A proof of Lee-Lee's conjecture about geometry of rigid modules
aa r X i v : . [ m a t h . C O ] F e b A PROOF OF LEE-LEE’S CONJECTURE ABOUT GEOMETRYOF RIGID MODULES
SON DANG NGUYEN
Abstract.
This paper proves Lee-Lee’s conjecture that establishes a coinci-dence between the set of associated roots of non-self-intersecting curves in a n -punctured disc and the set of real Schur roots of acyclic (valued) quiverswith n vertices. Introduction
Given an algebraically closed field F and an acyclic quiver Q with n vertices,an indecomposable module M is called by rigid if Ext F Q ( M, M ) = 0 over thepath algebra F Q . It is well-known that from the quiver Q one may construct theirreducible symmetric Cartan matrix companion A = A ( Q ), the root system Φ =Φ( A ) and the Coxeter-Weyl groups G n = G n ( A ) with ordered simple reflections { s , ..., s n } . In the root systems, positive real Schur roots are dimension vectorsof indecomposable rigid modules. Such rigid modules and their real Schur rootsplay very important roles in representation theory and cluster algebras, see [2], [6],[8], [9], [10], [16] just to name, hence it leads to necessarily characterize them amongall real roots of Φ. Remark that the real Schur roots depend on the orientation ofthe quiver Q , which makes it quite difficult to characterize them. In [11], K. H. Leeand K. Lee suggested a conjecture about their geometric property. Conjecture ([11]).
The real roots assigned by simple curves are precisely realSchur roots. (defined in Section 2).
In the paper, they proved it for acyclic quivers of 3 vertices with multiple arrowsbetween every pair of vertices. When the paper was published on arxiv, A. Feliksonand P. Tumarkin [2] proved it for all acyclic quivers of finite vertices with multiplearrows between every pair of vertices (called by 2 -complete acyclic quivers). In thepaper, we show the solution of the conjecture for all acyclic (valued) quivers offinite vertices.Before going to the proof, we importantly need to reform the conjecture inanother version. Given a quiver Q , we have a bijection between the orientationof the quiver and an admissible-sink (or admissible-source) order of its Coxeterelement c := s i s i ...s i n . Hence without any risk, in the paper we can fix thequiver Q such that its Coxeter element is c = s s ...s n . In [10], Igusa and Schifflershowed that reflections corresponding real Schur roots are precisely prefix reflectionsin the factorization of the Coxeter element c . In [1], Hubery and Krause gave acharacterization of real Schur roots in terms of simple partition. Their reflections Date : February 24, 2021.2000
Mathematics Subject Classification.
Key words and phrases. rigid modules, real Schur roots, cluster algebras, quiverrepresentations. are reflection elements belonging to the poset of generalized simple partitions
N C = N C ( Q ) := { w ∈ G n | ≤ w ≤ c } where ≤ denotes the absolute order on G n .In more general cases for a valued quiver Q , given A be an irreducible sym-metrizable generalized Cartan matrix with its symmetrizer D and orientation O , in [3] Geiß, Leclerc and Schr¨oer define a Iwanaga-Gorenstein F -algebra H := H F ( A, D, O ) for any field F in terms of a quiver with relations with a hereditaryalgebra ˜ H of the corresponding type and a Noetherian F [[ ǫ ]]-algebra ˆ H . In par-ticular, in [4], their main results showed that the indecomposable rigid locally free H -modules are parametrized via their rank vector, by the real Schur roots asso-ciated to ( A, O ). Moreover, the left finite bricks of H are parametrized via theirdimension vector by the real Schur roots associated to ( C T , O ). Also as the sym-metric case of acyclic quivers Q , real Schur roots were proven to be real roots inits associated root system such that their reflections are reflection elements of thesimple partition N C in its associated Weyl group. From the observations, Lee-Lee’sconjecture may be reformed as follow:
Conjecture ([11]) Three statements are equivalent:(1) A positive real root β is Schur.(2) Its reflection r β is a prefix of the Coxeter element c i.e. there exists n − r , r , ..., r n such that r β r , ..., r n = c .(3) The real root β may be presented by a simple curve and its reflection maybe presented by a simple closed curve.where [10] and [4] proved (1) ⇔ (2), so we only need to show (2) ⇔ (3).Section 2 is devoted to recall the reader the constructions used before giving theproof of Lee-Lee’s conjecture in section 3. In section 4, we present another proof onfinite types that also implies an one-side proof for affine types. We also give someother results and open problems that may be completed in the future. Acknowledgements:
This work is a part of my PhD thesis, supervised byProofessor Kygungong Lee. I would like to thank him for his guidance andpatience. The author is also deeply grateful to The University of Alabama forproviding ideal working conditions to write this note.2.
Reminders
Given an irreducible symmetrizable Cartan matrix A = ( a ij ) ∈ M n × n ( Z ), it iswell-known that one may construct its associated root system Φ = Φ( A ) as follows.We fix its simple roots { α , ..., α n } and define its simple reflections { s , ..., s n } by s i ( α j ) = α j − a ij α i , i, j = 1 , ..., n . These reflections generates a group G n = G n ( A ) = (cid:10) s , ..., s n (cid:12)(cid:12) s i = 1 , ( s i s j ) m ij = 1 (cid:11) called by the Coxeter / Weyl group where m ij are defined from the table: a ij a ji ≥ m ij ∞ Let B n be the braid group on n strands and abstractly presented B n = h σ , ..., σ n − | σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i if | i − j | > i with the action on n copies of an arbitrary group H as follow: σ i . ( g , ..., g i , g i +1 , ..., g n ) = ( g , ..., g i g i +1 g i − , g i , ..., g n ) ,σ − i . ( g , ..., g i , g i +1 , ..., g n ) = ( g , ..., g i +1 , g i +1 − g i g i +1 , ..., g n ) PROOF OF LEE-LEE’S CONJECTURE ABOUT GEOMETRY OF RIGID MODULES 3 for any sequence ( g , ..., g n ) ∈ H n . It is clear that the action fixes the product g g ...g n . This implies that the position of reflections appearing in the factorizationof the Coxeter element of G n may be ignored by the following lemma. Lemma 2.1.
Assume that a reflection r β belonging a factorization of the Coxeterelement c , then it is a prefix of c i.e. there exists n − reflections r , r , ..., r n suchthat r β r , ..., r n = c . Proof.
The smallest number of factorization of c into reflections is n (see [10]).Assume that c = r r ...r n and r j = r β for some 2 ≤ j ≤ n , then r β is a prefix of σ − ...σ − j . ( r , r , ..., r n ). The action of B n keeps the product invariant, hence theproof is completed. (cid:4) Now we will present real roots as curves and their reflections as closed curves ofby the following construction of D. Bessis in [5]. For convenience we will draw the disc D ⊂ C as an upper half-plane { z ∈ C | Im z > } with the punctured point p i placed from left to right on the horizontal line Imz = 1 and the point ∞ at infinity.Denote by ℓ i a vertical ray ℓ i = { z | Re z = Re p i , Im z > Im p i } . Definition 2.2.
A curve γ is a continuous map γ : [0 , → D such that γ (0) = O, γ (1) ∈ { p , p , ..., p n } and γ ( t ) / ∈ { O, p , p , ..., p n } for t ∈ (0 , simple curve is a non-self-intersecting curve.A closed curve ˆ λ is a continuous map ˆ λ : [0 , → D such that γ (0) = γ (1) = O and γ ( t ) / ∈ { O, p , p , ..., p n } for t ∈ (0 , simple loop .Denote Ins (ˆ λ ) be the interior region bounded by the closed curve ˆ λ . Remark . One may associate such a curve γ uniquely (up to isotopy) to a closedcurve ˆ γ as follows: starting from γ (0) = O , follow γ ; arriving close to γ (1), makea possible turn around a small circle centred on γ (1); return to O following γ backwards. Thus a simple curve associates to a simple loop and so the set of simplecurves may inject as a subset of the set of simple loops. Definition 2.4.
Given a closed curve ˆ λ , then the word w ˆ λ presents an element of G n constructed as follows: following ˆ λ , write s j each time it crosses some rays ℓ j . Definition 2.5.
Given a curve γ with γ (1) = p i and its closed curve ˆ γ , then theword α γ presents a real root constructed as follows: following γ , write s j each timeit crosses some rays ℓ j and add α i as last word. It is clear that the word w ˆ γ presentsits reflection, denoted by r ˆ γ .From the definition of curves and closed curves, now we may identify them andtheir words that present real roots and elements in G n . Since all real roots andelements in G n may be presented by words, the set of real roots is the set of curves(up to isotopy) and G n is the set of closed curves (up to isotopy). Note that twocurves may be isotopic together but the former presents a positive real root α , thelatter presents its negative real root − α as the following example. Example 2.6.
In the example of G , the simple red loop ˆ λ induces the word s s s of the Coxeter element c . In particular, the loop is isotopic to the boundary ∂D = S of the disc D . The simple dark curve γ presents a positive real root α γ = s α and its simple green loop ˆ γ presents a reflection r ˆ γ = s s s . Fromthe curve γ we may draw another simple curve β that is isotopic with γ by going SON DANG NGUYEN ℓ ℓ ℓ O p p p ˆ λ γ ˆ γ around p one time before ending at p . Then β presents a negative real root α β = s s α = − α γ .A real root of Φ and an element of G n may be presented by many differentwords i.e. curves and closed curves up to isotopy. However, if G n = W n := { s , ..., s n (cid:12)(cid:12) s = ... = s n = 1 } , called by universal Coxeter groups , then real rootsof Φ and elements of G n are determined uniquely by their words because Cayleygraphs of W n are trees. Hence real roots and elements of G n are presented uniquelyby curves and closed curves. Now we prove main results that lead to the Lee-Lee’sconjecture. Lemma 2.7.
Given two simple curves with distinct end points such that they donot intersect each other except for O . We order them in a clock wise order of theiremanating from O , written by the sequence ( γ , γ ) . Then(1) The product r ˆ γ r ˆ γ may be presented by a simple loop ˆ γ such that two simpleloops ˆ γ , ˆ γ ⊂ Ins (ˆ γ ) .(2) Braid group B acts on ( r ˆ γ , r ˆ γ ) to be σ . ( r ˆ γ , r ˆ γ ) = ( r ˆ γ r ˆ γ r ˆ γ , r ˆ γ ) (sim-ilarly for σ − . ( r ˆ γ , r ˆ γ ) = ( r ˆ γ , r ˆ γ r ˆ γ r ˆ γ ) ), then it is possible to constructa simple curve γ presenting the real root r ˆ γ α γ (thus the simple loop ˆ γ pre-senting r ˆ γ r ˆ γ r ˆ γ ) such that γ does not intersect with γ (thus ˆ γ does notintersect with ˆ γ ) and γ (1) = γ (1) . Hence braid group action preservesnon-intersecting property of simple curves and loops. Proof.
For (1), since two simple curves ( γ , γ ) do not intersect each other,neither do their simple loops (ˆ γ , ˆ γ ). Therefore we can connect them in a smallneighbor at O to become the loop ˆ γ qualified as the follow example. PROOF OF LEE-LEE’S CONJECTURE ABOUT GEOMETRY OF RIGID MODULES 5
O p p p ˆ γ ˆ γ O p p p ˆ γ For (2), we present ( r ˆ γ r ˆ γ r ˆ γ , r ˆ γ ) from ( r ˆ γ , r ˆ γ ) as follows: in a small neighborof O we connect the loop ˆ γ with the curve γ , then we get a new simple curve γ ′ presenting the real roots r ˆ γ α γ as qualified and its corresponding simple loop ˆ γ ′ presents r ˆ γ r ˆ γ r ˆ γ . Finally, we reorder two curve γ and γ ′ in clock wise order ofemanating from O, then we have the sequence (ˆ γ ′ , ˆ γ ) presenting for ( r ˆ γ r ˆ γ r ˆ γ , r ˆ γ ).The remaining part ( r ˆ γ , r ˆ γ r ˆ γ r ˆ γ ) is similar. (cid:4) O p p p γ γ ˆ γ O p p p γ γ ′ Corollary 2.8.
Given n simple curves with distinct end points such that their pairsdo not intersect each other except for O . We order them in a clock wise order ofemanating from O , possibly written by the sequence ( γ , ..., γ n ) . Then(1) All products r ˆ γ i ...r ˆ γ j , ≤ i ≤ j ≤ n may be presented by loops ˆ γ ij suchthat j − i + 1 simple loops ˆ γ i , ..., ˆ γ j ⊂ Ins (ˆ γ ij ) and Ins (ˆ γ ij ) ⊂ Ins (ˆ γ im ) for ≤ i ≤ j ≤ m ≤ n .In particular, the loop ˆ γ n presents the product r ˆ γ ...r ˆ γ n , all simple loops ˆ γ , ..., ˆ γ n ⊂ Ins (ˆ γ n ) and Ins (ˆ γ ij ) ⊂ Ins (ˆ γ n ) for ≤ i ≤ j ≤ n .(2) Braid group B n acting on ( r ˆ γ , ..., r ˆ γ n ) preserves non-intersecting propertyof simple curves and loops. Proof.
For (1), the proof is similar with (1) of Lemma 2.7 by applying itfor ( γ i , γ i +1 ) , ≤ i ≤ j − ≤ n . For (2), we only need to check actions of σ i ,1 ≤ i ≤ n − r ˆ γ , ..., r ˆ γ n ). But the actions only locally transform two loops inthe sequence and fix the remaining n − (cid:4) SON DANG NGUYEN
Proposition 2.9.
The simple loop ˆ γ n is isotopic with the simple loop presentingthe Coxeter element c = s ...s n . Hence r ˆ γ ...r ˆ γ n = c . Proof.
The idea of the proof is the same as Lemma 2.3 and 2.4 in [5]. Corollary2.8 implies that
Ins (ˆ γ n ) contains all punctured points { p , ..., p n } and presentstheir product r ˆ γ ....r ˆ γ n . In the other hand, c may be presented by a loop ˆ γ c withthe exact word that contains { p , ..., p n } and it is isotopic with ∂D = S . Thus itmay be chosen in its isotopic class such that Ins (ˆ γ n ) ⊂ Ins ( c ). Since the annulusbetween ˆ γ c and ˆ γ in contains no punched point in { p , ..., p n } , they are isotopic andso r ˆ γ ...r ˆ γ n = c . (cid:4) Proof of Lee-Lee’s conjecture
Theorem 3.1.
Three statements are equivalent:(1) A real root β is Schur.(2) Its reflection r β is a prefix of the Coxeter element c i.e. there exists n − reflections r , r , ..., r n such that r β r , ..., r n = c .(3) The real root β may be presented by a simple curve and its reflection maybe presented by a simple closed curve.where [10] and [4] proved (1) ⇔ (2) , so we only need to prove (2) ⇔ (3) . Proof.
A real Schur root α may be presented by a simple curve. Indeed, in[10] and [4], a real root α is Schur if and only if its reflection r α is a prefix of theCoxeter element c . Therefore it may be induced in a reflection sequence ( r , ..., r n )where r α = r j with some j such that r ...r n = c . Since the braid group B n acts transitively on the factorization of c (see [10]), there exists σ ∈ B n such that σ. ( s , ..., s n ) = ( r , ..., r n ). It is clear that ( s , ..., s n ) can be presented by n simpleloops without pairwise intersection each other. But Corollary 2.8 shows that theaction of B n preserves non-intersecting property of n simple loops presenting them,so the real Schur root α may be presented by a simple curve.Conversely, a simple curve can be induced in n simple curves with no pairwiseintersection because of induction on n by cutting the disc D along the curve givingrise to an n − c . Thus roots corresponding reflections of these simple curve areSchur roots because of [10] and [4]. (cid:4) Corollary 3.2.
In the case G n = W n := (cid:10) s , ..., s n (cid:12)(cid:12) s = ... = s n (cid:11) , we have abijective correspondence between simple curves (up to isotopy) and positive realSchur roots. Proof.
In the case, real roots and reflections are uniquely presented by theirreduced words, thus non-isotopic curves present distinct real roots. (cid:4)
This is the same result obtained in [2].Now we let Φ NC be the set of simple loops and define an equivalence ∼ as follows:ˆ γ ∼ ˆ λ if they present the same elements in G n and denote by ˜Φ NC := Φ NC / ∼ .Remark that any two isotopic loops present the same elements but conversely,it is not true because a reflection may be presented by many non-isotopic loops. Definition 3.3.
For ∀ ˆ γ, ˆ λ ∈ ˜Φ NC a partial order ⊆ in ˜Φ NC is defined byˆ γ ⊆ ˆ λ def ⇔ ˆ γ, ˆ λ may be chosen in their equivalent class such that Ins (ˆ γ ) ⊆ Ins (ˆ λ ) PROOF OF LEE-LEE’S CONJECTURE ABOUT GEOMETRY OF RIGID MODULES 7 and number of punctured points in
Ins (ˆ γ ) is fewer than in Ins (ˆ λ ).The absolute length l ( w ) of w ∈ G n is the minimal k ≥ w can bewritten as product w = t t ...t r of reflections t i ∈ G n . Definition 3.4.
For ∀ w, u ∈ G n an absolute order ≤ on N C is defined by w ≤ u def ⇔ l ( w ) + l ( w − u ) = l ( u ) . Recall the simple partition
N C := { w ∈ G n | ≤ w ≤ c } . Theorem 3.5.
We have an order-preserving isomorphism between ( ˜Φ NC , ⊆ ) and ( N C, ≤ ) . Proof.
Bijection between ˜Φ NC and N C is trivial from definition of ˜Φ NC .Given ˆ γ, ˆ λ ∈ ˜Φ NC such that ˆ γ ⊆ ˆ λ and w ˆ γ , w ˆ λ ∈ G n are elements that theypresent for, respectively. We may assume that all punctured points in Ins (ˆ λ ) are { p i , ..., p i k , p i k +1 , ..., p i m } and all punctured points in Ins (ˆ γ ) are { p i , ..., p i k } for1 ≤ i ≤ i k ≤ i m ≤ n . In the annulus between ˆ γ and ˆ λ we draw m − k simpleloops ˆ β i j such that each loop contains exact one punctured point p i j (thus each onepresents a reflection r i j ) with k + 1 ≤ j ≤ m , they are pairwise non-intersecting,and they do not intersect with ˆ λ . We order m − k + 1 loops { ˆ γ, β i k +1 , ..., β i m } ina clock wise order of their eliminating from O , then the product of their corre-sponding elements is w ˆ λ . Hence w ˆ γ ≤ w ˆ λ . Similarly, w ˆ λ ≤ c . Conversely, given1 ≤ u ≤ w ≤ c , then non-crossing partition N C implies that there are n reflections r i , ..., r i n such that r i j r i j +1 ...r i k = u , r i j r i j +1 ...r i k r i k +1 ...r i m = w and r i ...r i n = c for 1 ≤ j ≤ k ≤ m ≤ n . From Corollary 2.8, two simple loops ˆ γ, ˆ λ ∈ ˜Φ NC presenting u, w may be chosen such that ˆ γ ⊆ ˆ λ . (cid:4) Denote s ( c ) and t ( c ) be first and last simple reflections of c , respectively. Let s ( c )( Q ) be a new quiver obtained by conversing arrows adjacent to the vertexcorresponding to s ( c ) (similarly for t ( c )( Q )). The new quivers obtained by thisapproach correspond to mutation of quivers (see [2], [14]) at sink-source verticescorresponding to s ( c ) and t ( c ) in the theory of quiver representations. These mu-tations maintain root systems of the quivers but change their orientation, thuschange their set of real Schur roots. However, the set of real Schur roots of the newquivers and their corresponding simple curves may be obtained by the followingproposition. Proposition 3.6.
A real root β is Schur in the quiver Q if and only if the real root s ( c ) β is Schur in the quiver s ( c )( Q ) (similarly for t ( c )( Q ) ).The word w β is obtained from a simple curve γ β in the setting of the quiver Q .Then a simple curve presenting the real Schur root s ( c ) β in the setting of the quiver s ( c )( Q ) may be constructed from the word s ( c ) w β (similarly for t ( c )( Q ) ). Proof.
Since β is a real Schur root, there exists n − r , ..., r n − suchthat r β r ...r n − = c , thus ( s ( c ) r β s ( c ))( s ( c ) r s ( c )) ...s ( c ))( s ( c ) r n − s ( c )) = ( s ( c ) cs ( c )).The right hand side is the Coxeter element corresponding of the new quiver s ( c ))( sQ )so the first part of the proposition is proved. For the latter part, we may considerit when the Weyl group of the quiver Q is an universal Coxeter group. In thecase, all reduced words obtained from simple curves are unique, thus simple curvepresenting for s ( c ) β has to be constructed from the word s ( c ) β . (cid:4) SON DANG NGUYEN
Example 3.7.
We give an example for a rank-3 quiver with its Coxeter element c := s s s , so s ( c ) = s and t ( c ) = s . A real Schur root β := s s α is presentedby an simple curve γ β . Mutating the quiver at the vertex 1 corresponding to s ( c )we obtain the new quiver s ( c )( Q ) with its Coxeter element ¯ c := s s s = s ( c ) cs ( c ). O p p p γ β presents β = s s α O p p p γ s ( c ) β presents s ( c ) β = s s s α O p p p γ t ( c ) β presents t ( c ) β = s s s α Some Remarks on Finite, Affine and rank- Types
In the section, we give another proof for the Lee-Lee’s conjecture in the case offinite and affine types. It also yields an algorithm to construct simple curves for allreal Schur roots in finite, affine and rank-2 types. Assume G n is a Weyl group offinite types with the Dynkin diagram corresponding ∆. Lemma 4.1.
Given a simple curve γ with its real root α γ and its reflection r γ ,then the real roots cα γ and c − α γ may be presented by simple curves with the exactwords. Hence the action of the Coxeter element c preserves non-self-intersectingproperty of simple curves. This yields the equivalence between c -orbit of simplecurves and c -orbit of their real roots. Proof.
The proof is straight-forward from spiraling the simple curves clockwiseor counterclockwise as the figures:
PROOF OF LEE-LEE’S CONJECTURE ABOUT GEOMETRY OF RIGID MODULES 9
O p ℓ ℓ n p n α γ = wα n O p ℓ ℓ n p n cα γ = cwα n O p ℓ ℓ n p n c − α γ = c − wα n Remark . Since ( σ n − σ n − ...σ ) n . ( s , ..., s n ) = ( cs c − , ..., cs n c − ) and( σ n − σ n − ...σ ) − n . ( s , ..., s n ) = ( c − s c, ..., c − s n c ), the action of c on a simpleloop may be seen as a special case of the action of the braid group B n .Lemma 4.1 also shows a connection of Auslander-Reitein translation τ in acyclicquiver representation theory and the Coxeter element c . While the Auslander-Reitein translation preserves the rigid property of a rigid module (see [7]), theCoxeter element preserves non-self-intersecting property of their corresponding realroots. Corollary 4.3.
For n = 2 , Lee-Lee’s conjecture holds and the set of real Schurroots is c -orbit of { α , α } which contains all real roots. Therefore in the case allpositive real roots are Schur. Proof.
From [10], [4] and the transitive property of the action of B on ( s , s ),one knew that reflections of real Schur roots are precisely elements appear in theorbit of B on ( s , s ). In particular, for B = < σ > , we have σ h ( s , s ) = ( c h s c − h , c h s c − h ) for h ∈ Z ,σ k +11 ( s , s ) = ( c k s s s c − k , c k s c − k ) for k ∈ Z + ,σ − (2 k +1)1 ( s , s ) = ( c − k s c k , c − k s s s c k ) , for k ∈ Z + . Moreover, s s s = cs c − and s s s = c − s c . Hence Lemma 4.1 completed theproof. (cid:4) Corollary 4.4.
Lee-Lee’s conjecture holds for finite-type root systems.
Proof.
In finite-type cases, real roots are precisely real Schur roots, so theproof of simple curves presenting real Schur roots is trivial. Conversely, let β k = s s ...s k − α k , for 1 ≤ k ≤ n , then it is clear that these roots may be presented bysimple curves with the exact word. Moreover, finite-type root systems have exact n distinct c -orbits where β k belongs to the each one (see Proposition 33, chapterVI in [15]). Thus Lemma 4.1 implies Lee-Lee’s conjecture. (cid:4) In the case of affine types, the action of c also gives an one-side proof of Lee-Lee’sconjecture and the remain ones is delivered from Section 3. Corollary 4.5.
Lee-Lee’s conjecture holds for affine-type root systems.
Proof.
In [13] and [14], the authors show explicit Φ rec set of real Schur rootsas follows: the set has exact 2 n infinite c -orbits and n − c -orbits with n ≥
2. The transversal set of 2 n infinite c -orbits includes β k = s s ...s k − α k and δ k = s n s n − ...s k +1 α k , for 1 ≤ k ≤ n that clearly might be presented by simplecurves. Moreover, all real roots in n − c -orbits are of finite types that mightbe presented by simple curves because of Corollary 4.2. Hence Lemma 4.1 impliesthat all real Schur roots of affine types may be presented by simple curves. (cid:4) Let R be the set of reflections of G n and recall the action of B n on s =( s , ..., s n ) ∈ R n . We denote c = c (∆) and h = h (∆) respectively being theCoxeter element and the Coxeter number of Dynkin diagram ∆. Let B G n be thestabilizer subgroup of s and B n ( s ) is the B n -orbit at s . Transitivity action of B n onthe completed exceptional sequence implies that the orbit is all possibly completedexceptional sequences. Proposition 4.6.
We have a bijection between the coset B n /B G n and the factor-ization of the Coxeter element c , hence the index [ B n : B G n ] = n ! h n | G n | . Proof.
Bijection is trivial from the orbit-stabilizer theorem and the specificformula of number of the factorization of the Coxeter group is shown in [12]. (cid:4)
The following table in [12] exhibits the index formula for the connected Dynkindiagrams ∆:∆ A n B n , C n D n E E E F G n ! h n | G n | ( n + 1) n − n n n − n . . . . . . { σ , σ σ ...σ n − } generates B n and the group B G n is a finitely generatedsubgroup because it is a finite index subgroup of B n . Unfortunately, now we cannotyet find the generating set of B G n but introduce some elements of B G n . Let ∆( i −− j ) to be subdiagrams of the Dynkin diagram ∆ with vertices { i, i + 1 , ..., j } and h ij := h (∆( i − − j )) to be their corresponding Coxeter number; particularly h i := h ii +1 . Lemma 4.7.
For ≤ i < j ≤ n − , ( σ j − σ j − ...σ i ) ( j − i +1) h ij , ( σ n − ...σ ) nh ∈ B G n . In particular, if ( s i s i +1 ) m ij = 1 , then σ m ij i ∈ B G n with m i,j = 2 , , , . Proof.
We have σ i . ( s , ...s i , s i +1 , ...s n ) = ( s , ..., c i s i c i − , c i s i +1 c i − , ...s n ) , ( σ j − σ j − ...σ i ) ( j − i + 1) . ( s , ...s i , ..., s i +1 , ..., s n ) = ( s , ...c ij s i c − ij , ..., c ij s i +1 c − ij , ..., s n ) , ( σ n − σ n − ...σ ) n . ( s , ..., s n ) = ( cs c − , ..., cs n c − ) , where c i = c (∆( i − − i + 1)) = s i s i +1 , c ij = c (∆( i − − j )) = s i s i +1 ...s j . Since c h i i = c h ij ij = c h = id , the proof is completed. (cid:4) Remark that ( σ n − σ ...σ ) nh belongs to the center of B n , hence ( σ n − σ ...σ ) nh ∈ N n := T σ ∈ A n σB G n σ − i.e. B n acts unfaithfully on s . Since B G n is a finite-indexsubgroup, so is N n . PROOF OF LEE-LEE’S CONJECTURE ABOUT GEOMETRY OF RIGID MODULES 11
Example 4.8.
For G = A we have [ B : B A ] = 3 and B A = < σ > .For G = G we have [ B : B G ] = 6 and B G = < σ > . Corollary 4.9.
Assume that there exists a proper normal subgroup M that isstrictlly larger than B G n for n ≥ , then n = 2 ( mod . Hence B n and B n / N n isnot solvable for n ≥ . Proof.
We consider the natural projection π : B n → B n / M . Since σ ∈ B A n and M is proper, the order of π ( σ ) is 3 in B n / M so 3 divides [ B n : B A n ] = ( n + 1) n − .Thus n = 2 ( mod B A in B . Moreover, B A is not a normal subgroup of B ,hence B is not solvable. Since B may be embedded in B n for n ≥ B n is notsolvable and so is B n / N n . (cid:4) Finally, we finish the paper with several further questions which might be ofinterest.
Question 1:
How can we find the finite generating set of B G n and N n := T σ ∈ A n σB G n σ − when G n is a Weyl group of finite types? Question 2:
What is the classification of the finite non-solvable group B n / N n for n ≥ References
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