+1 for r = 2, . . . , k − 1.

Since (I) and (b) hold, if (II) were false, (a) would be false—a contradiction. There(II) holds forX1.


tat

nf

(II).

s

,

t

10.10)

A similar argument shows that the inequalities (III) and (IV) hold forX1 in the casewhenX1 is a right component.

Suppose we are still in the case whereX1 is a left component. LetX2 be the componenimmediately to the right ofX1 (if it exists); this must be a right component. Note ththe inequality (II) forX1 is inequality (III) forX2. We can then argue as above forX2 todeduce (IV) forX2.

We can argue similarly in the case whereX1 is a right component. In this way, we cadeduce the relevant inequalities for all components ofQ, by induction on the number othe component, starting from the left, and we are done.✷Proposition 8.2. LetQ be an arbitrary quiver of typeAn, and letLPBW(Q) andCPBW(Q)

be the cones as above. ThenLPBW(Q)⊆ CPBW(Q).

Proof. Let c ∈ LPBW(Q). Suppose thatX is any left component ofQ, and thatSz(X) =Tz(X) andSz+1(X)= Tz+1(X). The following inequalities hold:

(c)∑k

r=1 cir ,jr �∑k

r=1 cir+1,jr+1, and(d) cir ,jr � cir+1,jr+1 for r = 1, . . . , k − 1 andcik,jk � cik+1,jk+1.

Inequality (c) comes from (L1), and inequalities (d) come from (L2) and Lemma 8.1We show that the inequalities (C1) all hold. We already have the inequalitycik,jk �cik+1,jk+1, from (d). If we also hadcik−1,jk−1 + cik,jk � cik−1+1,jk−1+1 + cik+1,jk+1, thenthis, together with the inequalitiescir ,jr � cir+1,jr+1 for r = 1, . . . , k − 2 would give∑k

r=1 cir ,jr �∑k

r=1 cir+1,jr+1, contradicting (c). Similarly, if we had∑k

r=a cir ,jr �∑kr=a cir+1,jr+1, for some 1� a � k − 2, the inequalities in (d) would give u∑kr=1 cir ,jr �

∑kr=a cir+1,jr+1, contradicting (c). Hence

∑kr=a cir ,jr �

∑kr=a cir+1,jr+1,

for any 1� a � k − 1, and we see that the defining inequalities (C1) ofCPBW(Q) aresatisfied.

Suppose next thatX is any right component ofQ. Then the following inequalities holdusing (L4) and Lemma 8.1(III):

(e) cir ,jr � cir+1,jr+1 for r = 1, . . . , l − 1.

In this case, (e) contains all of the defining inequalities (C2) ofCPBW(Q). We thus see thaLPBW(Q)⊆ CPBW(Q). ✷

9. The image of the Lusztig cone Lst(Q) under EΓk

Q is LPBW(Q)

We show in this section that we haveEΓkQ (Lst(Q))= LPBW(Q), i.e., thatEΓk

Q takes theLusztig cone to the cone defined in Section 7. We shall see in Section 10 (Theoremthat Sk

i(Q)(Lst(Q)) = LPBW(Q), since we shall see thatSki(Q) andEΓk

Q are identical onLst(Q).


2)ing

ingays

finingtion

n

es

We suppose thata = (a1, a2, . . . , aN) ∈ Lst(Q), and letc = (cij ) = EΓkQ (a). We will

show thata ∈ Lst(Q) if and only if c ∈ LPBW(Q). We translate the linear inequalities (definingLst(Q) using the linear functionEΓk

Q , and show that they become the defininequalities ofLPBW(Q). We also show thatai � 0 for all i if and only if cij � 0 for all

1 � i � j � n. SinceEΓkQ is a linear function, it will follow thatEΓk

Q (Lst(Q))= LPBW(Q).Recall that

i(Q)= (l1↘ 1)(l2↘ 1) · · · (la↘ 1)(n↘ 1)(n↘ n+ 1− rb)

· (n↘ n+ 1− rb−1) · · · (n↘ n+ 1− r1)

is compatible withQ (see the end of Section 3).The inequalities definingLst(Q) arise from pairs of equal simple reflections occurr

in i(Q). Note that, for the firsta + 1 factors appearing in the above, each factor is alwcontained in the one immediately to the right, and that, for the lastb + 1 factors, eachfactor is always contained in the one immediately to the left. It is clear that the deinequalities forLst(Q) always arise from such pairs of factors. To make the notaclearer, let us definela+1= rb+1= n.

Let us consider such a pair,(lp ↘ 1)(lp+1↘ 1), and suppose that 1� s � lp , so soccurs both in(lp ↘ 1) and in (lp+1↘ 1). Supposes appears in positiont of i in thefirst factor; it then must appear in the second factor—suppose that this is in positiou inthe second factor. Let us first assume thats > 1. The defining inequality ofLst(Q) arisingfrom this pair ofs ’s is

at+1+ au−1 � at + au. (3)

We can rewrite this as

at − at+1 � au−1− au. (4)

SinceDΓkQ (c)= a, we have (by Remark 4.6)

at =∑

α∈Tn+1−ps∈α

cα, at+1=∑

α∈Tn+1−ps−1∈α

cα,

au−1=∑

α∈Tn−ps+1∈α

cα, au =∑

α∈Tn−ps∈α

cα.

Suppose first, thats � p. Thens − 1 ∈ α implies thats ∈ α for α ∈ Tn+1−p , ands ∈ αimplies thats+1∈ α for α ∈ Tn−p , ass+1� p+1 also. Thus, the inequality (4) becom∑

α∈Tn+1−ps∈α

cα �∑

α∈Tn−ps+1∈α

cα.

s−1/∈α s /∈α


t the

)

t

vetive

)

rwe

This is an inequality of type (L2) or (L3) forLPBW(Q), depending on whats is.Next suppose thats = 1. Then the defining inequality (4) is the same, except tha

termat+1 does not appear. So we haveat � au−1− au. This translates to

∑α∈Tn+1−p

1∈α

cα �∑

α∈Tn−p2∈α1/∈α

cα,

and we again get an inequality of type (L2) or (L3) forLPBW(Q).Suppose next, thats > p. Then s ∈ α implies thats − 1 ∈ α for α ∈ Tn+1−p , and

s + 1∈ α implies thats ∈ α for α ∈ Tn−p , ass + 1 � p+ 1 also. We rewrite inequality (4asat+1− at � au − au−1, and we have

∑α∈Tn+1−ps−1∈αs /∈α

cα �∑

α∈Tn−ps∈α

s+1/∈α

cα.

It is clear that this is an inequality of type (L1) or (L4) forLPBW(Q), depending on whas is. We do not get a boundary case to consider in this case.

For consecutive factors appearing after(n↘ 1), a similar argument shows that they gidefining inequalities forLPBW(Q). It is easy to see that, if all possible pairs of consecufactors are taken, we get precisely the defining inequalities forLPBW(Q).

Thus, the defining inequalities forLst(Q) correspond, underEΓkQ , to the defining

inequalities forLPBW(Q). It remains to check thatai � 0 for all i if and only if cij � 0 for

all 1� i � j � n (if EΓkQ (a)= c).

By the description of the coordinates of the functionDΓkQ = (E

ΓkQ )−1 (see Remark 4.6

as nonnegative combinations of the coordinate entries ofc, it is clear that if allcij � 0 thenall ai � 0. Now, suppose that allai � 0. Then we know by [16, 4.1] that, ifa is labelled(aα)α∈Φ+ according to the ordering on the positive roots induced byi(Q), then, wheneveα,β ∈ Φ+ andα � β (i.e., α = β plus a nonnegative combination of simple roots),haveaα � aβ . By the description of the functionEΓk

Q , we know that the entries oncij ,regarded as functions of the coordinates ofa, are all of the formaα−aβ whereα = β+γ ,for some simple rootγ . It follows thatcij � 0 for all 1� i � j � n.

We have proved:

Theorem 9.1. EΓkQ (Lst(Q))= LPBW(Q).

Example. We return to our running example, withQ = RLRL in typeA5. Let a ∈ NN .ThenEΓk

Q (a)= (c)= (cij ) wherec is given by the diagram in Fig. 7.We give in Table 1 the correspondence between the defining inequalities ofLst(Q) and

those ofLPBW(Q). Recall thati(Q)= (2,1,4,3,2,1,5,4,3,2,1,5,4,3,5).


erationa

ee

at

eving

entt

a2− a1 a5− a4 a9− a8 a13− a12 a15

a1 a4− a6 a8− a10 a12− a14

a6− a3 a10− a7 a14

a3 a7− a11

a11

Fig. 7. The functionSki(Q)

(a).

Table 1

Inequality ofLst(Q) Inequality ofLPBW(Q)

a2+ a4 � a1+ a5 c11 � c22a5 � a2+ a6 c22+ c23 � c11+ c12

a4+ a7 � a3+ a8 c23+ c13 � c34+ c24a5+ a8 � a4+ a9 c22 � c33a6+ a9 � a5+ a10 c33+ c34 � c22+ c23a10 � a6+ a11 c24+ c25 � c13+ c14a8 � a7+ a12 c34+ c24 � c45+ c35

a9+ a12 � a8+ a13 c33 � c44a10+ a13 � a9+ a14 c44+ c45 � c33+ c34a13 � a12+ a15 c44 � c55

10. Description of the function Ski(Q)

We now show that the functionSki(Q) coincides with the functionEΓk

Q = (DΓkQ )−1

arising from representations of quivers, as described in Section 4, on the degenconeCst(Q). We first of all note that each edge ofQ corresponds in a natural way toslice ofA(Γk); such a slice generates the factor ofi(Q) corresponding to this edge (sSection 3).

Suppose thata ∈Cst(Q)=DΓkQ (CPBW(Q)) (the degeneration cone). We will show th

a ∈ Xst(Q) (i.e., the string coneXst(i(Q))) and thatSki(Q)(a) = E

ΓkQ (a). It then follows

that, asEΓkQ andSk

i(Q) are bijective,(Ski(Q))

−1(c) = a = DΓkQ (c) for any c ∈ CPBW(Q).

This is the result we would like to prove, and this section is mainly devoted to achithis aim. In the following analysis,a shall denote an arbitrary element ofCst(Q), andc=E

ΓkQ (a).

In applyingSki(Q) to a = (aij ), we need to computec such thatF a1

i1· · · F aN

iN· 1≡ F c

k

modvL′, wherei = i(Q). Recall (see the end of Section 3) that

i(Q)= (l1↘ 1)(l2↘ 1) · · · (la↘ 1)(n↘ 1)(n↘ n+ 1− rb)

· (n↘ n+ 1− rb−1) · · · (n↘ n+ 1− r1).

Let m be an edge ofQ. Then in the Berenstein–Fomin–Zelevinsky arrangemcorresponding toQ (see, for example, Fig. 1), the line corresponding tom passes firs


e

ot

l

through the lines corresponding toR’s to the left ofm, starting from the right, then the linfrom top left to bottom right, followed by theL’s to the left ofEΓk

Q in Q, from left to right.This is how the reduced expressioni(Q) is built up. Thus, we apply the product

F (l1)F (l2) · · · F (la)F (n)F (rb)F (rb−1) · · · F (r1) (5)

to 1= F 0k where 0 denotes the zero vector, andF(m) is a monomial of Kashiwara

operators defined as follows. Givenm ∈ [1, n], let d be maximal so thatld < m and lete be maximal so thatre < m. Then we have thatd + e = m− 1, som− e − d = 1, andn− e− d = n−m+m− e− d = n+ 1−m. Then, if edgem is anL, we have

F (m)= Fare+1,m+1m F

are−1+1,m+1

m−1 · · · F ar1+1,m+1

m−e+1 Fa1,m+1m−e F

al1+1,m+1

m−e−1 Fal2+1,m+1

m−e−2 · · · F ald+1,m+1

m−e−d ,

and if edgem is anR, orm= n, we have

F (m)= Fare+1,m+1n F

are−1+1,m+1

n−1 · · · F ar1+1,m+1

n−e+1 Fa1,m+1n−e F

al1+1,m+1

n−e−1 Fal2+1,m+1

n−e−2 · · · F ald+1,m+1

n−e−d .

We can regard the vectorc ∈ NN as the vector(cα)α∈Φ+ (see Section 4.3). Each roα ∈Φ+ will lie in a slice Tz of Φ+. Givenz ∈N, let c(z) denote the vectorc but with cαset to zero for everyα lying in a sliceTz′ with z′ < z; thusc(n+ 1)= 0 andc(1)= c.

For i = 1,2, . . . , n, let Pi denote theith product appearing in (5). Thus, fori =1,2, . . . , a, Pi = F (li ), Pa+1 = F (n), and fori = a + 2, . . . , n, Pi = F (ra+2−i+b), andwe have

P1P2 · · · Pn = F (l1)F (l2) · · · F (la)F (n)F (rb)F (rb−1) · · · F (r1).

We will show that, forz = 1,2, . . . , n, Pz · F c(z+1)k ≡ F

c(z)k modvL′, from which it will

follow that

P1P2 · · · Pn · 1≡ F ck modvL′,

as required.We also need to show thata ∈Xst(Q). We will use the following definition: a monomia

actionF b1j1Fb2j2· · · F bt

jt· F x

k is said to satisfy (STRING) provided that

Eju Fbu+1ju+1

Fbu+2ju+2· · · F bt

jt· F x

k ≡ 0 modvL′,

for u = 1,2, . . . , t . We will also use the description of the action of theFi ’s on a PBWbasis modulovL as proved by the second author in [20]:

Proposition 10.1 (Reineke).Supposec= (cij ) ∈NN . For each1� i � j � n, define

fij =i∑ckj −

i−1∑ck,j−1.

k=1 k=1


n of

le

r

Let i0 be maximal so thatfi0j =maxi fij . ThenFj increasesci0j by 1, decreasesci0,j−1by 1 (unlessi0 = j , when this latter effect does not occur), and leaves the othercij ’sunchanged.

It is easy to see, using Theorem 2.2(ii), that this implies the following descriptiotheEi ’s.

Proposition 10.2. Supposec= (cij ) ∈NN . For each1 � i � j � n, define

fij =i∑

k=1

ckj −i−1∑k=1

ck,j−1.

Let i0 be minimal so thatfi0j =maxi fij . ThenEj decreasesci0j by 1, increasesci0,j−1by 1 (unlessi0 = j , when this latter effect does not occur), and leaves the othercij ’sunchanged, except, ifci0j = 0, then it acts as zero modulovL.

We will need the following technical lemmas, describing the action ofFi and Ei incertain circumstances.

Lemma 10.3. Fix 1 � i � j � n and s ∈ N. Suppose the following hold for a triang(ckl) ∈NN :

(a) For k = 1, . . . , i − 1, we have∑i

l=k+1 clj �∑i−1

l=k cl,j−1.

(b) For k = i + 1, . . . , j , we haves �∑k−1

l=i cl,j−1−∑kl=i+1 clj .

ThenF sj acts on(ckl) by increasingcij by s, decreasingci,j−1 by s, and leaving the othe

ckl ’s unchanged.

Proof. For t = 0, . . . , s − 1, define a new trianglectkl by

ctkl ={cij + t, k = i, l = j,

ci,j−1− t, k = i, l = j − 1,ckl, otherwise.

The lemma clearly holds if for allt = 0, . . . , i−1, the indexi0 of Proposition 10.1 equalsi.This translates into the following inequalities for thefkl ’s of Proposition 10.1:

fkj � fij for k = 1, . . . , i − 1, fkj < fij for k = i + 1, . . . , j.

Using the definitions of(ctkl) andfkl , this translates into the following conditions:

t �i−1∑

cl,j−1−i∑

clj for t = 0, . . . , s − 1, k = 1, . . . , i − 1,
l=k l=k+1


le

my

ma

ly

which is equivalent to condition (a), and

t <

k−1∑l=i

cl,j−1−k∑

l=i+1

clj for t = 0, . . . , s − 1, k = i + 1, . . . , j,

which is equivalent to condition (b).✷Lemma 10.4. Fix 1 � i � j � n and s ∈ N. Suppose the following hold for a triang(ckl) ∈NN :

(a) For k = 1, . . . , i, we haveckj = 0.

(b) For k = i + 1, . . . , j , we haves �∑k−1

l=i cl,j−1−∑kl=i+1 clj .

Then fors := s +∑i−1k=1 ck,j−1, we have

F sj (ckl)kl

=

0, l = j − 1, k < i,

ck,j−1, l = j, k < i,

ci,j−1− s, l = j − 1, k = i,

s, l = j, k = i,

ckl, otherwise.

Proof. We proceed by induction oni. If i = 1, the claimed statement follows directly froLemma 10.3. For arbitraryi, we seti ′ = i − 1, s′ = ci−1,j−1 and claim that we can applthe lemma—which we assume to be already true fori ′—with the same triangle(ckl), butwith (i, j, s) replaced by(i ′, j, s′). We thus have to show that the assumptions of the lemare satisfied.

Condition (a) is satisfied trivially. Fork = i+1, . . . , j , condition (b) gives∑k−1

l=i cl,j−1−∑kl=i+1 clj � s � 0, which (usingcij = 0 by condition (a)) means

k−1∑l=i′

cl,j−1−k∑

l=i′+1

clj � ci−1,j−1= s′.

For k = i ′ + 1= i, the desired condition iss′ � ci−1,j−1− cij , which clearly holds sincecij = 0. Thus, we can apply the lemma to(i ′, j, s′). Setting(c′kl)= F s ′

j (ckl), we find:

c′kl =

0, l = j − 1, k < i ′ck,j−1, l = j, k < i ′ci′,j−1− s, l = j − 1, k = i ′s, l = j, k = i ′ckl, otherwise

=

{0, l = j − 1, k < i,

ck,j−1, l = j, k < i,

ckl, otherwise.

Sinces′ =∑i−1k=1 ck,j−1, it thus remains to show thatFs

j acts on(c′kl) by decreasingci,j−1

by s, increasingcij = 0 bys, and leaving the rest of(c′ ) unchanged. To prove this, we on
kl


,e.

le

have to check the assumptions of Lemma 10.3 for(c′kl): for condition (a) of Lemma 10.3this is trivial sincec′k,j−1 = 0 for k < i; for condition (b) of Lemma 10.3, we just uscondition (b) of the present lemma. Applying Lemma 10.3, we see that we are done✷Lemma 10.5. Fix 1 � i � j � n and s ∈ N. Suppose the following hold for a triang(ckl) ∈NN :

(a) For k = 1, . . . , i, we haveckj = 0.(b) For k = i + 1, . . . , j , we have0 �

∑k−1l=i cl,j−1−∑k

l=i+1 clj .

Then we haveEj(ckl)= 0.

Proof. For each 1� p � j � n, define

fpj =p∑

k=1

ckj −p−1∑k=1

ck,j−1,

as in Proposition 10.2. By assumption (a),fpj = −∑p−1k=1 ck,j−1, for p = 1,2, . . . i, and,

using both assumptions,

fpj =p∑

k=i+1

ckj −p−1∑k=1

ck,j−1= fij +p∑

k=i+1

ckj −p−1∑k=i

ck,j−1 � fij

for p = i + 1, i + 2, . . . , j . Hence, ifp0 is minimal so thatfp0j = maxp fpj , we musthavep0 � i. It follows from (a) thatcp0j = 0, so we conclude thatEj acts as zeromodulovL. ✷

We start by considering the action of the monomialF (rp) (where 1� p � b+1—recallthatrb+1= n) on c(n+ 2− p). Recall that, forp = 1,2, . . . , b+ 1, we have

F (rp)= Farp−1+1,m+1n F

arp−2+1,m+1

n−1 · · · F ar1+1,m+1

n−p+2 Fa1,m+1n−p+1F

al1+1,m+1n−p F

al2+1,m+1

n−p−1 · · · Fald+1,m+1

n+1−p−d .

The initial part of the computation is reasonably easy:

Lemma 10.6. Suppose that1 � p � b. Letd ∈NN be the triangle given by

dn+1−p,n+1−p = a1,rp+1− al1+1,rp+1,

dn−p,n+1−p = al1+1,rp+1− al2+1,rp+1,

dn−p−1,n+1−p = al2+1,rp+1− al3+1,rp+1,

...


yse

at the

p, and

ht.

-dges

dn+2−p−d,n+1−p = ald−1+1,rp+1− ald+1,rp+1,

dn+3−p−d,n+1−p = ald+1,rp+1.

Then we have

Fa1,m+1n−p+1F

al1+1,m+1n−p F

al2+1,m+1

n−p−1 · · · Fald+1,m+1

n+1−p−d · F c(n+2−p)k ≡ F

c(n+2−p)+dk .

Furthermore, the action

Fa1,m+1n−p+1F

al1+1,m+1n−p F

al2+1,m+1

n−p−1 · · · Fald+1,m+1

n+1−p−d · F c(n+2−p)k

satisfies(STRING).

Proof. We first note that we havea1,r+1 � al1+1,r+1 � al2+1,r+1 � · · ·� alb+1,r+1. These

follow from the fact thata = DΓkQ (c), since we know that all the coordinates ofc are

nonnegative: we use Remark 4.6. Letl0= 0. We know that, fort = 0,1, . . .d − 1,

alt ,r+1=∑

α∈Tn+1−pn−p+1−t∈α

cα and alt+1,r+1=∑

α∈Tn+1−pn−p−t∈α

cα. (6)

Let t ∈ {0,1, . . . , d−1}. Sincen−p+1− t � n−p+1, it follows that ifαn−p−t appearsin a root of slicep, so doesαn−p+1−t . Thus, by Eqs. (6) , we havealt ,r+1 � alt+1,r+1 asrequired.

Let x = c(n + 2− p). It is easy to see, using Proposition 10.1, thatFalb+1,rp+1

n+1−rp setsxn+1−rp,n+1−rp to bealb+1,rp+1, and leaves the otherxij ’s unchanged. Similarly, it is easto see, using Proposition 10.2, thatEn+1−rp acts as zero onx. In both cases this is becauxk,n+1−rp = 0 for k = 1,2, . . .n+ 1− r, andxk,n−rp = 0 for k = 1,2, . . .n− rp , so thatall of thefij in Propositions 10.1 or 10.2 are zero.

Next, Falb−1+1,rp+1

n+2−rp setsxn+1−rp,n+1−rp = 0 and setsxn+1−rp,n+2−rp = alb+1,rp+1. It

also setsxn+2−rp,n+2−rp to bealb−1+1 − alb+1,rp+1. The rest of theFi ’s act in a similarway, until we finally get the vector described in the lemma. It is also easy to see thgiven monomial action satisfies (STRING) using Proposition 10.2.✷

The next part of the computation, which involves computingFarp−1+1,m+1n F

arp−2+1,m+1

n−1

· · · F ar1+1,m+1

n−p+2 · F c′(n−p+2)k (wherec′(n − p + 2) = c(n − p + 2) + d whered is as in

Lemma 10.6) is more involved. We need to consider what happens step-by-steto understand what is happening in detail along each sliceTz = Tn+1−p of A(Γk). LetTz = {βz1, βz2, . . . , βzkz}, where thekz roots inTz are listed according to increasing heigThusβz1= αn+1−p .

Given q such that 1� q � p − 1, we define a vectorc(p, q) as follows. (In or-der to simplify notation, we use the numbering arising from the edges ofQ orientedto the right, rather than the usual slice numbering.) Ifα belongs to a slice numbered n + 2 − p or greater (i.e., corresponding to one of the right-oriented e


t

.ig. 8

he

qualityside ofve

tion

numberedr1, r2, . . . , rp−1), then cα(p, q) = cα . We also set, fori = 1,2, . . . , rq − 1,cβzi (p, q) = ai,rp+1 − ai+1,rp+1, and setcβrq ,i (p, q) = arq+1,rp+1 − als+1,rp+1, wherelt is the number of the edge of the firstL to the right of edgerq . Suppose thaβzrq = αij . Then we also setci−1,j (p, q) = alt+1,rp+1 − alt+1+1,rp+1, ci−2,j (p, q) =alt+2+1,rp+1, . . . , ct+i+1−b,j (p, q) = alb−1+1,rp+1 − alb+1,rp+1, and ct+i−b,j (p, q) =alb+1,rp+1. We write c(p,0) for the vectorc(n + 2− p) + d appearing in Lemma 10.6The vectorc(p, q − 1) (which appears in the lemma below) can be visualised as in F(note that the case displayed is where edgerq − 1 is oriented to the right). We have

Lemma 10.7. Suppose that1 � p � b and that1 � q � p− 1. We have that

(a) Farq+1,rp+1

n+1−p+q · F c(p,q−1)k ≡ F

c(p,q)k modvL′, and that

(b) Earq+1,rp+1

n+1−p+q · F c(p,q−1)k ≡ 0 modvL′.

Proof. Suppose thatβz,rq−1 = αij . It is easy to check thati = n+ 1− p − (t − 1) andj = n+1−p+ q −1, wherelt is the number of the edge of the firstL to the right of edgerq − 1. Let s = arq+1,rp+1− alt+1,rp+1 (note that these notations were used forc(p, q) inthe above, rather thanc(p, q − 1)).

We apply Lemma 10.4 with the pairi, j + 1 ands as above. We need to check tassumptions of the lemma. It is clear from the definition ofc(p, q − 1) that ck,j+1(p,

q − 1)= 0 for k = 1,2, . . . , i. Suppose thati + 1� k � j + 1. We consider

k−1∑l=i

cl,j (p, q − 1)−k∑

l=i+1

cl,j+1(p, q − 1).

This is a sum of terms, each of which is the sum of elements on one side on an ine(C1) or (C2) (see Section 6), minus the corresponding sum of elements on the otherthe inequality. Since the vectorc ∈ CPBW(Q) satisfies inequalities (C1) and (C2), we hathat

k−1∑l=i

cl,j −k∑

l=i+1

cl,j+1 � 0 (7)

(indeed, the inequalities ofCPBW(Q) are designed to ensure this holds). By the definiof c(p, q − 1), it is clear that forl = i + 1, . . . , j , cl,j = cl,j (p, q − 1) and that forl = i + 1, . . . , k, cl,j+1 = cl,j+1(p, q − 1). Furthermore,cij (p, q − 1)= arq−1+1,rp+1 −alt+1,rp+1= s + (arq−1+1,rp+1− arq+1,rp+1)= s + cij . It follows from Eq. (7) that

k−1∑cl,j (p, q − 1)−

k∑cl,j+1(p, q − 1)� s (8)

l=i l=i+1


Fig. 8. The vectorc(p,q − 1).


and it

byfzero),

e

as required in Lemma 10.4. Thus, the conditions of the lemma are all satisfied,applies. Let

s = s +i−1∑k=1

ck,j = arq+1,rp+1− alt+1,rp+1+ alt+1,rp+1− alt+1+1,rp+1+ alt+1+1,rp+1

− alt+2+1,rp+1+ · · · + alb−1+1,rp+1− alb+1,rp+1+ alb+1,rp+1

= arq+1,rp+1.

Then it is easy to check that Lemma 10.4 tells us that

Farq+1,rp+1

n+1−p+q · F c(p,q−1)k ≡ F

c(p,q)k modvL′,

giving us (a) above. This can easily be visualised in Fig. 8; the value ofcij (p, q − 1) isdisplayed atA in the diagram, and this is replaced byarq−1+1,rp+1− arq+1,rp+1; the valueof ci,j+1(p, q − 1) (which is zero) is displayed atB in the diagram, and this is replaceds = arq+1,rp+1− alt+1,rp+1. Finally, the values on the diagonal below and to the left oA

are all moved one step down and to the right (to places where the value is currentlyforming the diagonal below and to the left ofB. The new values in the diagonal belowAare all zero.

In order to show (b), we apply Lemma 10.5 toc(p, q − 1), again with the pairi, j + 1.It is clear that the conditions of this lemma are satisfied byc(p, q−1), since they are samas those in Lemma 10.4 but withs taken to be zero. The lemma tells us that

Earq+1,rp+1

n+1−p+qc(p, q − 1)≡ 0 modvL′,

giving us (b) above. ✷We can now prove our main result.

Theorem 10.8. LetQ be an arbitrary quiver of typeAn, and suppose thatc ∈ CPBW(Q).Then (

Ski(Q)

)−1(c)=D

ΓkQ (c).

Proof. Let a ∈ Cst(Q) = DΓkQ (CPBW(Q)), and suppose 1� p � b. Lemma 10.6 tells us

that

Fa1,m+1n−p+1F

al1+1,m+1n−p F

al2+1,m+1

n−p−1 · · · Fald+1,m+1

n+1−p−d · F c(n+2−p)k ≡ F

c(p,0)k .

Lemma 10.7(a) tells us that, for 1� q � p− 1, we have

Farq+1,rp+1 · F c(p,q−1) ≡ F

c(p,q) modvL′.
n+1−p+q k k


r the

d

one

Repeated application of this second result (forq = 1,2, . . . , p− 1) tells us that

F (rp) · F c(n+2−p)k ≡ F

c(n+1−p)k modvL′, for p = 1,2, . . . , rb.

It follows that

F (n)F (rb)F (rb−1) · · · F (r1) · F c(n+1)k ≡ F

c(a+1)k modvL′.

A proof similar to the one given above can be used to show that

F (li) · F c(i+1)k ≡ F

c(i)k modvL′ for i = 1,2, . . . , a.

It then follows that

F (l1)F (l2) · · · F (la)F (n)F (rb)F (rb−1) · · · F (r1) · F c(n+1)k ≡ F

c(1)k ≡ F c

k ≡ FEΓkQ (a)

k

modvL′,

as required. It also follows from Lemmas 10.6 and 10.7 (and a similar proof fomonomialF (l1)F (l2) · · · F (la)) that the monomial action

F (l1)F (l2) · · · F (la)F (n)F (rb)F (rb−1) · · · F (r1) · F 0k ≡ F c

k modvL′

satisfies (STRING), from which it follows thata ∈Xst(Q).We thus have that, for alla ∈ Cst(Q) = D

ΓkQ (CPBW(Q)), Sk

i(Q)(a) = EΓkQ (a) and

a ∈Xst(Q). It follows that for allc ∈ CPBW(Q), (Ski(Q))

−1(c)=DΓkQ (c), as required. ✷

We conjecture that this theorem holds in the case whenk is replaced by any reduceexpression forw0 compatible with a quiver.

We have the following corollary.

Corollary 10.9. LetQ be an arbitrary quiver of typeAn. Then(Sk

i(Q)

)−1(CPBW(Q)

)= Cst(Q).

Proof. This follows from the theorem and the definition of the degeneration cCst(Q)=D

ΓkQ (CPBW(Q)). ✷

Finally, since (by Theorem 9.1)EΓkQ (Lst(Q)) = LPBW(Q) and sinceLPBW(Q) ⊆

CPBW(Q), we have, by Theorem 10.8, thatSki(Q) andEΓk

Q coincide onLst(Q). It followsthat:

Theorem 10.10. LetQ be an arbitrary quiver of typeAn. Then

Ski(Q)

(Lst(Q)

)= LPBW(Q).


22]

bel

Example. We return to our running example, withQ = RLRL in type A5. ByTheorem 10.8, fora such thatEΓk

Q (a) ∈ CPBW(Q), we haveSki(Q)

(a)= (c)= (cij ) wherecij is given in Fig. 7.

Using the description ofDΓkQ , we have, forc ∈LPBW(Q), that(Sk

i(Q))−1(c)= a, where

a = (c12, c11+ c12, c14, c23+ c13+ c14, c22+ c23+ c13+ c14, c13+ c14, c25+ c15,

c34+ c24+ c25+ c15, c33+ c34+ c24+ c25+ c15, c24+ c25+ c15, c15,

c45+ c35, c44+ c45+ c35, c35, c55).

11. Connection with the canonical basis

Recall thatα1, α2, . . . , αN is the ordering onR+ induced byk. Given akΓk-moduleM = X

c1α1 ⊕ X

c2α2 ⊕ · · · ⊕ X

cNαN , write the PBW-basis elementF c

k asF[M]. Let D(M) =D

ΓkQ (M)= (d1, d2, . . . , dN), and letFi(Q)(D(M))= F

(d1)i1

F(d2)i2· · ·F (dN)

iN. Let�degdenote

the degeneration ordering onkΓk-modules (see [4]). A special case of Lemma 4.5 of [states that:

Lemma 11.1 (Reineke).LetM be akΓk-module as above. Then

Fi(Q)

(D(M)

)=∑[N]

γMN F[N],

whereγMN ∈ Z[v, v−1] is zero unlessM �degN , andγM

M = 1.

This means that the monomialsFi(Q)(D(M)), like the canonical basis, can naturallyindexed by thekΓk-modules. ForM akΓk-module, letFi(Q)(D(M)) denote the monomiain Kashiwara operators obtained by replacing each divided powerF

(a)i in Fi(Q)(D(M))

with F ai . Theorem 10.8 implies that:

Corollary 11.2. LetM be akΓk-module as above. Then, ifc ∈ CPBW(Q), we have

Fi(Q)

(D(M)

) · 1≡E[M] modvL′.

It thus follows that, ifB(M) denotes the canonical basis element corresponding toE[M],then

Fi(Q)

(D(M)

) · 1≡ B(M) modvL′,


s of thet

is in

l basis

ity ofcester,ree for

. Adv.

atrices,

) 245–

)

ieties,

antized

(2)

. 71 (3)

al bases

indicating a curious connection between the canonical basis and the monomial basisecond author. The situation is even nicer in small cases. For example, suppose thag is oftypeA4. Then it is known (see [7, 8.4]) that, forc ∈LPBW(Q), we have

Fi(Q)

(D(M)

) · 1≡ Fi(Q)

(D(M)

)modvL′,

and thatFi(Q)(D(M)) ∈B . Thus, part of the monomial basis lies in the canonical basthis case.

In general, it is at least known that the monomial basis is related to the canonicain the following way:

Proposition 11.3 [23, Proposition 5.4].LetM be akΓk-module as above. Then

Fi(Q)

(D(M)

)=∑[N]

δMN B(N),

whereδMN ∈N[v, v−1] is zero unlessM �degN , andδMM = 1.

Acknowledgments

The authors gratefully acknowledge support for this research from a UniversLeicester Research Fund Grant, and thank the Universities of Wuppertal and Leiwhere this research was carried out, for their hospitality. The authors thank the refesome helpful and interesting comments.

References

[1] M. Auslander, I. Reiten, S.O. Smalø, Representation Theory of Artin Algebras, in: Cambridge StudMath., Vol. 36, Cambridge Univ. Press, Cambridge, 1997.

[2] R. Bedard, On the spanning vectors of Lusztig cones, Represent. Theory 4 (2000) 306–329.[3] A. Berenstein, S. Fomin, A. Zelevinsky, Parametrizations of canonical bases and totally positive m

Adv. Math. 122 (1996) 49–149.[4] K. Bongartz, On degenerations and extensions of finite dimensional modules, Adv. Math. 121 (1996

287.[5] A. Berenstein, A. Zelevinsky, String bases for quantum groups of typeAr , Adv. Soviet Math. 16 (1) (1993

51–89.[6] A. Berenstein, A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive var

Invent. Math. 143 (1) (2001) 77–128.[7] R.W. Carter, R.J. Marsh, Regions of linearity, Lusztig cones and canonical basis elements for the qu

enveloping algebra of typeA4, J. Algebra 234 (2) (2000) 545–603.[8] M. Kashiwara, On crystal bases of theq-analogue of universal enveloping algebras, Duke Math. J. 63

(1991) 465–516.[9] M. Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J

(1993) 839–858.[10] B. Leclerc, M. Nazarov, J.-Y. Thibon, Induced representations of affine Hecke algebras and canonic

of quantum groups, Preprint, arXiv: math.QA/0011074, 2000.


0) 447–

ppl. 102

. Amer.

as and

.732.ype

ebras,

) 705–

Appl.

(2001)

tschler

(2000)

[11] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (199498.

[12] G. Lusztig, Canonical bases arising from quantized enveloping algebras, II, Progr. Theoret. Phys. Su(1990) 175–201.

[13] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, JMath. Soc. 3 (1990) 257–296.

[14] G. Lusztig, Introduction to Quantum Groups, Birkhäuser, Boston, 1993.[15] G. Lusztig, Tight monomials in quantized enveloping algebras, in: Quantum Deformations of Algebr

Their Representations, in: Israel Math. Conf. Proc., Vol. 7, 1993, pp. 117–132.[16] R.J. Marsh, The Lusztig cones of quantized enveloping algebras of typeA, J. Algebra 244 (1) (2001) 59–75[17] R.J. Marsh, More tight monomials in quantized enveloping algebras, J. Algebra 204 (2) (1998) 711–[18] R.J. Marsh, Rectangle diagrams for the Lusztig cones of quantized enveloping algebras of tA,

J. Algebra 249 (2) (2002) 377–401.[19] T. Nakashima, A. Zelevinsky, Polyhedral realizations of crystal bases for quantized Kac–Moody alg

Adv. Math. 131 (1) (1997) 253–278.[20] M. Reineke, On the coloured graph structure of Lusztig’s canonical basis, Math. Ann. 307 (4) (1997

723.[21] M. Reineke, Monomials in canonical bases of quantum groups and quadratic forms, J. Pure

Algebra 157 (2001) 301–309.[22] M. Reineke, Feigin’s map and monomial bases for quantized enveloping algebras, Math. Z. 237

639–667.[23] M. Reineke, Quivers, desingularizations, and canonical bases, in: A. Joseph, A. Melnikov, R. Ren

(Eds.), Studies in Memory of Issai Schur, in: Progr. Math., Vol. 210, Birkhäuser, Boston, 2002.[24] C.M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990) 583–592.[25] N. Xi, Canonical basis for typeA3, Comm. Algebra 27 (11) (1999) 5703–5710.[26] A. Zelevinsky, Connected components of real double Bruhat cells, Internat. Math. Res. Notices 21

1131–1154.

Documents

Canonical basis linearity regions arising from quiver representations