Preprojective algebras and cluster algebras
aa r X i v : . [ m a t h . R T ] A p r Prepro jective algebras and cluster algebras
Christof Geiss, Bernard Leclerc and Jan Schr¨oer ∗ Abstract.
We use the representation theory of preprojective algebras to construct andstudy certain cluster algebras related to semisimple algebraic groups.
Mathematics Subject Classification (2000).
Primary 16G20; Secondary 14M15,16D90, 16G70, 17B10, 20G05, 20G20, 20G42.
Keywords. preprojective algebra, cluster algebra, flag variety, rigid module, mutation,Frobenius category, semicanonical basis
1. Introduction
Cluster algebras were invented by Fomin and Zelevinsky in 2001 [10]. One of themain motivations for introducing this new class of commutative algebras was toprovide a combinatorial and algebraic framework for studying the canonical basesof quantum groups introduced by Lusztig and Kashiwara [26, 22] and the notionof total positivity for semisimple algebraic groups developed by Lusztig [29].A first attempt to understand cluster algebras in terms of the representationtheory of quivers was done by Marsh, Reineke, and Zelevinsky [33], using a categoryof decorated representations. This was quickly followed by the seminal paper ofBuan, Marsh, Reiten, Reineke, and Todorov [4] who introduced a new family oftriangulated categories attached to hereditary algebras, called cluster categories,and showed that the combinatorics of cluster mutations arises in the tilting theoryof these cluster categories. This yielded a categorification of a large family ofcluster algebras: the acyclic cluster algebras.In this review, we will explain a different but somewhat parallel developmentaimed at giving a representation-theoretic treatment for another class of cluster al-gebras, namely those discovered by Berenstein, Fomin and Zelevinsky [3] in relationwith their series of works on total positivity and the geometry of double Bruhatcells in semisimple groups. Instead of the cluster categories we have used themodule categories of the preprojective algebras corresponding to these semisimplegroups, and more generally certain Frobenius subcategories of these module cate-gories. This allowed us to prove that the cluster monomials of the cluster algebraswe consider belong to the dual of Lusztig’s semicanonical basis, and in particularare linearly independent [14]. It also enabled us to introduce new cluster algebrastructures on the coordinate rings of partial flag varieties [17]. ∗ We are grateful to A. Skowro´nski and the ICRA committee for encouraging us to preparethis survey for the ICRA XII book.
C. Geiss, B. Leclerc and J. Schr¨oer
2. Total positivity, canonical bases and cluster al-gebras
Before reviewing our construction, we would like to illustrate by means of somesimple examples why total positivity, canonical bases and cluster algebras areintimately connected.Lusztig has defined the totally positive part X > for several classes of complexalgebraic varieties X attached to a semisimple algebraic group G . The definitionuses the theory of canonical bases for irreducible G -modules. It is not our intentionin this survey to explain this definition, neither to recall the construction of canon-ical bases (for excellent reviews of these topics, we refer the reader to [28, 31]).Instead of this, we shall present a few examples for which both the totally positivevarieties and the canonical bases can be described in an explicit and elementaryway. We shall then see that a cluster algebra structure on the coordinate ringnaturally arises from this description.Our first example is trivial, but nevertheless useful to get started. Let V = C n be an even dimensional vector space with natural coordinates ( y , . . . , y n ). Let X = P ( V ) = P n − be the corresponding projective space. In this case our group G is SL ( V ). The totally positive part of V is simply the orthant V > = { v ∈ V | y ( v ) > , . . . , y n ( v ) > } , and the totally positive part of X is the subset of X consisting of points having asystem of homogeneous coordinates ( y : · · · : y n ) with all y i positive. The coordi-nate ring of V (or the homogeneous coordinate ring of X ) is R = C [ y , . . . , y n ]. Ithas a natural C -basis given by all monomials in the y i ’s. The natural action of G on V makes R into a linear representation of G , which decomposes into irreduciblerepresentations as R = M k ≥ R k , where R k ∼ = S k ( V ∗ ) is the degree k homogeneous part of R . For every irreduciblerepresentation of G , Lusztig and Kashiwara have introduced a canonical basis anda dual canonical basis (also called lower global basis and upper global basis byKashiwara). It is not difficult to check that, in the case of the simple representa-tions R k , the dual canonical basis coincides with the basis of monomials in the y i ’sof total degree k .We now pass to a more interesting example. Consider the nondegeneratequadratic form on V given by q ( y , . . . , y n ) = n X i =1 ( − i − y i y n +1 − i . Let C = { v ∈ V | q ( v ) = 0 } be its isotropic cone and Q = P ( C ) the corresponding smooth quadric in X = P ( V ).The quadric Q can be seen as a partial flag variety for the special orthogonal group reprojective algebras and cluster algebras H attached to the form q , and so it has, following Lusztig [30], a well-defined totallypositive part. Let us try to guess what is Q > , or equivalently what is C > .It seems natural to require that C > is contained in V > . But this is not enough,and in general C > is going to be a proper subset of C ∩ V > . To see this, we mayuse the known fact that the totally positive part of a variety of dimension k ishomeomorphic to R k> , hence we might expect that it is described by a system of k inequalities. However, C ∩ V > is the subset of C given by the 2 n inequalities y i >
0, and since C has dimension 2 n −
1, we would like to define C > by a systemof only 2 n − n − f , . . . , f n − ) on C such that C > = { v ∈ C | f ( v ) > , . . . , f n − ( v ) > } . (1)Such a system ( f , . . . , f n − ) is called a positive coordinate system .So we are looking for a “natural” set of 2 n − f , . . . , f n − ) suchthat the positivity of f ( v ) , . . . , f n − ( v ) implies the positivity of y ( v ) , . . . , y n ( v ).Let us try this idea in the case n = 3. We have q ( y , . . . , y ) = y y − y y + y y . On C ∩ V > we therefore have the relation y = y y + y y y . Hence the positivity of the 5 coordinates in the right-hand side implies the posi-tivity of y , that is, the defining equation of C allows to eliminate the inequality y > C ∩ V > . So we could take( y , y , y , y , y )as a positive coordinate system. Note that y and y play the same role and areexchangeable: we could also take ( y , y , y , y , y ). This would define the samesubset C > , which in this case is simply C ∩ V > .Already in the case n = 4 the same trick no longer works. Indeed, the definingequation of C is now y y − y y + y y − y y = 0, which does not allow us toexpress any of the y i ’s as a subtraction-free expression in terms of the 7 remainingones. To overcome this problem, we introduce a new quadratic function p = y y − y y = y y − y y on C . On C ∩ V > we then have y = y y + py , y = y y + py , and this leads us to take ( y , y , y , y , p, y , y ) (2) C. Geiss, B. Leclerc and J. Schr¨oer as a positive coordinate system. Again, y and y are exchangeable, as are y and y . So we would obtain the same subset C > by using, instead of (2), each ofthe 3 alternative systems of coordinates( y , y , y , y , p, y , y ) , ( y , y , y , y , p, y , y ) , ( y , y , y , y , p, y , y ) . (3)Note that in this case our candidate for C > is a proper subset of C ∩ V > , sincethe positivity of p does not follow from the positivity of the y i ’s.It turns out that this naive candidate coincides with the totally positive partof C defined by Lusztig. To explain this, let us consider the coordinate ring A = C [ y , . . . , y ] / ( y y − y y + y y − y y )of C , or in other words the homogeneous coordinate ring of the quadric Q . Asbefore, A is in a natural way a representation of the special orthogonal group H ,and the homogeneous components A k ( k ≥
0) coincide with the irreducible directsummands of this representation. Hence by putting together the dual canonicalbases of all summands A k , we get a dual canonical basis of A . We claim that inthis easy situation, the dual canonical basis can be explicitly computed and hasthe following simple description. Namely, the dual canonical basis of A k consistsof all the degree k monomials in y , . . . , y , p containing only variables of one ofthe 4 coordinate systems displayed in (2), (3). Here, y , . . . , y have degree 1 and p has degree 2.For example, the dual canonical basis of A ∼ = V ∗ is { y , y , y , y , y , y , y , y } ,and the dual canonical basis of A consists of p and of all the degree 2 monomialsin the y i ’s except y y and y y .Now, Lusztig has shown [30, Prop. 3.2, Th. 3.4] that C > has the followingcharacterization: it consists of all elements v of C n such that, for every element b of the dual canonical basis of A k and for every k , one has b ( v ) >
0. Because ofthe monomial description of the dual canonical basis, we see that this agrees withour naive definition of C > . Exercise 2.1.
Guess in a similar way what is the definition of C > for n ≥ Answer.
For s = 1 , , . . . , n −
3, put p s = s +1 X k =1 ( − s +1 − k y k y n +1 − k . Then C > is the subset of C given by the following n + 1 inequalities y > , y n > , y n +1 > , y n > , p s > , ( s = 1 , . . . , n − , together with one (it does not matter which one) of the two inequalities y k > , y n +1 − k > , for each k = 2 , , . . . , n − reprojective algebras and cluster algebras n , C > can be described as in (1) by a positive coordinatesystem, and there are 2 n − different but equivalent such systems. In fact, one canalso check that the dual canonical basis of the coordinate ring of C consists of allmonomials in the y i ’s and p s ’s supported on one of these 2 n − coordinate systems.The definition of a cluster algebra will be recalled in Section 14 below. Areader already familiar with it will immediately recognize an obvious cluster algebrastructure on the coordinate ring of C emerging from this discussion. Its 2( n − cluster variables are y k , y n +1 − k , ( k = 2 , , . . . , n − . Its coefficient ring is generated by y , y n , y n +1 , y n , p s , ( s = 1 , . . . , n − . Its clusters are the 2 n − possible choices of 2 n − cluster monomials are all the monomials supportedon a single cluster, and its exchange relations are y k y n +1 − k = p + y y n if k = 2, p k − + p k − if 3 ≤ k ≤ n − y n y n +1 + p n − if k = n − A n − .To summarize, the cone C and the corresponding quadric Q are examples ofalgebraic varieties for which Lusztig has described a natural totally positive subset C > or Q > . What we have found is that their coordinate ring is endowed withthe structure of a cluster algebra such that(1) each cluster gives rise to a positive coordinate system;(2) the dual canonical basis of the coordinate ring coincides with theset of cluster monomials.This is the prototype of what one would like to do for each variety X having atotally positive part X > in Lusztig’s sense. But in general, things become morecomplicated. First, the cluster algebra structure, when it is known, is usually well-hidden, and its description requires a lot of difficult (but beautiful) combinatorics.As an example, one may consult the paper of Scott [34] and in particular thecluster structures of the Grassmannians Gr(3 , ,
7) and Gr(3 , all positive coordinate systems on X . But it becomes a challenging issue if oneaims at a monomial description of the dual canonical basis of the coordinate ring,because that would likely involve infinitely many families of monomials. In fact C. Geiss, B. Leclerc and J. Schr¨oer such a monomial description may not even be possible, since, as shown in [25],there may exist elements of the dual canonical basis whose square does not belongto the basis. In any case, even if the cluster structure is known, more work iscertainly needed to obtain from it a full description of the dual canonical basis.Finally, there is no universal recipe for getting a cluster structure on the coordinatering. Actually, the existence of such a structure is not guaranteed by any generaltheorem, so it often seems kind of miraculous when it eventually comes out of somecomplicated calculations.The aim of this review is to explain some recent progress made in these di-rections by means of the representation theory of preprojective algebras. We willchoose as our main example the partial flag varieties X attached to a simple alge-braic group G of type A, D, E . Thus X is a homogeneous space G/P , where P isa parabolic subgroup of G . To G one can attach the preprojective algebra Λ withthe same Dynkin type. To P (or rather to its conjugacy class) one can attach acertain injective Λ-module Q , and the subcategory Sub Q of the module categorymod Λ cogenerated by Q . We will show that Sub Q can be regarded as a categori-fication of the multihomogeneous coordinate ring of X , and that the rigid modulesin Sub Q give rise to a cluster structure on this ring. In particular, this yields auniform recipe for producing explicit cluster structures, many of which were firstdiscovered in this way. The cluster structure is of finite type when Sub Q has finiterepresentation type, and in these exceptional cases, the Auslander-Reiten quiverof Q is quite helpful for understanding the ensuing combinatorics. Finally, thisapproach allows to show that the cluster monomials of these algebras belong tothe dual of Lusztig’s semicanonical basis. Unfortunately, the relation between thesemicanonical and the canonical basis is a subtle question (see [12]). Nevertheless,as predicted by the general conjecture of Fomin and Zelevinsky [10, p.498], webelieve that the cluster monomials also belong to the dual canonical basis, that is,we conjecture that they lie in the intersection of the dual canonical basis and thedual semicanonical basis (see below §
3. Preprojective algebras
We start with definitions and basic results about preprojective algebras of Dynkintype.Let ∆ be a Dynkin diagram of type A , D or E . We denote by I the setof vertices and by n its cardinality. Let Q be the quiver obtained from ∆ byreplacing each edge, between i and j say, by a pair of opposite arrows a : i −→ j and a ∗ : j −→ i . Let C Q denote the path algebra of Q over C . We can form thefollowing quadratic element in C Q , c = X ( a ∗ a − aa ∗ ) , where the sum is over all edges of ∆. Let ( c ) be the two-sided ideal generated by c .Following Gelfand and Ponomarev [20], we define the preprojective algebra Λ := C Q/ ( c ) . reprojective algebras and cluster algebras A n ( n ≤ A or D ,and wild type in all other cases (see [7]).We denote by S i ( i ∈ I ) the simple Λ-modules, and by Q i ( i ∈ I ) their injectiveenvelopes. Example 3.1.
Let Λ be of type D . We label the 3 external nodes of the Dynkindiagram of type D by 1, 2, 4, and the central node by 3. With this convention,the socle filtration of Q is S S S ⊕ S S S and the socle filtration of Q is S S ⊕ S ⊕ S S ⊕ S S ⊕ S ⊕ S S The structure of Q and Q can be obtained from that of Q by applying the order3 diagram automorphism 1 D denote duality with respect to the field C . We haveExt ( M, N ) ∼ = D Ext ( N, M ) , ( M, N ∈ mod Λ) , (4)and this isomorphism is functorial with respect to M and N (see [16]).
4. Regular functions on maximal unipotent sub-groups
We turn now to semisimple algebraic groups. For unexplained terminology, thereader can consult standard references, e.g. [6], [8], [21].Let G be a simply connected simple complex algebraic group with the sameDynkin diagram ∆ as Λ. Let N be a fixed maximal unipotent subgroup of G .If G = SL( n + 1 , C ), we can take N to be the subgroup of upper unitriangularmatrices. In general N is less easy to describe. To perform concrete calculations,one can use the one-parameter subgroups x i ( t ) ( i ∈ I, t ∈ C ) associated with thesimple roots, which form a distinguished set of generators of N . Example 4.1.
In type A n , if N is the subgroup of upper unitriangular matricesof SL( n + 1 , C ), we have x i ( t ) = I + tE i,i +1 , where I is the identity matrix and E ij the matrix unit with a unique nonzero entry equal to 1 in row i and column j . C. Geiss, B. Leclerc and J. Schr¨oer
Example 4.2.
In type D n , N can be identified with the subgroup of the group ofupper unitriangular matrices of SL(2 n, C ), generated by x i ( t ) = (cid:26) I + t ( E n − i +1 ,n − i +2 + E n + i − ,n + i ) if 2 ≤ i ≤ n,I + t ( E n − ,n +1 + E n,n +2 ) if i = 1 . Thus in type D we can take for N the subgroup of SL(8 , C ) generated by x ( t ) = t t , x ( t ) = t t ,x ( t ) = t t
00 0 0 0 0 0 1 00 0 0 0 0 0 0 1 , x ( t ) = t t . As an algebraic variety, N is isomorphic to an affine space of complex dimensionthe number r of positive roots of ∆. Hence its coordinate ring C [ N ] is isomorphicto a polynomial ring in r variables. For example in type A n if N is the group ofunitriangular matrices, each matrix entry n ij (1 ≤ i < j ≤ n + 1) is a regularfunction on N and C [ N ] is the ring of polynomials in the n ( n + 1) / n ij .In the general case, the most convenient way of specifying a regular function f ∈ C [ N ] is to describe its evaluation f ( x i ( t ) · · · x i k ( t k )) at an arbitrary productof elements of the one-parameter subgroups. In fact one can restrict to certainspecial products. Namely, let W denote the Weyl group of G and s i ( i ∈ I ) itsCoxeter generators. Let w be the longest element of W and let w = s i · · · s i r bea reduced decomposition. Then it is well-known that the image of the map( t , . . . , t r ) ∈ C r x i ( t ) · · · x i r ( t r ) ∈ N is a dense subset of N . It follows that a polynomial function f ∈ C [ N ] is completelydetermined by its values on this subset.
5. A map from mod Λ to C [ N ] In [27, Section 12], Lusztig has given a geometric construction of the envelopingalgebra U ( n ) of the Lie algebra of N . It is very similar to Ringel’s realization of reprojective algebras and cluster algebras U ( n ) as the Hall algebra of mod( F q Q ) “specialized at q = 1”. Here Q denotes anyquiver obtained by orienting the edges of the Dynkin diagram ∆.There are two main differences between Ringel’s and Lusztig’s constructions.First, in Lusztig’s approach one works directly at q = 1 by replacing the countingmeasure for varieties over finite fields by the Euler characteristic measure for con-structible subsets of complex algebraic varieties. The second difference is that onereplaces the module varieties of Q by the module varieties of Λ, in order to obtaina construction independent of the choice of an orientation of ∆.As a result, one gets a model of U ( n ) in which the homogeneous piece U ( n ) d ofmultidegree d = ( d i ) ∈ N I is realized as a certain vector space of complex-valuedconstructible functions on the variety Λ d of Λ-modules with dimension vector d .It follows that to every M ∈ mod Λ of dimension vector d , one can attach anatural element of the dual space U ( n ) ∗ d , namely the linear form δ M mapping aconstructible function ψ ∈ U ( n ) d to its evaluation at M (by regarding M as apoint of Λ d ). Let U ( n ) ∗ gr := M d ∈ N I U ( n ) ∗ d be the graded dual of U ( n ) endowed with the dual Hopf structure. The follow-ing result is well-known to the experts, but we were unable to find a convenientreference. We include a sketch of proof for the convenience of the reader. Proposition 5.1. U ( n ) ∗ gr is isomorphic, as a Hopf algebra, to C [ N ] .Proof. (Sketch.) H = U ( n ) ∗ gr is a commutative Hopf algebra, and therefore it canbe regarded as the coordinate ring of the affine algebraic group Hom alg ( H, C ) ofalgebra homomorphisms from H to C , or equivalently as the coordinate ring ofthe group G ( H ◦ ) of all group-like elements in the dual Hopf algebra H ◦ (see e.g. [1, § H being the graded dual of U ( n ), the dual H ∗ of H is thecompletion [ U ( n ) of U ( n ) with respect to its grading. A simple calculation showsthat for every e ∈ n the exponential exp( e ) = P k ≥ e k /k ! ∈ [ U ( n ) is a group-likeelement in H ◦ . Let e i ( i ∈ I ) be the Chevalley generators of n . Then the map x i ( t ) exp( te i ) ( i ∈ I ) extends to a homomorphism from N to G ( H ◦ ). One cancheck that this is an isomorphism using the fact that H is a polynomial algebrain r variables. This induces the claimed Hopf algebra isomorphism from H to C [ N ].Let ι : U ( n ) ∗ gr → C [ N ] denote this isomorphism. Let ω denote the automor-phism of C [ N ] described in [15, § N leaves invariant the one-parametersubgroups x i ( t ). In other words, for f ∈ C [ N ] we have( ωf )( x i ( t ) · · · x i k ( t k )) = f ( x i k ( t k ) · · · x i ( t )) , ( i , . . . i k ∈ I, t , . . . , t k ∈ C ) . Define ϕ M = ω ◦ ι ( δ M ). We have thus obtained a map M ϕ M from mod Λ to C [ N ]. Let us describe it more explicitly.0 C. Geiss, B. Leclerc and J. Schr¨oer
Consider a composition series f = ( { } = M ⊂ M ⊂ · · · ⊂ M d = M )of M with simple factors M k /M k − ∼ = S i k . We call i := ( i , . . . , i d ) the type of f . Let Φ i ,M denote the subset of the flag variety of M (regarded as a C -vectorspace) consisting of all flags which are in fact composition series of M (regardedas a Λ-module) of type i . This is a closed subset of the flag variety, hence aprojective variety. We denote by χ i ,M = χ (Φ i ,M ) ∈ Z its Euler characteristic.By unwinding Lusztig’s construction of U ( n ), dualizing it, and going through theabove isomorphisms, one gets the following formula for ϕ M . Proposition 5.2.
For every i = ( i , . . . , i k ) ∈ I k we have ϕ M ( x i ( t ) · · · x i k ( t k )) = X a ∈ N k χ i a ,M t a · · · t a k k a ! · · · a k ! , where we use the short-hand notation i a = ( i , . . . , i | {z } a , . . . , i k , . . . , i k | {z } a k ) .Proof. (Sketch.) Using the above embedding of N in [ U ( n ), we have x i ( t ) · · · x i k ( t k ) = X a ∈ N k t a · · · t a k k a ! · · · a k ! e a i · · · e a k i k , as an element of [ U ( n ). Now, for a fixed j = ( j , . . . , j d ), consider the constructiblefunction χ j : M χ j ,M defined on Λ d , where d = ( d i ) and d i is the number of s ’ssuch that j s = i . In Lusztig’s Lagrangian construction of U ( n ) [27], the functions χ j span the vector space U ( n ) d . More precisely, χ j is identified with the monomial e j d · · · e j . By the definition of ϕ M , we thus get ϕ M ( x i ( t ) · · · x i k ( t k )) ≡ δ M X a ∈ N k t a · · · t a k k a ! · · · a k ! χ i a = X a ∈ N k χ i a ,M t a · · · t a k k a ! · · · a k ! , as claimed. Note that the twist by ω and the twist χ j ,...,j d ≡ e j d · · · e j canceleach other. Remark 5.3.
In [14] we have denoted by ϕ M the function ι ( δ M ), without twistingby ω (in the definition of Φ i ,M we were using descending flags f instead of ascendingones). On the other hand, in [13, §
7] we have defined a left U ( n )-module structureon U ( n ) ∗ gr . The twisting by ω is needed if we want this structure to agree with theusual left U ( n )-module structure on C [ N ] given by( e i f )( x ) = ddt f ( xx i ( t )) | t =0 , ( f ∈ C [ N ] , x ∈ N ) . This is the convention which we have taken in [17] and which we follow here. reprojective algebras and cluster algebras Example 5.4.
In type A , we have w = s s s , hence every f ∈ C [ N ] is deter-mined by its values at x ( t ) x ( t ) x ( t ) for ( t , t , t ) ∈ C . One calculates x ( t ) x ( t ) x ( t ) = t
00 1 00 0 1 t t
00 1 00 0 1 = t + t t t t . On the other hand, using the formula of Proposition 5.2 one gets easily ϕ S ( x ( t ) x ( t ) x ( t )) = t + t ,ϕ S ( x ( t ) x ( t ) x ( t )) = t ,ϕ Q ( x ( t ) x ( t ) x ( t )) = t t ,ϕ Q ( x ( t ) x ( t ) x ( t )) = t t . It follows that, in terms of matrix entries, we have ϕ S = n , ϕ S = n , ϕ Q = n , ϕ Q = (cid:12)(cid:12)(cid:12)(cid:12) n n n (cid:12)(cid:12)(cid:12)(cid:12) . Exercise 5.5.
In type A n , for 1 ≤ i ≤ j ≤ n , let M [ i,j ] denote the indecomposableΛ-module of dimension j − i +1 with socle S i and top S j . ( M [ i,j ] is in fact uniserial.)Show that ϕ M [ i,j ] = n i,j +1 , the matrix entry on row i and column j + 1. Exercise 5.6.
In type A n , show that ϕ Q k is equal to the k × k minor of n . . . n ,n +1 . . . n ,n +1 ... ... . . . ...0 0 . . . with row indices 1 , , . . . , k and column indices n − k + 2 , n − k + 3 , . . . , n + 1.More generally, show that for every submodule M of Q k , ϕ M is equal to a k × k minor with row indices 1 , , . . . , k , and that conversely, every nonzero k × k minorwith row indices 1 , , . . . , k is of the form ϕ M for a unique submodule M of Q k . Exercise 5.7.
In type D a reduced decomposition of w is for example w = s s s s s s s s s s s s . Using the realization of N as a group of unitriangular 8 × x = x ( t ) x ( t ) x ( t ) x ( t ) x ( t ) x ( t ) x ( t ) x ( t ) x ( t ) x ( t ) x ( t ) x ( t ) . C. Geiss, B. Leclerc and J. Schr¨oer
Check that the first row of the matrix x is equal to[1 , t + t + t , t t + t t + t t + t t + t t + t t ,t t t + t t t + t t t + t t t , t t t + t t t + t t t + t t t ,t t t t + t t t t + t t t t + t t t t + t t t t + t t t t ,t t t t t + t t t t t + t t t t t + t t t t t + t t t t t + t t t t t + t t t t t , t t t t t t ] . Check that the 8 entries on this row are equal to ϕ M ( x ) where M runs over the8 submodules of Q , including the zero and the full submodules (see Example 3.1).Express in a similar way all the entries of x as the evaluations at x of functions ϕ M where M runs over the subquotients of Q .Investigate the relations between the 2 × x and the values ϕ M ( x ) where M is a submodule of Q .
6. Multiplicative properties of ϕ In the geometric realization of U ( n ) given in [27], only the multiplication is con-structed, or equivalently the comultiplication of C [ N ]. For our purposes though,it is essential to study the multiplicative properties of the maps ϕ M . The mostimportant ones are Theorem 6.1 ([12, 16]).
Let
M, N ∈ mod Λ . Then the following hold: (1) ϕ M ϕ N = ϕ M ⊕ N . (2) Assume that dim Ext ( M, N ) = 1 . Let → M → X → N → , → N → Y → M → , be two non-split short exact sequences (note that this determines X and Y uniquely up to isomorphism). Then ϕ M ϕ N = ϕ X + ϕ Y . Note that in [16] a formula is proved which generalizes (2) to any pair (
M, N )of Λ-modules with dim Ext ( M, N ) >
0. It involves all possible middle terms ofnon-split short exact sequences with end terms M and N , weighted by certainEuler characteristics. It was inspired by a similar formula of Caldero and Keller inthe framework of cluster categories [5]. We will not need this general multiplicationformula here. Example 6.2.
Type A . Using the formulas of Example 5.4, one checks easilythat ϕ S ϕ S = ϕ Q + ϕ Q , in agreement with the short exact sequences0 → S → Q → S → , → S → Q → S → . reprojective algebras and cluster algebras Exercise 6.3.
Type A . Consider the following indecomposable Λ-modules de-fined unambiguously by means of their socle filtration: M = S , N = S ⊕ S S , X = S S ⊕ S S = Q , Y = S S , Z = S S . Check that ϕ M ϕ N = ϕ X + ϕ Y ⊕ Z . Using Exercise 5.6, show that this identity isnothing else than the classical Pl¨ucker relation[1 , × [2 ,
4] = [1 , × [3 ,
4] + [1 , × [2 , × n n n n n n , where [ i, j ] denotes the 2 × ,
2) and columns ( i, j ).
7. The dual semicanonical basis
The functions ϕ M ( M ∈ mod Λ) satisfy many linear relations. For example ifdim Ext ( M, N ) = 1, combining (1) and (2) in Theorem 6.1 we get ϕ M ⊕ N = ϕ X + ϕ Y . It is possible, though, to form bases of C [ N ] consisting of functions ϕ M where M is taken in a certain restricted family of modules M . For example, let Q be a fixedorientation of ∆. Every C Q -module can be regarded as a Λ-module in an obviousway. It is easy to check that { ϕ M | M ∈ mod( C Q ) } is a C -basis of C [ N ]. In fact this is the dual of the PBW-basis of U ( n ) constructedfrom Q by Ringel (see [12, § Q . Usingsome geometry, one can obtain a more “canonical” basis of C [ N ]. Let us fix adimension vector d ∈ N I and regard the map ϕ as a map from the module varietyΛ d to C [ N ]. This is a constructible map, hence on every irreducible componentof Λ d there is a Zariski open set on which M ϕ M is constant. Let us say thata module M in this open set is generic . Then, dualizing Lusztig’s construction in[32], one gets Theorem 7.1. { ϕ M | M is generic } is a basis of C [ N ] . C. Geiss, B. Leclerc and J. Schr¨oer
This is the dual of Lusztig’s semicanonical basis of U ( n ). We shall call itthe dual semicanonical basis of C [ N ]. By construction it comes with a naturallabelling by the union over all d ∈ N I of the sets of irreducible components of thevarieties Λ d .Important examples of generic modules are given by rigid modules. We saythat a Λ-module M is rigid if Ext ( M, M ) = 0, or equivalently if the orbit of M in its module variety is open (see [14]). Corollary 7.2. If M is a rigid Λ -module then ϕ M belongs to the dual semicanon-ical basis of C [ N ] . The converse does not hold in general. More precisely every generic Λ-moduleis rigid if and only if Λ has type A n ( n ≤
4) (see [12]).
Example 7.3.
In type D , there is a one-parameter family of indecomposableΛ-modules with socle series and radical series S S ⊕ S ⊕ S S These modules are generic, but they are not rigid. For example there is a self-extension with middle term Q .
8. Dual Verma modules
Let g denote the Lie algebra of G with its triangular decomposition g = n ⊕ h ⊕ n − .Any G -module can also be regarded as a g -module. We shall denote by L ( λ ) theirreducible finite-dimensional module with highest weight λ . It is convenientlyconstructed as the unique top factor of the infinite-dimensional Verma g -module M ( λ ) (see e.g. [6, I, § U ( n − )-module, M ( λ ) is naturally isomorphicto U ( n − ), hence we have a natural projection U ( n ) ∼ = U ( n − ) ∼ = M ( λ ) → L ( λ ) forevery weight λ . Dualizing and taking into account that L ( λ ) is self-dual, we thusget an embedding L ( λ ) → M ( λ ) ∗ ∼ = C [ N ]. This embedding has a nice descriptionin terms of the functions ϕ M , as we shall now see.Let λ = P i ∈ I a i ̟ i be the decomposition of λ in terms of the fundamentalweights ̟ i . As L ( λ ) is finite-dimensional, the a i ’s are nonnegative integers. Set Q λ = ⊕ i ∈ I Q ⊕ a i i , an injective Λ-module. Theorem 8.1 ([13]).
In the above identification of M ( λ ) ∗ with C [ N ] , the irre-ducible representation L ( λ ) gets identified with the linear span of { ϕ M | M is a submodule of Q λ } . We refer to [13] for an explicit formula calculating the images of ϕ M ∈ L ( λ )under the action of the Chevalley generators e i and f i of g . reprojective algebras and cluster algebras Example 8.2.
In type A n , consider the fundamental representation L ( ̟ k ). Itis isomorphic to the natural representation of SL( n + 1 , C ) in ∧ k C n +1 . UsingExercise 5.6, we recover via Theorem 8.1 that L ( ̟ k ) can be identified with thesubspace of C [ N ] spanned by the k × k -minors taken on the first k rows of n . . . n ,n +1 . . . n ,n +1 ... ... . . . ...0 0 . . . Example 8.3.
In type D , consider the fundamental representation L ( ̟ ). It isisomorphic to the defining representation of G in C . If we realize N as a groupof 8 × L ( ̟ ) can be identified with the subspace of C [ N ] spannedby the coordinate functions mapping an 8 × x ∈ N to the entries of itsfirst row.
9. Parabolic subgroups and flag varieties
Let us fix a proper subset K of I . Denote by y i ( t ) ( i ∈ I, t ∈ C ) the one-parametersubgroups of G attached to the negatives of the simple roots. Let B be the Borelsubgroup of G containing N . The subgroup of G generated by B and the elements y k ( t ) ( k ∈ K, t ∈ C ) is called the standard parabolic subgroup attached to K . Weshall denote it by B K . In particular, B ∅ = B . It is known that every parabolicsubgroup of G is conjugate to a standard parabolic subgroup. The unipotentradical of B K will be denoted by N K . In particular, N ∅ = N . Example 9.1.
Let G = SL(5 , C ), a group of type A . We choose for B thesubgroup of upper triangular matrices. Take K = { , , } . Then B K and N K arethe subgroups of G with the following block form: B K = ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , N K = ∗ ∗ ∗ ∗ ∗ ∗ . Geometrically, N K is an affine space. It can be identified with an open cellin the partial flag variety B − K \ G , where B − K is the opposite parabolic subgroup(defined as B K but switching the x i ( t )’s and the y i ( t )’s). More precisely, therestriction to N K of the natural projection G → B − K \ G is an open embedding.6 C. Geiss, B. Leclerc and J. Schr¨oer
Example 9.2.
Let us continue Example 9.1. We have B − K = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ . Let ( u , . . . , u ) be the standard basis of C . We regard vectors of C as rowvectors and we let G act on the right on C , so that the k th row of the matrix g is u k g . Then B − K is the stabilizer of the 2-space spanned by u and u for theinduced transitive action of G on the Grassmann variety of 2-planes of C . Hence B − K \ G is the Grassmannian Gr(2 ,
5) of dimension 6.The unipotent subgroup N K can be identified with the open subset of Gr(2 , ,
2] does not vanish.
10. Coordinate rings
Let J be the complement of K in I . The partial flag variety B − K \ G can be naturallyembedded as a closed subset in the product of projective spaces Y j ∈ J P ( L ( ̟ j ))[24, p.123]. This is a generalization of the classical Pl¨ucker embedding of theGrassmannian Gr( k, n + 1) in P ( ∧ k C n +1 ) = P ( L ( ̟ k )). We denote by C [ B − K \ G ]the multi-homogeneous coordinate ring of B − K \ G coming from this embedding. LetΠ J ∼ = N J denote the monoid of dominant integral weights of the form λ = X j ∈ J a j ̟ j , ( a j ∈ N ) . Then, C [ B − K \ G ] is a Π J -graded ring with a natural G -module structure. Thehomogeneous component with multi-degree λ ∈ Π J is an irreducible G -modulewith highest weight λ . In other words, we have C [ B − K \ G ] = M λ ∈ Π J L ( λ ) . Moreover, C [ B − K \ G ] is generated by its subspace L j ∈ J L ( ̟ j ).In particular, C [ B − \ G ] = L λ ∈ Π L ( λ ), where the sum is over the monoid Πof all dominant integral weights of G . This is equal to the affine coordinate ring C [ N − \ G ] of the multi-cone N − \ G over B − \ G , that is, to the ring C [ N − \ G ] = { f ∈ C [ G ] | f ( n g ) = f ( g ) , n ∈ N − , g ∈ G } of polynomial functions on G invariant under N − . We will identify C [ B − K \ G ] withthe subalgebra of C [ N − \ G ] generated by the homogeneous elements of degree ̟ j ( j ∈ J ). reprojective algebras and cluster algebras Example 10.1.
We continue Example 9.2. The Pl¨ucker embedding of the Grass-mannian Gr(2 ,
5) consists in mapping the 2-plane V of C with basis ( v , v ) tothe line spanned by v ∧ v in ∧ C , which is isomorphic to L ( ̟ ).This induces an embedding of Gr(2 ,
5) into P ( L ( ̟ )). The homogeneous co-ordinate ring for this embedding is isomorphic to the subring of C [ G ] generatedby the functions g ∆ ij ( g ), where ∆ ij ( g ) denotes the 2 × g taken oncolumns i, j and on the first two rows. The ∆ ij are called Pl¨ucker coordinates. Asa G -module we have C [Gr(2 , M k ∈ N L ( k̟ ) , where the degree k homogeneous component L ( k̟ ) consists of the homogeneouspolynomials of degree k in the Pl¨ucker coordinates.Some distinguished elements of degree ̟ j in C [ N − \ G ] are the generalizedminors ∆ ̟ j ,w ( ̟ j ) , ( w ∈ W ) , (see [9, § N K in B − K \ G under the natural projection is the opensubset defined by the non-vanishing of the minors ∆ ̟ j ,̟ j ( j ∈ J ). Therefore theaffine coordinate ring of C [ N K ] can be identified with the subring of degree 0 ho-mogeneous elements in the localized ring C [ B − K \ G ][∆ − ̟ j ,̟ j , j ∈ J ]. Equivalently, C [ N K ] can be identified with the quotient of C [ B − K \ G ] by the ideal generated bythe elements ∆ ̟ j ,̟ j − j ∈ J ). Example 10.2.
We continue Example 10.1. The coordinate ring of C [ N K ] isisomorphic to the ring generated by the ∆ ij modulo the relation ∆ = 1. Thisdescription may seem unnecessarily complicated since N K is just an affine spaceof dimension 6 and we choose a presentation with 9 generators and the Pl¨uckerrelations. But these generators are better adapted to the cluster algebra structurethat we shall introduce.Let pr J : C [ B − K \ G ] → C [ N K ] denote the projection obtained by modding outthe ideal generated by the elements ∆ ̟ j ,̟ j − j ∈ J ). If C [ B − K \ G ] is identified asexplained above with a subalgebra of C [ G ], this map pr J is nothing else than therestriction of functions from G to N K . The restriction of pr J to each homogeneouspiece L ( λ ) ( λ ∈ Π J ) of C [ B − K \ G ] is injective and gives an embedding of L ( λ ) into C [ N K ]. Moreover, any f ∈ C [ N K ] belongs to pr J ( L ( λ )) for some λ ∈ Π J .Summarizing this discussion and taking into account Theorem 8.1, we get thefollowing description of C [ N K ], convenient for our purpose. Theorem 10.3 ([17]).
Let R J be the subspace of C [ N ] spanned by { ϕ M | M is a submodule of Q λ for some λ ∈ Π J } . The restriction to R J of the natural homomorphism C [ N ] → C [ N K ] (given byrestricting functions from N to N K ) is an isomorphism. C. Geiss, B. Leclerc and J. Schr¨oer
11. The category Sub Q J Set Q J = ⊕ j ∈ J Q j . Let Sub Q J denote the full subcategory of mod Λ whose objectsare the submodules of a direct sum of a finite number of copies of Q J . Example 11.1.
Type D . We have seen in Example 3.1 the structure of theindecomposable injective Q . It is easy to see that Q has only seven nonzerosubmodules S , S S , S S S , S S S , S ⊕ S S S , S S ⊕ S S S , S S S ⊕ S S S , which are all indecomposable. It turns out that Sub Q contains a unique otherindecomposable object, which is a submodule of Q ⊕ Q , and has the followingsocle series S S ⊕ S S S ⊕ S . Every object of Sub Q is a sum of copies of these eight indecomposable objects.Since Q J is an injective Λ-module, the category Sub Q J has good homologicalproperties [2]. In particular, it is closed under extensions, has enough injectives,enough projectives and almost split sequences. Moreover, the injectives coincidewith the projectives (it is a Frobenius category) and its stable category is a 2-Calabi-Yau triangulated category. Clearly, the algebra R J of Theorem 10.3 isnothing but the linear span of { ϕ M | M ∈ Sub Q J } . Hence we may regard Sub Q J as a kind of “categorification” of C [ N K ]. We aregoing to make this statement more precise by studying the rigid modules in Sub Q J . Theorem 11.2 ([17]).
Let T be a rigid module in Sub Q J . The number of pairwisenonisomorphic indecomposable direct summands of T is at most equal to dim N K . In the case when J = I , that is, Sub Q J = mod Λ, this result was first obtainedin [19].We shall say that a rigid module T in Sub Q J is complete if it has this maximalnumber of nonisomorphic summands. Note that in this case, T obviously containsthe n indecomposable injective objects of Sub Q J .In order to construct explicitly some complete rigid modules, we shall introducecertain functors. reprojective algebras and cluster algebras
12. The functors E i and E † i For i ∈ I , we define an endo-functor E i of mod Λ as follows. Given an object M ∈ mod Λ we define E i ( M ) as the kernel of the surjection M → S ⊕ m i ( M ) i , where m i ( M ) denotes the multiplicity of S i in the top of M . If f : M → N is ahomomorphism, f ( E i ( M )) is contained in E i ( N ), and we define E i ( f ) : E i ( M ) → E i ( N )as the restriction of f to E i ( M ). Clearly, E i is an additive functor. It acts on amodule M by removing the S i -isotypical part of its top. Similarly, one can definea functor E † i acting on M by removing the S i -isotypical part of its socle. Proposition 12.1 ([17]).
The functors E i , E † i ( i ∈ I ) satisfy the following relations: (i) E i E i = E i , E † i E † i = E † i . (ii) E i E j = E j E i , E † i E † j = E † j E † i , if i and j are not connected by an edge in ∆ . (iii) E i E j E i = E j E i E j , E † i E † j E † i = E † j E † i E † j , if i and j are connected by an edge. Relations (ii) and (iii) are the braid relations for ∆. It follows that for anyelement w of the Weyl group W of G , we have well-defined functors E w := E i · · · E i k , E † w := E † i · · · E † i k , where w = s i · · · s i k is an arbitrary reduced decomposition of w .Consider now the parabolic subgroup W K of W generated by the s k ( k ∈ K ).This is a finite Coxeter group. Let w K denote its longest element. One cancheck that E † w K ( M ) ∈ Sub Q J for every M ∈ mod Λ, and that E † w K ( M ) = M if M ∈ Sub Q J . In other words, the subcategory Sub Q J can be described as theimage of mod Λ under the endo-functor E † w K [17].
13. Construction of complete rigid modules
The relevance of these functors for constructing rigid modules comes from thefollowing property.
Proposition 13.1 ([17]).
The functors E w and E † w preserve rigid modules, i.e. if M is rigid then E w ( M ) and E † w ( M ) are also rigid. Let w be the longest element of W , and let w = s i · · · s i r be a reduceddecomposition such that the first r K factors form a reduced decomposition of w K .Set u ≤ p = s i · · · s i p , M p = E † u ≤ p ( Q i p ) , ( p = 1 , . . . , r ) . C. Geiss, B. Leclerc and J. Schr¨oer
For k ∈ K , let q k = max { q ≤ r K | i q = k } . Finally, define T = M r K +1 ⊕ M r K +2 ⊕ · · · ⊕ M r ⊕ ( ⊕ k ∈ K M q k ) ⊕ Q J . Theorem 13.2 ([17]). T is a complete rigid module in Sub Q J . Note that by construction the modules M p with p > r K are in the image of thefunctor E † w K , hence in Sub Q J . Note also that M q k = E † w K ( Q k ) for k ∈ K . Themodules M q k together with the summands of Q J are the indecomposable injectivesof Sub Q J . Finally, if t l = max { t ≤ r | i t = l } , then M t l = E † w ( Q l ) = 0 for every l ∈ I . It follows that T has in fact r − r K + | K | + | J | − | I | = r − r K = dim N K indecomposable summands, in agreement with Theorem 11.2. Example 13.3.
Type D . We take again K = { , , } , J = { } (remember thatthe central node of ∆ is labelled by 3). Here r K = 6. We choose the reduceddecomposition w = s s s s s s s s s s s s . We then have M s = M = S S S , M s = M = S S ⊕ S S S ⊕ S , M s = M = S S S ,M = S , M = S S , M = M = M = M = 0 . The module T = M ⊕ M ⊕ M ⊕ M ⊕ M ⊕ Q is complete rigid in Sub Q .
14. Cluster algebras of geometric type
Our next aim will be to associate to the category Sub Q J certain cluster algebras ofgeometric type. We refer the reader to [10, 11, 3] for a detailed exposition of theirproperties and of the motivating examples of coordinate rings of double Bruhatcells. Here we shall merely recall their definition.Let d and n be integers with d ≥ n ≥
0. If B = ( b ij ) is any d × ( d − n )-matrixwith integer entries, then the principal part of B is obtained by deleting from B thelast n rows. Given some k ∈ [1 , d − n ] define a new d × ( d − n )-matrix µ k ( B ) = ( b ′ ij )by b ′ ij = − b ij if i = k or j = k,b ij + | b ik | b kj + b ik | b kj | , where i ∈ [1 , d ] and j ∈ [1 , d − n ]. One calls µ k ( B ) the mutation of B in direction k .If B is an integer matrix whose principal part is skew-symmetric, then it is easyto check that µ k ( B ) is also an integer matrix with skew-symmetric principal part. reprojective algebras and cluster algebras A ( B ) as follows. Let F = C ( y , . . . , y d ) be the field of rational functions in d commuting variables y = ( y , . . . , y d ). One calls ( y , B ) the initial seed of A ( B ). For 1 ≤ k ≤ d − n define y ∗ k = Q b ik > y b ik i + Q b ik < y − b ik i y k . (5)Let µ k ( y ) denote the d -tuple obtained from y by replacing y k by y ∗ k . The pair( µ k ( y ) , µ k ( B )) is the mutation of the seed ( y , B ) in direction k .Now one can iterate this process and mutate again each seed ( µ k ( y ) , µ k ( B )) in d − n directions. The collection of all seeds obtained from the initial seed ( y , B )via a finite sequence of mutations is called the mutation class of ( y , B ). Each seedin this class consists of a d -tuple of algebraically independent elements of F calleda cluster and of a matrix called the exchange matrix . The elements of a cluster areits cluster variables . One does not mutate the last n elements of a cluster. Theyare called coefficients and belong to every cluster. The cluster algebra A ( B ) isby definition the subalgebra of F generated by the set of all the cluster variablesappearing in a seed of the mutation class of ( y , B ). The subring of A ( B ) generatedby the coefficients is called the coefficient ring . The integer d − n is called the rank of A ( B ). A monomial in the cluster variables is called a cluster monomial if all itsvariables belong to a single cluster. Example 14.1.
Take d = 7 and n = 5. Let B = −
11 0 − − −
10 1 . Then the mutation in direction 1 reads µ ( B ) = − − − −
10 1 , y ∗ = y y + y y y . In this simple example, it turns out that A ( B ) has only a finite number of clustervariables. In fact A ( B ) is isomorphic to the homogeneous coordinate ring of theGrassmannian Gr(2 ,
5) of 2-planes of C [11, § y , . . . , y to the following Pl¨ucker coordinates: y [1 , , y [1 , , y [1 , , y [2 , , y [3 , , y [4 , , y [1 , . C. Geiss, B. Leclerc and J. Schr¨oer
The other cluster variables obtained by mutation from this initial seed are theremaining Pl¨ucker coordinates [2 , , [2 , , [3 ,
15. Mutation of complete rigid modules
We shall now introduce an operation of mutation for complete rigid modules inSub Q J , inspired by the cluster algebra mutation of Fomin and Zelevinsky.Let T = T ⊕ · · · ⊕ T d be an arbitrary basic complete rigid module in Sub Q J .Thus the T i ’s are indecomposable and pairwise non isomorphic, and d = r − r K = ℓ ( w ) − ℓ ( w K ) = ℓ ( w K w ) . Assume that the injective summands of T are the last n ones, namely T d − n +1 , . . . , T d .Relying on the results of [14] we show in [17] the following Theorem 15.1.
Let k ≤ d − n . There exists a unique indecomposable module T ∗ k = T k in Sub Q J such that ( T /T k ) ⊕ T ∗ k is a basic complete rigid module in Sub Q J . Moreover, dim Ext ( T k , T ∗ k ) = 1 and if → T k g → X k f → T ∗ k → , → T ∗ k i → Y k h → T k → are the unique non-split short exact sequences between T k and T ∗ k , then f, g, h, i are minimal add( T /T k ) -approximations, and X k and Y k have no isomorphic inde-composable summands. In this situation, we say that (
T /T k ) ⊕ T ∗ k is the mutation of T in the directionof T k , and we write µ k ( T ) = ( T /T k ) ⊕ T ∗ k . Since X k and Y k belong to add( T ), wecan describe this mutation by means of the multiplicities of each indecomposablesummand of T in X k and Y k . This leads to associate to T a matrix of integers calledits exchange matrix encoding the mutations of T in all possible d − n directions.More precisely, define b ik = [ X k : T i ] if X k has summands isomorphic to T i , b ik = − [ Y k : T i ] if Y k has summands isomorphic to T i , and b ik = 0 otherwise. Notethat these conditions are disjoint because X k and Y k have no isomorphic directsummands. The d × ( d − n ) matrix B ( T ) = [ b ik ] is called the exchange matrix of T . We can now state: Theorem 15.2 ([14, 17]).
Let T = T ⊕ · · · ⊕ T d be a complete rigid module in Sub Q J as above, and let k ≤ d − n . Then B ( µ k ( T )) = µ k ( B ( T )) , where on the right-hand side µ k stands for the Fomin-Zelevinsky matrix mutation. In other words, our mutation of complete rigid modules induces at the level ofexchange matrices the Fomin-Zelevinsky matrix mutation. reprojective algebras and cluster algebras
16. Cluster algebra structure on C [ N K ] Let T be one of the complete rigid modules of §
13. Consider the mutation class R of T , that is, the set of all complete rigid modules of Sub Q J which can beobtained from T by a finite sequence of mutations. One can show that R containsall the rigid modules of §
13 corresponding to all possible choices of a reduceddecomposition of w K w , hence R does not depend on the choice of T .We can now project R on R J ∼ = C [ N K ] using the map M ϕ M . Moreprecisely, for U = U ⊕ · · · ⊕ U d ∈ R , let x ( U ) = ( ϕ U , . . . , ϕ U d ), (where again d = r − r K ). The next result follows from Theorem 15.1 and Theorem 6.1. Theorem 16.1. (i) { x ( U ) | U ∈ R} is the set of clusters of a cluster algebra A J ⊆ R J ∼ = C [ N K ] of rank d − n . (ii) The coefficient ring of A J is the ring of polynomials in the n variables ϕ L i ( i ∈ I ) , where the L i are the indecomposable injective objects in Sub Q J . (iii) All the cluster monomials belong to the dual semicanonical basis of C [ N ] ,and are thus linearly independent. The rigid modules T of §
13 project to initial seeds of the cluster algebra A J that we are going to describe.For i ∈ I and u, v ∈ W , let ∆ u ( ̟ i ) ,v ( ̟ i ) denote the generalized minor intro-duced by Fomin and Zelevinsky [9, § G . We shallmainly work with the restriction of this function to N , that we shall denote by D u ( ̟ i ) ,v ( ̟ i ) . It is easy to see that D u ( ̟ i ) ,v ( ̟ i ) = 0 unless u ( ̟ i ) is less or equal to v ( ̟ i ) in the Bruhat order, and that D u ( ̟ i ) ,u ( ̟ i ) = 1 for every i ∈ I and u ∈ W .It is also well known that D ̟ i ,w ( ̟ i ) is a lowest weight vector of L ( ̟ i ) in its re-alization as a subspace of C [ N ] explained in §
8. Therefore, using Theorem 8.1, weget ϕ Q i = D ̟ i ,w ( ̟ i ) , ( i ∈ I ) . More generally, it follows from [15, Lemma 5.4] that for u, v ∈ W we have ϕ E † u E v Q i = D u ( ̟ i ) ,vw ( ̟ i ) . Thus the elements of C [ N ] attached to the summands M p of the complete rigidmodule T of Theorem 13.2 are given by ϕ M p = D ̟ ip ,u ≤ p w ( ̟ ip ) , ( p ∈ { r K + 1 , . . . , r } ∪ { s k | k ∈ K } ) ,ϕ Q j = D ̟ j ,w ( ̟ j ) , ( j ∈ J ) . Moreover the matrix B ( T ) can also be described explicitly by means of a graphsimilar to those arising in the Chamber Ansatz of Fomin and Zelevinsky (see [17, § C. Geiss, B. Leclerc and J. Schr¨oer
Example 16.2.
We continue Example 13.3. We have ϕ M = D ̟ ,s s s s w ( ̟ ) , ϕ M = D ̟ ,s s s s s w ( ̟ ) ,ϕ M = D ̟ ,s s s s s s w ( ̟ ) , ϕ M = D ̟ ,s s s s s s s w ( ̟ ) ,ϕ M = D ̟ ,s s s s s s s w ( ̟ ) , ϕ Q = D ̟ ,w ( ̟ ) . It turns out that in the matrix realization of N given in Example 4.2 the generalizedminors above can be expressed as ordinary minors of the unitriangular 8 × x ∈ N . Indeed, denoting the matrix entries of x by n ij ( x ) one can check that ϕ M = n , ϕ M = (cid:12)(cid:12)(cid:12)(cid:12) n n n (cid:12)(cid:12)(cid:12)(cid:12) , ϕ M = n ,ϕ M = n , ϕ M = n , ϕ Q = n . The cluster variables of this seed are ϕ M and ϕ M . The exchange relations comefrom the following exact sequences0 → M → M → M ∗ → , → M ∗ → Q → M → , → M → M ⊕ M → M ∗ → , → M ∗ → M → M → , where M ∗ = S S ⊕ S S S M ∗ = S ⊕ S S S . The exchange matrix is therefore B ( T ) = − − −
11 0 where the rows are labelled by ( M , M , M , M , M , Q ) and the columns by( M , M ).A priori, we only have an inclusion of our cluster algebra A J in R J ∼ = C [ N K ],but we believe that Conjecture 16.3.
We have A J = R J . The conjecture is proved for G of type A n and of type D . It is also proved for J = { n } in type D n , and for J = { } in type D (see [17]). Moreover it followsfrom [18] that it is also true whenever w K w has a reduced expression adapted toan orientation of the Dynkin diagram. reprojective algebras and cluster algebras
17. Cluster algebra structure on C [ B − K \ G ] Let us start by some simple remarks. Consider the affine space C r and the pro-jective space P ( C r +1 ). The coordinate ring of C r is R = C [ x , . . . , x r ], and thehomogeneous coordinate ring of P ( C r +1 ) is S = C [ x , . . . , x r +1 ]. Moreover C r canbe regarded as the open subset of P ( C r +1 ) given by the non-vanishing of x r +1 .Given a hypersurface Σ ⊂ C r of equation f ( x , . . . , x r ) = 0 for some f ∈ R , itscompletion b Σ ⊂ P ( C r +1 ) is described by the equation b f ( x , . . . , x r +1 ) = 0, where b f is the homogeneous element of S obtained by multiplying each monomial of f by an appropriate power of x r +1 .Similarly, consider the open embedding N K ⊂ B − K \ G given by restricting thenatural projection G → B − K \ G to N K . By this embedding, N K is identifiedwith the subset of B − K \ G given by the simultaneous non-vanishing of the gen-eralized minors ∆ ̟ j ,̟ j ( j ∈ J ). To an element f ∈ C [ N K ] we can associate aΠ J -homogeneous element e f ∈ C [ B − K \ G ] by multiplying each monomial in f byan appropriate monomial in the ∆ ̟ j ,̟ j ’s. More precisely, using the notation of § e f is the homogeneous element of C [ B − K \ G ] with smallest degree such thatpr J ( e f ) = f .With this notation, we can now state the following result of [17]. Theorem 17.1. (i) { ] x ( U ) | U ∈ R} is the set of clusters of a cluster algebra e A J ⊆ C [ B − K \ G ] of rank d − n . (ii) The coefficient ring of e A J is the ring of polynomials in the n + | J | variables g ϕ L i ( i ∈ I ) and ∆ ̟ j ,̟ j ( j ∈ J ) . (iii) The exchange matrix e B attached to ] x ( U ) is obtained from the exchange ma-trix B of x ( U ) by adding | J | new rows (in the non-principal part) labelled by j ∈ J , where the entry in column k and row j is equal to b jk = dim Hom Λ ( S j , X k ) − dim Hom Λ ( S j , Y k ) . Here, if U k denotes the k th summand of U , U ∗ k its mutation, then X k , Y k arethe middle terms of the non-split short exact sequences → U k → X k → U ∗ k → , → U ∗ k → Y k → U k → . Example 17.2.
We continue Example 13.3 and Example 16.2. So we are in type D with K = { , , } and J = { } . The cluster ] x ( T ) consists of the 7 functions g ϕ M , g ϕ M , g ϕ M , g ϕ M , g ϕ M , g ϕ Q , ∆ ̟ ,̟ . The exchange matrix ] B ( T ) is obtained from the matrix B ( T ) of Example 16.2 byadding a new row labelled by the extra coefficient ∆ ̟ ,̟ . Sincedim Hom Λ ( S , M ) − dim Hom Λ ( S , Q ) = 2 − , C. Geiss, B. Leclerc and J. Schr¨oer dim Hom Λ ( S , M ⊕ M ) − dim Hom Λ ( S , M ) = 2 − , this new row is equal to [1 , ] B ( T ) = − − −
11 01 0 . Note that in this example, the variety B − K \ G is isomorphic to a smooth quadricin P ( C ). Its homogeneous coordinate ring C [ B − K \ G ] coincides with the affinecoordinate ring of the isotropic cone in C of the corresponding non-degeneratequadratic form. Thus we recover an example of §
2. The precise identification isvia the following formulas (see Exercise 5.7): y = ∆ ̟ ,̟ , y = g ϕ M , y = g ϕ M , y = g ϕ M , y = g ϕ M ,y = g ϕ M ∗ , y = g ϕ M ∗ , y = g ϕ Q , p = g ϕ M . Note that since C [ B − K \ G ] is generated by the y i (1 ≤ i ≤ e A J , we have in this case that e A J = C [ B − K \ G ].When J = { j } and G is of type A , B − K \ G is a Grassmannian and the clusteralgebra e A J coincides with the one defined by Scott in [34].When K = ∅ , C [ B − K \ G ] = C [ N − \ G ] and J = I the cluster algebra e A J isessentially the same as the one attached by Berenstein, Fomin and Zelevinsky tothe big cell of the base affine space N − \ G in [3, § upper cluster algebra, and they assume that the coefficients g ϕ L i = g ϕ Q i = ∆ ̟ i ,w ( ̟ i ) , ∆ ̟ i ,̟ i , ( i ∈ I ) , are invertible, i.e. the ring of coefficients consists of Laurent polynomials.Let Σ J be the multiplicative submonoid of e A J generated by the set { ∆ ̟ j ,̟ j | j ∈ J and ̟ j is not a minuscule weight } . Conjecture 17.3.
The localizations of e A J and C [ B − K \ G ] with respect to Σ J areequal. Note that if J is such that all the weights ̟ j ( j ∈ J ) are minuscule, then Σ J is trivial and the conjecture states that the algebras e A J and C [ B − K \ G ] coincidewithout localization. This is in particular the case for every J in type A n .The conjecture is proved for G of type A n and of type D . It is also provedfor J = { n } in type D n , and for J = { } in type D (see [17]). Note also thatConjecture 17.3 implies Conjecture 16.3. reprojective algebras and cluster algebras G J
Type of A J A n ( n ≥ { } — A n ( n ≥ { } A n − A n ( n ≥ { , } A n − A n ( n ≥ { , n } ( A ) n − A n ( n ≥ { , n − } A n − A n ( n ≥ { , , n } A n − A { , } D A { , , } D A { , , , } D A { } D A { , } E A { , } E A { , , } E A { } E A { , } E A { } E D n ( n ≥ { n } ( A ) n − D { , } A D { } A Table 1.
Algebras A J of finite cluster type.
18. Finite type classification
Recall that a cluster algebra is said to be of finite type if it has finitely manycluster variables, or equivalently, finitely many clusters. Fomin and Zelevinskyhave classified the cluster algebras of finite type [11], attaching to them a finiteroot system called their cluster type .Note that the clusters of A J and e A J are in natural one-to-one correspondence,and that the principal parts of the exchange matrices of two corresponding clustersare the same. This shows that A J and e A J have the same cluster type, finite orinfinite.Using the explicit initial seed described in §
16 it is possible to give a completelist of the algebras A J which have a finite cluster type [17]. The results are sum-marized in Table 1. Here, we label the vertices of the Dynkin diagram of type D n as follows: We have only listed one representative of each orbit under a diagram automor-phism. For example, in type A n we have an order 2 diagram automorphism map-8 C. Geiss, B. Leclerc and J. Schr¨oer ping J = { , } to J ′ = { n − , n } . Clearly, A J ′ has the same cluster type as J ,namely A n − .The classification when J = I (that is, in the case of C [ N ] or C [ B − \ G ]) wasgiven by Berenstein, Fomin and Zelevinsky [3]. The only finite type cases are A n ( n ≤ J = { j } is a singleton and G is of type A n (the Grassmannian Gr( j, n + 1)) was given by Scott [34]. When J = { } (theprojective space P ( C n +1 )), the cluster algebra is trivial, since every indecomposableobject of Sub Q is a relative projective.Note that if Sub Q J has finitely many isomorphism classes of indecomposableobjects then by construction A J has finite cluster type. The converse is also truealthough not so obvious. Indeed, if A J has finite cluster type, then by usingthe classification theorem of Fomin and Zelevinsky [11] there exists a completerigid object of Sub Q J whose endomorphism ring has a Gabriel quiver with stablepart of Dynkin type. Using a theorem of Keller and Reiten [23], it follows thatthe stable category Sub Q J is triangle equivalent to a cluster category of Dynkintype, hence Sub Q J has finitely many indecomposable objects. Therefore the aboveclassification is also the classification of all subcategories Sub Q J with finitely manyindecomposable objects.
19. Canonical bases, total positivity and open pro-blems
Since we started this survey with a discussion of total positivity and canonicalbases, it is natural to ask if the previous constructions give a better understandingof these topics.So let B and S denote respectively the dual canonical and dual semicanonicalbases of C [ N ]. We have seen (see §
7) that for every rigid Λ-module M , the function ϕ M belongs to S . Conjecture 19.1.
For every rigid Λ -module M , the function ϕ M belongs to B . The conjecture holds in type A n ( n ≤
4) [12], that is, when Λ has finite repre-sentation type. In this case one even has B = S .As explained in §
8, each finite-dimensional irreducible G -module L ( λ ) has acanonical embedding in C [ N ]. It is known that the subsets B ( λ ) = B ∩ L ( λ ) , S ( λ ) = S ∩ L ( λ ) , are bases of L ( λ ). Using the multiplicity-free decomposition (see § C [ B − K \ G ] = M λ ∈ Π J L ( λ ) , we therefore obtain a dual canonical and a dual semicanonical basis of C [ B − K \ G ]: B J = ∪ λ ∈ Π J B ( λ ) , S J = ∪ λ ∈ Π J S ( λ ) . reprojective algebras and cluster algebras e A J ⊂ C [ B − K \ G ] belong to S J . Conjecture 19.1 would imply that they also belongto B J . In particular, when e A J has finite cluster type, B J should be equal to theset of cluster monomials.Regarding total positivity, we propose the following conjecture, inspired byFomin and Zelevinsky’s approach to total positivity via cluster algebras. Let X denote the partial flag variety B − K \ G and let X > be the totally positive part of X [30]. Lusztig has shown that it can be defined by dim L ( λ ) algebraic inequalitiesgiven by the elements of B ( λ ) for a “sufficiently large” λ ∈ Π J [30, Th. 3.4]. Infact X > ⊂ N K , where N K is embedded in X as in §
17. We propose the followingalternative descriptions of X > by systems of d = dim X algebraic inequalities. Conjecture 19.2.
Let T = T ⊕ · · · ⊕ T d be a basic complete rigid Λ -module in Sub Q J . Then x ∈ N K belongs to X > if and only if ϕ T i ( x ) > , ( i = 1 , . . . , d ) . Example 19.3.
We consider again type D and J = { } , so that X can beidentified with the Grassmannian of isotropic lines in C , as in §
2. In this caseΠ J = N ̟ and Lusztig’s description involves dim L ( λ ) inequalities where λ = k̟ with k ≥
5. For example dim L (5 ̟ ) = 672.On the other hand the category Sub Q has 4 basic complete rigid modules: M ⊕ M ⊕ M ⊕ M ⊕ M ⊕ Q , M ⊕ M ⊕ M ⊕ M ∗ ⊕ M ⊕ Q ,M ⊕ M ⊕ M ⊕ M ⊕ M ∗ ⊕ Q , M ⊕ M ⊕ M ⊕ M ∗ ⊕ M ∗ ⊕ Q , where we have used the notation of Examples 13.3 and 16.2. Each of them gives riseto a positivity criterion consisting of dim X = 6 inequalities. Using the notationof §
2, these are respectively y > , p > , y > , y > , y > , y > y > , p > , y > , y > , y > , y > y > , p > , y > , y > , y > , y > y > , p > , y > , y > , y > , y > . Note that since we regard X > as a subset of N K , the additional relation y = 1is understood. Thus the conjecture holds in this case, and more generally in type D n when J = { n } .When X = B − \ G and M is a rigid module in R , the conjecture follows fromthe work of Berenstein, Fomin and Zelevinsky and our construction. When X is atype A Grassmannian and M is a rigid module in R , the conjecture follows fromthe work of Scott [34] and our construction.0 C. Geiss, B. Leclerc and J. Schr¨oer
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