Cremmer–Gervais cluster structure on SLn SPECIAL FEATURE
Michael Gekhtmana, Michael Shapirob, and Alek Vainshteinc,1
aDepartment of Mathematics, University of Notre Dame, Notre Dame, IN 46556, bDepartment of Mathematics, Michigan State University, East Lansing, MI 48823, and cDepartment of Mathematics and Department of Computer Science, University of Haifa, Haifa, Mount Carmel 31905, Israel
Edited by Bernard Leclerc, University of Caen, Caen, France, and accepted by the Editorial Board December 19, 2013 (received for review August 13, 2013)
We study natural cluster structures in the rings of regular functions Cremmer–Gervais case. Our result allows equipping SLn, GLn, on simple complex Lie groups and Poisson–Lie structures compat- and the affine space Matn of n × n matrices with an alternative ible with these cluster structures. According to our main conjec- cluster structure, CCG. ture, each class in the Belavin–Drinfeld classification of Poisson–Lie In the first section below, we collect the necessary information structures on G corresponds to a cluster structure in OðGÞ. We have on cluster algebras, compatible Poisson brackets, and the toric shown before that this conjecture holds for any G in the case of the action. In the next section, we formulate the main conjecture standard Poisson–Lie structure and for all Belavin–Drinfeld classes from ref. 1, present the definition of the Cremmer–Gervais in SLn, n < 5. In this paper we establish it for the Cremmer–Gervais Poisson bracket, and formulate our main result. We introduce – SL Poisson Lie structure on n, which is the least similar to the stan- the cluster structure CCG and outline the proof of the main SL dard one. Besides, we prove that on 3 the cluster algebra and the theorem by breaking it into a series of intermediate results about upper cluster algebra corresponding to the Cremmer–Gervais cluster CCG. In the following section, we discuss the relation between structure do not coincide, unlike the case of the standard cluster cluster algebras and upper cluster algebras on SLn. In the stan- structure. Finally, we show that the positive locus with respect to dard case these two algebras coincide. We show that for the the Cremmer–Gervais cluster structure is contained in the set of Cremmer–Gervais cluster structure on SL3 this is not the case. totally positive matrices. The next section treats positivity for the exotic cluster structure C . Finally, in the last section we formulate several directions – – CG Poisson Lie group | Belavin Drinfeld triple for future research.
n ref. 1 we initiated a systematic study of multiple cluster Cluster Structures and Compatible Poisson Brackets MATHEMATICS Istructures in the rings of regular functions on simple Lie groups We start with the basics on cluster algebras of geometric type. following an approach developed and implemented in refs. 2–4 The definition that we present below is not the most general one, for constructing cluster structures on algebraic Poisson varieties. see, e.g., refs. 5, 7 for a detailed exposition. In what follows, we It is based on the notion of a Poisson bracket compatible with a will use a notation ½i; j for an interval fi; i + 1; ...; jg in N, and we cluster structure. The key point is that if an algebraic Poisson will denote ½1; n by ½n . variety ðM; f·;·gÞ possesses a coordinate chart that consists of The coefficient group P is a free multiplicative abelian group of regular functions whose logarithms have pairwise constant Poisson finite rank m with generators g1; ...; gm.Anambient field is the brackets, then one can try to use this chart to define a cluster field F of rational functions in n independent variables with ·;· structure CM compatible with f g. Algebraic structures cor- coefficients in the field of fractions of the integer group ring responding to (the cluster algebra and the upper cluster al- ± ± ± − CM ZP = Z½ g 1; ...; g 1 (here we write x 1 instead of x; x 1). gebra) are closely related to the ring OðMÞ of regular functions 1 m ~ A seed (of geometric type)inF is a pair Σ = ðx; BÞ, where on M. In fact, under certain rather mild conditions, OðMÞ can x = ðx ; ...; x Þ is a transcendence basis of F over the field of be obtained by tensoring one of these algebras with C. 1 n fractions of Z and B~ is an n × n + m integer matrix whose This construction was applied in ref. 4, Ch. 4.3 to double P ð Þ Bruhat cells in semisimple Lie groups equipped with (the re- principal part B is skew-symmetrizable (recall that the principal part of a rectangular matrix is its maximal leading square sub- striction of) the standard Poisson–Lie structure. It was shown ~ that the resulting cluster structure coincides with the one built in matrix). Matrices B and B are called the exchange matrix and the ref. 5. Recall that it was proved in ref. 5 that the corresponding extended exchange matrix, respectively. upper cluster algebra is isomorphic to the ring of regular func- tions on the double Bruhat cell. Because the open double Bruhat Significance cell is dense in the corresponding Lie group, one can equip the ring of regular functions on the Lie group with the same cluster Coexistence of diverse mathematical structures supported on structure. The standard Poisson–Lie structure is a particular case the same variety leads to deeper understanding of its features. of Poisson–Lie structures corresponding to quasitriangular Lie If the manifold is a Lie group, endowing it with a Poisson bialgebras. Such structures are associated with solutions to the structure that respects group multiplication (Poisson–Lie struc- classical Yang–Baxter equation. Their complete classification ture) is instrumental in a study of classical and quantum- was obtained by Belavin and Drinfeld in ref. 6. In ref. 1 we mechanical systems with symmetries. In turn, a Poisson struc- conjectured that any such solution gives rise to a compatible ture on a variety can be compatible with a cluster structure— cluster structure on the Lie group and provided several examples a useful combinatorial tool that organizes generators of the ring supporting this conjecture by showing that it holds true for the of regular functions into a collection of overlapping clusters class of the standard Poisson–Lie structure in any simple com- connected via rational transformations. We conjectured that this plex Lie group, and for the whole Belavin–Drinfeld classification is the case for an important class of Poisson–Lie groups. The in SLn for n = 2; 3; 4. We call the cluster structures associated paper verifies this conjecture for a group of invertible matrices with the nontrivial Belavin–Drinfeld data exotic. equipped with a nonstandard Poisson–Lie structure. In this paper, we outline the proof of the conjecture of ref. 1 in the case of the Cremmer–Gervais Poisson structure on SLn.We Author contributions: M.G., M.S., and A.V. performed research and wrote the paper. chose to consider this case because, in a sense, the Poisson The authors declare no conflict of interest. structure in question differs the most from the standard: the This article is a PNAS Direct Submission. B.L. is a guest editor invited by the Editorial discrete data featured in the Belavin–Drinfeld classification are Board. trivial in the standard case and have the “maximal size” in the 1To whom correspondence should be addressed. Email: [email protected].
www.pnas.org/cgi/doi/10.1073/pnas.1315283111 PNAS Early Edition | 1of8 Downloaded by guest on October 3, 2021 The n-tuple x is called a cluster, and its elements x1; ...; xn are counterparts in F C obtained by extension of scalars; they are called cluster variables. Denote xn+i = gi for i ∈ ½m . We say that denoted AC and AC. If, moreover, the field isomorphism φ can ~x = ðx1; ...; xn+mÞ is an extended cluster, and xn+1; ...; xn+m are be restricted to an isomorphism of AC (or AC) and OðVÞ, we say stable variables. It is convenient to think of F as of the field of that AC (or AC)isnaturally isomorphic to OðVÞ. rational functions in n + m independent variables with rational Let f·;·g be a Poisson bracket on the ambient field F, and C be coefficients. a cluster structure in F. We say that the bracket and the In what follows, we will only deal with the case when the exchange cluster structure are compatible if, for any extended cluster matrix is skew-symmetric. In this situation the extended ex- ~x = ðx1; ...; xn+mÞ, one has change matrix can be conveniently represented by a quiver = ~ ; ...; + Q QðBÞ. It is a directed graph on the vertices 1 n m cor- xi; xj = ωij xi xj; [2] responding to all variables; the vertices corresponding to stable > ~ variables are called frozen.Eachentrybij 0ofthematrixB where ωij ∈ Z are constants for all i; j ∈ ½n + m .Thematrix ~x gives rise to bij edges going from the vertex i to the vertex j; each Ω = ðωijÞ is called the coefficient matrix of f·;·g (in the basis ~ ~x such edge is denoted i → j. Clearly, B can be restored uniquely ~x); clearly, Ω is skew-symmetric. The notion of compatibility from Q. extends to Poisson brackets on F C without any changes. Given a seed as above, the adjacent cluster in direction k ∈ ½n is A complete characterization of Poisson brackets compatible ~ ~ defined by with a given cluster structure C = CðBÞ in the case rank B = n is S given in ref. 2; see also ref. 4, Ch. 4. A different description of x = ðxnfx gÞ fx′g; k k k compatible Poisson brackets on F C is based on the notion of a ~x = ; ...; ′ toric action. Fix an arbitrary extended cluster ðx1 xn+mÞ where the new cluster variable xk is given by the exchange relation W and define a local toric action of rank r as the map T d : F C → F C − givenonthegeneratorsofF C = Cðx ; ...; x + Þ by the formula ′ = ∏ bki + ∏ bki ; [1] 1 n m xkxk xi xi 1≤i≤n+m 1≤i≤n+m r n+m b >0 b <0 ki ki W wiα p r T d ~x = xi ∏ dα ; d = ðd1; ...; drÞ ∈ ðC Þ ; [3] α=1 i=1 here, as usual, the product over the empty set is assumed to be equal to 1. where W = ðw αÞ is an integer ðn + mÞ × r weight matrix of full ~ ~ i We say that B′ is obtained from B by a matrix mutation in rank, and extended naturally to the whole F C. ~′ = μ ~ ~x′ direction k and write B kðBÞ if Let be another extended cluster, then the corresponding 8 local toric action defined by the weight matrix W′ is compatible <> −bij; if i = k or j = k; with the local toric action 3 if it commutes with the cluster ′ = transformation that takes ~x to ~x′. If local toric actions at all bij > jb jb + b b : + ik kj ik kj ; : clusters are compatible, they define a global toric action T d on bij otherwise ~ 2 F C called the extension of the local toric action 3. If rank B = n, μ μ ~ = ~ then the maximal possible rank of a global toric action equals m. It can be easily verified that kð kðBÞÞ B. Σ = x; ~ Σ′ = x′; ~′ Any global toric action can be obtained from a toric action of the Given a seed ð BÞ, we say that a seed ð B Þ is ad- maximal rank by setting some of the values of d equal to 1. jacent to Σ (in direction k)ifx′ is adjacent to x in direction k and i ~′ = μ ~ Main Results B kðBÞ. Two seeds are mutation equivalent if they can be connected by a sequence of pairwise adjacent seeds. The set of Let G be a Lie group equipped with a Poisson bracket f·;·g. Σ all seeds mutation equivalent to is called the cluster structure G is called a Poisson–Lie group if the multiplication map (of geometric type) in F associated with Σ and denoted by CðΣÞ; ~ in what follows, we usually write CðBÞ, or even just C instead. G × G ∋ðx; yÞ # xy ∈ G ~ Following refs. 5, 7, we associate with CðBÞ two algebras of rank n over the ground ring A, Z ⊆ A ⊆ ZP: the cluster algebra is Poisson. The tangent Lie algebra g of a Poisson–Lie group G ~ A = AðCÞ = AðBÞ, which is the A-subalgebra of F generated by has a natural structure of a Lie bialgebra. We are interested in the ~ all cluster variables in all seeds in CðBÞ, and the upper cluster case when G is a simple complex Lie group and its tangent Lie ~ algebra A = AðCÞ = AðBÞ, which is the intersection of the rings of bialgebra is factorizable. Laurent polynomials over A in cluster variables taken over all A factorizable Lie bialgebra structure on a complex simple Lie ~ seeds in CðBÞ. The famous Laurent phenomenon (ref. 8) claims algebra can be described in terms of a classical R-matrix. This is the inclusion AðCÞ ⊆ AðCÞ. The natural choice of the ground ring an element r ∈ g ⊗ g that satisfies the classical Yang–Baxter for the geometric type is the polynomial ring in stable variables equation and an additional condition that r + r21 defines an in- 21 A = ZP+ = Z½xn+1; ...; xn+m ; this choice is assumed unless ex- variant nondegenerate inner product on g. (Here r is obtained plicitly stated otherwise. from r by switching factors in tensor products.) Classical R- Let V be a quasiaffine variety over C, CðVÞ be the field of matrices were classified, up to an automorphism, by Belavin and rational functions on V, and OðVÞ be the ring of regular func- Drinfeld in ref. 6. Let h be a Cartan subalgebra of g, Φ be the + tions on V. Let C be a cluster structure in F as above. Assume root system associated with g, Φ be the set of positive roots, and Δ ⊂ Φ+ that f f1; ...; fn+mg is a transcendence basis of CðVÞ. Then the be the set of positive simple roots. The Killing form on g ; – = Γ ; Γ ; γ map φ : xi # fi,1≤ i ≤ n + m, can be extended to a field iso- is denoted by h i.ABelavin Drinfeld triple T ð 1 2 Þ consists morphism φ : F C → CðVÞ, where F C = F ⊗ C is obtained from F of two subsets Γ1; Γ2 of Δ and an isometry γ : Γ1 → Γ2 nilpotent in by extension of scalars. The pair ðC; φÞ is called a cluster struc- the following sense: for every α ∈ Γ1 there exists m ∈ N such that γj α ∈ Γ = ; ...; − γm α ∉ Γ γ ture in CðVÞ (or just a cluster structure on V); ff1; ...; fn+mg is ð Þ 1 for j 0 m 1, but ð Þ 1. The isometry called an extended cluster in ðC; φÞ. Sometimes we omit direct extends in a natural way to a map between root systems Φ1; Φ2 indication of φ and say that C is a cluster structure on V. A cluster generated by Γ1; Γ2. This allows one to define a partial ordering ; φ φ Φ : α β β = γj α ∈ N = structure ðC Þ is called regular if ðxÞ is a regular function for on T if ð Þ for some j ,andtosethT any cluster variable x. The two algebras defined above have their fh ∈ h : αðhÞ = βðhÞ if α T βg.
2of8 | www.pnas.org/cgi/doi/10.1073/pnas.1315283111 Gekhtman et al. Downloaded by guest on October 3, 2021 To each T there corresponds a set RT of classical R-matrices ½n−i+1;n θ = ; ∈ − ; SPECIAL FEATURE that we call the Belavin-Drinfeld class corresponding to T.Two iðXÞ det X½n−i+1;n i ½n 1 + − + ; + R-matrices in the same Belavin–Drinfeld class RT differ by an φ ; = ; ½kðn 1Þ p 1 kðn 1Þ ; ∈ ; [5] pðX YÞ det UðX YÞ − + ; p ½N h ⊗ h ½N p 1 N element from satisfying a linear relation specified by T. + − + ; + + ·;· – ψ ; = ; ½kðn 1Þ q 2 kðn 1Þ 1 ; ∈ : We denote by f gr the Poisson Lie bracket associated with qðX YÞ det UðX YÞ½N−q+1;N q ½M r ∈ RT . Given a Belavin–Drinfeld triple T for G, define the torus = = − + HT = exp h ⊂ G. Note that the dimension of the torus equals In the last family, M N if n is even and M N n 1ifn T k = ΔnΓ . is odd. T 1 Theorem 3. θ φ ; ψ ; In ref. 1 we conjectured that there exists a classification of The functions iðXÞ, pðX XÞ, qðX XÞ form a log- – regular cluster structures on G that is completely parallel to the canonical family with respect to the Cremmer Gervais bracket. – Consequently, we choose family 5 restricted to X = Y as an Belavin Drinfeld classification. φ Conjecture 1. initial (extended) cluster for CCGðnÞ. Furthermore, functions N Let G be a simple complex Lie group. For any ψ and M are the only stable variables for CCGðnÞ. Belavin–Drinfeld triple T = ðΓ1; Γ2; γÞ there exists a cluster structure ; φ Let us explain the motivation behind this choice and the in- ðCT T Þ on G such that tuition it provides for constructing the initial cluster for CCGðnÞ (i) the number of stable variables is 2kT , and the corresponding presented above. To this end, we first need to recall the notion extended exchange matrix has a full rank; of the Drinfeld double DðGÞ of a factorizable Poisson–Lie group ; φ ; ·;· – (ii) ðCT T Þ is regular, and the corresponding upper cluster algebra ðG f grÞ. The double DðGÞ is endowed with a Poisson Lie ; ; ACðCT Þ is naturally isomorphic to OðGÞ; structure associated with the Manin triple ðDðgÞ gd grÞ.Here p 2kT = ⊕ = +_ (iii) the global toric action of ðC Þ on CðGÞ is generated by the DðgÞ g g gd gr is equipped with the invariant nondegen- ξ; η ; ξ′; η′ = ξ; ξ′ − η; η′ action of HT × HT on G given by ðH1; H2ÞðXÞ = H1XH2; erate inner product hhð Þ ð Þii h i h i, an isotro- ∈ ·;· g g g (iv) for any r RT , f gr is compatible with CT ; pic subalgebra d is the image of in Dð Þ under the diagonal (v) a Poisson–Lie bracket on G is compatible with CT only if it is a embedding, and gr is an isotropic subalgebra of DðgÞ given by ·;· ∈ = + ξ ; − ξ : ξ ∈ ± ∈ scalar multiple of f gr for some r RT . gr fðR ð Þ R ð ÞÞ gg, where R End g are defined by hR+η; ζi = − hR−ζ; ηi = hr; η ⊗ ζi. The embedding g↪DðgÞ whose The Belavin–Drinfeld data (triple, class) are said to be trivial if σ ; ·;· image is gr is denoted by . The group ðG f grÞ becomes a Γ1 = Γ2 = ∅. In this case, HT = H is the Cartan subgroup in G. – Poisson Lie subgroup of DðGÞ under the diagonal embedding. MATHEMATICS – The resulting Poisson bracket is called the standard Poisson Lie Another Poisson–Lie subgroup of DðGÞ is the group Gr whose Lie × structure on G. Conjecture 1 in this case was verified in ref. 1. algebra is gr. Orbits of the two-sided action of Gr Gr on DðGÞ In this paper we consider the case G = SLn and the Belavin– play an important role in the description of symplectic leaves of Drinfeld data that are “the farthest” from the trivial data, DðGÞ and G (see refs. 9 and 10). This action is relevant to our namely, Γ = fα ; ...; α − g, Γ = fα ; ...; α − g,andγðα Þ = α − tackling of Conjecture 1 due to the following observation. Let f be 1 2 n 1 2 1 n 2 i i 1 × for i = 2; ...; n − 1. The resulting Poisson–Lie bracket on SL is a semiinvariant of the two-sided action of Gr Gr, that is, for any n ; ∈ ; ∈ ; = ξ ; ξ called the Cremmer–Gervais bracket. The main result of this g1 g2 Gr and ðX YÞ DðGÞ, fðg1ðX YÞg2Þ lðg1ÞfðX YÞ rðg2Þ, where ξ ; ξ are two characters of G trivial on elements of the paper is l r r form exp σ eα for any root vector eα. Then the Hamiltonian Theorem 2. Conjecture 1 is valid for the Cremmer–Gervais ð ð ÞÞ flow generated by fðX; XÞ on G with respect to f·;·g is given by Poisson–Lie structure. r XðtÞ = expðth ÞXð0Þexpðth Þ, where h ; h are two elements of the Theorem 2 is proved by producing a cluster structure C = l r l r CG Cartan subalgebra. CCGðnÞ that possesses all of the needed properties. In fact, we This observation makes regular functions on G obtained by × will construct a cluster structure in the space Matn of n n ma- restriction to the diagonal subgroup of semiinvariants described trices compatible with a natural extension of the Cremmer–Gervais above natural candidates for stable variables in the cluster Poisson bracket and derive the required properties of CCG from structure we are trying to construct. Indeed, if y is a stable var- similar features of the latter cluster structure. In what follows we iable in a cluster algebra admitting a compatible Poisson bracket ·;· ; = use the same notation CCG for both cluster structures and in- f g, then, for any cluster variable x, flog y xg cxx for some ∈ Z dicate explicitly which one is meant when needed. cx , and thus the Hamiltonian flow generated by log y gives = To describe the initial cluster for CCG, we need to introduce rise to a global toric action T tðxÞ x expðcxtÞ. On the other hand, ... some notation. For a matrix A, we denote by Aj1 jm its submatrix part (iii)ofConjecture 1 predicts that the global toric action in i1...il formed by rows i ; ...; i and columns j ; ...; j . If all rows (re- CCGðnÞ is induced by the right and left action on G by elements of 1 l 1 m – spectively, columns) of A are involved, we will omit the lower the subgroup HT of the Cartan group specified by the Belavin Drinfeld data. (respectively, upper) list of indices. For any two n × n matrices The stable variables for C ðnÞ were constructed using the X, Y,denotebyX and Y two ðn − 1Þ × ðn + 1Þ matrices CG strategy indicated above: The action ðX; YÞ # g1ðX; YÞg2 trans- lates into a transformation UðX; YÞ#Mlðg1ÞUðX; YÞMrðg2Þ,where Mlðg1Þ, Mrðg2Þ are certain invertible block diagonal matrices. This X = X½2;n 0 ; Y = 0 Y½1;n−1 : φ ; ψ ; allows identification of N ðX YÞ and M ðX YÞ as semiinvariants, and also hints at a possibility of constructing an initial cluster = n + 1 = − − × + + Put k b 2 c and N kðn 1Þ. Define a kðn 1Þ ðk 1Þðn 1Þ consisting of minors of UðX; YÞ. It is this intuition that eventually matrix led us to formulation of Theorem 3. Moreover, the proof of the theorem relies on computation of Poisson brackets in the double. 2 3 Next, we need to describe the quiver that corresponds to the YX 0 ⋯ 0 initial cluster above. In fact, it will be convenient to do this in the 6 ⋯ 7 ; = 6 0 YX 0 7: [4] Matn rather than SLn situation, in which case the initial cluster is UðX YÞ 4 ⋱⋱⋱ 5 0 0 augmented by the addition of one more stable variable, θnðXÞ = 0 ⋯ 0 YX det X. The Cremmer–Gervais Poisson structure is extended to Matn by requiring that θnðXÞ is a Casimir function. We denote Define three families of functions via the quiver corresponding to the augmented initial cluster by
Gekhtman et al. PNAS Early Edition | 3of8 Downloaded by guest on October 3, 2021 2 QCGðnÞ. Its vertices are n nodes of the n × n rectangular grid Theorem 5. The cluster structure CCG is regular. indexed by pairs ði; jÞ, i; j ∈ ½n ,withi increasing top to bottom The proof of Theorem 5 relies on Dodgson-type identities and j increasing left to right. Before describing edges of the applied to submatrices of UðX; YÞ while taking into account its quiver, let us explain the correspondence between the cluster shift-invariance properties. As a corollary, we get parts (iii) and variables and the vertices of QCGðnÞ. (v)ofConjecture 1. For any cluster or stable variable f in the augmented initial By Theorem 5, the upper cluster algebra ACG = ACðCCGÞ is a cluster, consider the upper-left matrix entry of the submatrix of subalgebra in the ring of regular functions. To complete the UðX; XÞ (or X) associated with this variable. This matrix entry proof of Theorem 2, it remains to establish the opposite in- ; ∈ is xij for some i j ½n . Thus, we define a correspondence clusion, which will settle part (ii)ofConjecture 1. The proof ρ : f↔ði; jÞ ∈ ½n × ½n .Wehaveprovedthatρ is a one-to-one relies on induction on n. Its main ingredient is a construction correspondence between the augmented initial cluster and of two distinguished sequences of cluster transformations. The × ½n ½n . first sequence, S, followed by freezing some of the cluster vari- Let us assign each cluster variable f to the vertex indexed by ables and localization at a single cluster variable φ − ðXÞ, leads ði; jÞ = ρð fÞ. In particular, the stable variable θ is assigned to n 1 n to a map ζ : Mat X : φ − X = 0 → Mat − that “respects” ; φ ; nnf n 1ð Þ g n 1 vertex ð1 1Þ, the stable variable N is assigned to vertex ð2 1Þ if n – ζ ; ψ the Cremmer Gervais cluster structure. The map is needed to is odd and ð1 nÞ if n is even, and the stable variable M is perform an induction step. However, because of the localization assigned to vertex 1; n if n is odd and 2; 1 if n is even. ð Þ ð Þ mentioned above, we also need a second sequence T of trans- Now, let us describe the arrows of Q ðnÞ. There are hori- CG formations that can be viewed as a cluster-algebraic realization zontal arrows ði; j + 1Þ → ði; jÞ for all i ∈ ½n ; j ∈ ½n − 1 except ði; jÞ = of the anti-Poisson involution X # W XW on Mat equipped ð1; n − 1Þ; vertical arrows ði + 1; jÞ → ði; jÞ for all i ∈ ½n − 1 ; j ∈ ½n 0 0 n with the Cremmer–Gervais Poisson bracket (here W is the n × n except ði; jÞ = ð1; 1Þ; diagonal arrows ði; jÞ → ði + 1; j + 1Þ for all 0 ; ∈ − matrix corresponding to the longest permutation). This allows i j ½n 1 . In addition, there are arrows between vertices of the ζ first and the last rows: ðn; jÞ → ð1; jÞ and ð1; jÞ → ðn; j + 1Þ for one to apply to W0XW0 as well and then invoke certain gen- j ∈ ½2; n − 1 ; and arrows between vertices of the first and the eral properties of cluster algebras to fully use the induction assumption. last columns: ði; nÞ → ði + 2; 1Þ and ði + 2; 1Þ → ði + 1; nÞ for i ∈ ^ Denote by Q ðnÞ the quiver obtained by adding to QCGðnÞ ½1; n − 2 . This concludes the description of QCGðnÞ. The quiver CG ′ ; → ; ; → ; Q ðnÞ that corresponds to the SL case is obtained from Q ðnÞ two additional arrows: ð1 1Þ ðn 2Þ and ðn 1Þ ð1 1Þ. CG n CG Theorem 6. by deleting the vertex ð1; 1Þ and erasing all arrows incident to this There exists a sequence S of cluster transformations in C ðnÞ such that SðQ ðnÞÞ contains a subquiver isomorphic to vertex. Quiver QCGð5Þ is shown in Fig. 1. ^CG CG Theorem 4. ′ Q ðn − 1Þ and cluster variables indexed by vertices of this sub- (i) The quivers QCGðnÞ and QCGðnÞ define cluster CG structures compatible with the Cremmer–Gervais Poisson structure quiver satisfy on Matn and SLn, respectively. « = φ ij ζ ; ; ∈ − ; (ii) The corresponding extended exchange matrices are of full rank. Sð fÞijðXÞ n−1ðXÞ fijð ðXÞÞ i j ½n 1 Note that kT = 1 in the Cremmer–Gervais case, and hence the corresponding Belavin–Drinfeld class contains a unique R-matrix. where «ij = 1 if ði; jÞ is a ψ-vertex in QCGðn − 1Þ and «ij = 0 otherwise. w0 Therefore, Theorem 4 establishes parts (i) and (iv)ofConjecture 1 For any function g on Matn define g ðXÞ = gðW0XW0Þ. Be- in the Cremmer–Gervais case. sides, for any quiver Q denote by Qopp the quiver obtained from Another property of the cluster structure CCG is given by the Q by reversing all arrows. following theorem. Theorem 7. There exists a sequence T of cluster transformations in opp CCG such that TðQCGðnÞÞ is isomorphic to QCGðnÞ and cluster variables indexed by vertices of TðQCGðnÞÞ satisfy
= w0 : TðfÞijðXÞ fij ðXÞ
Each cluster transformation in the sequences S and T can be written as a Dodgson-type identity for minors of an augmenta- tion of the matrix UðX; XÞ. To complete the proof of Theorem 2, it remains to show that functions xij ∈ OðMatnÞ belong to the upper cluster algebra ACGðnÞ, which is done by induction on n that relies on Theorems 6 and 7. The details of the proofs can be found in ref. 11.
Relation Between ACG and ACG
Recall that the standard cluster algebra on SLn, that is, the one that corresponds to the standard Poisson–Lie structure, coin- cides with the standard upper cluster algebra. Such results for unipotent groups were proved by Geiss et al. in ref. 12, implying the above statement via a special embedding of SLn as a uni- potent subgroup of SL2n. In this section we will prove that this is not the case for the Cremmer–Gervais cluster structure. More exactly, we prove the following theorem. Theorem 8. The cluster algebra ACGð3Þ is a proper subalgebra of the upper cluster algebra ACGð3Þ. Proof: It will be convenient to consider regular functions on Mat3, instead of SL3. Recall that the Cremmer–Gervais cluster structure CCG in Mat3 has six cluster variables and three stable ; ; Fig. 1. Quiver QCGð5Þ. variables s1 s2 s3, where
4of8 | www.pnas.org/cgi/doi/10.1073/pnas.1315283111 Gekhtman et al. Downloaded by guest on October 3, 2021 Lemma 2. Let a; b; c; x; y; u1; ...um be independent variables, = ψ = x13 x21 ; 2 2 SPECIAL FEATURE s1 2ðXÞ and let polynomials g = abx + ðay − bcÞ and f ∈ C½a; b; c; x23 x31 ; ...; ; ; u1 um x y abcg satisfy the following condition: for any choice of 0 0 0 0 m+3 x21 x22 x23 0 values a = a ; b = b ; c = c ; u = u in an open subset of C and i i x31 x32 x33 0 any σ ∈ C there exists τ ∈ C such that s2 = φ ðXÞ = ; n o n o 4 0 x x x 11 12 13 = σ ⊂ = τ gja=a0;b=b0;c=c0;u =u0 fja=a0;b=b0;c=c0;u =u0 0 x21 x22 x23 i i i i ⊂ C2: s3 = θ3ðXÞ = jXj: Then f = FðgÞ where F is a Laurent polynomial over C½a; b; c; Matrix entries x11, x12, x13, x21, x22, x23 together with stable var- ; ...; u1 um abc. iables s1, s2, s3 form a coordinate system on an open subset in Proof: Multiplying f if needed by a positive power of g, we can Mat3. It will be important for the future to calculate the Poisson assume without loss of generality that we consider a function bracket of the matrix entry x12 with all of the other coordinates: p ∈ C ; ; ; ; ...; ; ; = f ½a b c u1 um x y abc. IntroduceP a new variable z 2 2 ay − bc,theng = abx2 + z2, whereas f p = h xkzl,where fx ; x g = − x x ; fx ; x g = x x ; k;l≥0 kl 12 11 3 11 12 12 13 3 12 13 ∈ C ; ; ; ; ...; p 0 = hkl ½a b c u1 um abc. Because the level curves of ðf Þ p 0 = 2 f ja=a0;b=b0;c=c0;u =u0 and g gja=a0;b=b0;c=c0;u =u0 coincide, the fx ; x g = x x ; fx ; x g = 2x x ; i i i i 12 21 3 12 21 12 23 13 22 Jacobian [6] p 4 ∂f =∂x ∂g=∂x fx12; x22g = x12x22 + 2x13x21 − 2x11x23; 3 ∂f p=∂z ∂g=∂z 2 4 fx ; s g = x s ; fx ; s g = x s ; fx ; s g = 0: Cm+5 12 1 3 12 1 12 2 3 12 2 21 3 vanishes in an open subset of . Consequently, ∂f p ∂f p Consider the polynomial z − abx = 0; ∂x ∂z MATHEMATICS x11 x12 x13 0 and hence, = x21 x22 x23 0 : p X 0 x11 x12 x13 k l ðk + 1Þhk+1;l−1 − ðl + 1Þabhk−1;l+1 x z = 0: 0 x21 x22 x23 k;l Note that it can be written as p = p + x p + x2 p with 0 12 1 12 2 Equivalently, for all k; l the equality 2 2 p0 = ðx11x13Þx + ðx11x23 − x13x21Þ ; 22 ðk + 1Þhk+1;l−1 − ðl + 1Þabhk−1;l+1 = 0 p1 = x22ðx13x21 + x11x23Þ; p2 = x21x23: holds. Solving this recurrence relation in k and taking into ac- In what follows we consider the following rings: count that hkl = 0 for k < 0orl < 0, we see that hkl = 0 if at least = = − M = C½x11; x13; x21; s1; s2; s3 ; one of k and l is odd, whereas for even k 2r, l 2n 2r, x11x13x21 ^ = ; ; ~ = ^ ; M M½x22 x23 p M M½x12 p r n 0 h ; − = ðabÞ h ; : 2r 2n 2r r 0 2n where the subscript stands for the localization. Expressing the P P − ; ; p = N k = N N1 k last row of matrix X through the first two rows and s1 s2 s3,we Therefore, f k=0 h0;2kg , and hence f k=−N h0;2kg , = C ⊂ ~ ∈ C ; ; ; ; ...; 1 □ observe that ACGð3Þ ½Mat3 M. where h0;2k ½a b c u1 um abc. Define two differential operators Lemma 3. Let f ∈ M~ , ord f = 0, and 2 ∂ ∂ ∂ ∂ ∂ ∂ fx12; fg = ωx12 f: [7] D1 = x12 − + + + 2 + + 2 ; 3 ∂x11 ∂x13 ∂x21 ∂x22 ∂s1 ∂s2 P = = n2 i ∈ Then ½ f Fðp0Þ i=−n aip0, where ai M. ∂ ∂ 1 ~ = − + : Proof: By Lemma 1, fx ; fg = fx ; ½ f g + x fx ; fg, therefore D2 2ðx13x21 x11x23Þ ∂ 2x13x22 ∂ 12 12 12 12 x22 x23 ½fx12; fg = ½fx12; ½ f g , and hence 7 yields ½fx12; ½ f g = 0. On the other hand, ½fx ;½ f g =½ðD + D Þð½ f Þ = D ð½ f Þ, and therefore 6 ; = + ∈ ~ 12 1 2 2 It follows from Eq. that fx12 fg ðD1 D2Þf for any f M. D ð½ f Þ=0. Solving the differential equation ~ ~ ⊃ ~ ⊃ 2 ~ ⊃ ⋯ 2 The ring M has a filtration by ideals M x12M x12M . ∩ ∞ k ~ = ∈ ~ ∂½ f ∂½ f Clearly, k=0 x12M 0. Define for any nonzero f M the order ð2x x − 2x x Þ + ð2x x Þ = 0 ν ≥ ∈ ν ~ = ∞ 13 21 11 23 ∂ 13 22 ∂ ord f as the maximal 0 such that f x12M. Set ord 0 . x22 x23 Note that for any f ∈ M~ the order satisfies inequalities by the method of characteristics, we obtain a parametric equation ≥ + ; ≥ : ord D1ð fÞ ord f 1 ord D2ð fÞ ord f 8 rffiffiffiffiffiffiffiffiffiffiffiffi > ρ ffiffiffiffiffiffiffiffiffiffiffiffi ~ > = · p · ; Lemma 1. Let f ∈ M and ord f = 0, then there is a unique decom- < x22ðtÞ cos 2 x11x13 t ~ ^ ~ ~ x11x13 position f = ½ f + x12 f with ½ f ∈ M and f ∈ M. [8] Proof: > ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi The existence of the decomposition follows from the :> = 1 pρ · p · + : = = = − + = x23ðtÞ sinð2 x11x13 tÞ x13x21 identity 1 p 1 p0 x12ðp1 x12p2Þ ðpp0Þ. Uniqueness follows x11 from the observation that no element of the subring M^ depends on x . □ Therefore, characteristics are level curves Eρ described by the 12 ~ For f ∈ M such that ord f > 0 we put ½ f = 0. equation p0 = ρ. Consequently, Lemma 2 applies with a = x11,
Gekhtman et al. PNAS Early Edition | 5of8 Downloaded by guest on October 3, 2021 b = x13, c = x21, x = x22, y = x23, m = 3, ui = si for i = 1; 2; 3, and = 2 g p0. AccordingP to this lemma, [f] is a Laurent polynomial ω = αki + k [10] in p , ½ f = Fðp Þ = n2 a pk with a ∈ M, which accomplishes 3 0 0 k=−n1 k 0 k the proof. □ Lemma 4. = for all k and i. P Let f satisfy all conditions of Lemma 3, so that ½ f Let us compute the Poisson bracket of x and FðpÞ = = k ; = ω P P 12 Fðp0Þ kakp0. Then fx12 FðpÞg x12FðpÞ. k ; = 2 k iakip . Note that fx12 pg 3 x12p,hence Proof: By the proof of Lemma 3, fx12; ð½ f Þg = D1ð½ f Þ. Note that ( ) D2 = d=dt,wheret is the parameter of the level sets Eρ = fp0 = ρg X X 8 k 2 k in . In particular, fixing real values of all variables except for x12; akip = x12 αki + k akip : ; ; 3 x22 and x23 and assuming x11x13 > 0andρ > 0, the level set Eρ k i k i becomes an ellipse in the real plane x , x . Condition fx ; fg = 22 23 12 10 ; = ω + ~ = ω d ~ = Taking into account ,weconcludethatfx12 FðpÞg x12 f implies D1ð½ f Þ D2ðx12 fÞ x12 ½ f , or, equivalently, dt x12 f ωx12FðpÞ. □ ω − ~ Corollary 9. ∈ ~ ; = ω x12 ½ f D1ð½ f Þ. Recall that x12 f is a rational function in x22, x23, Let f M and fx12 fg x12f. Then forP any positive which means that its integral along any level set Eρ vanishes: integer d > 0 there exists a Laurent polynomial F ðξÞ = a ξk such Z d k dk that adk ∈ M, fx12; FdðpÞg = ωx12FdðpÞ and ord ð f − FdðpÞÞ ≥ d. ðωx12½ f − D1ð½ f ÞÞdt = 0: [9] Proof: The proof goes by induction on d. For d = 1, define = = ord f Eρ f1 f x12 . Then f1 satisfies all conditions of Lemma 3, and P hence ½f1 = F1ðp0Þ. Therefore, by Lemma 4, F1ðpÞ possesses all of = = k Assuming ½ f Fðp0Þ k;iakip0 where each aki is a Laurent the desired properties. monomial in M, we get Assume that we have built Fd−1. Define fd = ðf − Fd−1ðpÞÞ= ! − ordðf Fd−1ðpÞÞ = X X x12 . Clearly, ord fd 0, and by the inductive hypothesis = α k + k−1 D1ð½ f Þ x12 kiakip0 kakip0 D1ðp0Þ fd satisfies 7.Therefore,½fd = Fðp0Þ and ord ð fd − FðpÞÞ ≥1. Con- ; ; − k i k i = + ordðf Fd−1ðpÞÞ □ sequently, it suffices to put FdðpÞ Fd−1ðpÞ x12 FðpÞ. Lemma 5. ; ∈ ~ ; = ω for some rational constants αki determined by the equation Let f g M, and assume that fx12 fg 1x12fand ; = ω fx12; akig = αkix12aki;by6, such a constant exists for any mo- fx12 gg 2x12g. Then nomial in M. ∂f=∂x ∂g=∂x Let us find D1ðp0Þ: J = 22 22 = 0: ∂f=∂x ∂g=∂x 3 1 23 23 D ðp Þ = 8x x x2 + 8Δ − x x − x x x 1 0 11 13 22 2 11 23 13 21 12 Proof: Assume that J ≠ 0, then d = ord J ≥ 0isfinite.ByCorollary = 8p − 12Δ2 − 12Δx x 9, there exist Laurent polynomials F and G such that f = 0 13 21 ~ ~ ~d ~ d FdðpÞ + Fd,ord Fd > d and g = GdðpÞ + Gd,ord Gd > d.Therefore, with Δ = x11x23 − x13x21. Note that ∂~ ∂ ~ Z ∂FdðpÞ F ∂GdðpÞ G pffiffiffiffiffiffiffiffiffiffiffiffi + d + d pkdt = πρk= x x ∂x ∂x ∂x ∂x 0 11 13 22 22 22 22 J = Eρ ~ ~ ∂F ðpÞ ∂F ∂G ðpÞ ∂G ρ d + d d + d because p0 equals on Eρ, ∂ ∂ ∂ ∂ Z x23 x23 x23 x23 ∂ ∂ 4ðx11x23 − x13x21Þx13x21dt = 0 FdðpÞ GdðpÞ ∂ ∂ Eρ x22 x22 = + ~J ∂ ∂ by the symmetry of the level set Eρ, and FdðpÞ GdðpÞ Z ∂ ∂ ffiffiffiffiffiffiffiffiffiffiffiffi x23 x23 2 p 4ðx11x23 − x13x21Þ dt = 2πρ= x11x13: ~ > Eρ with ord J d. Notice that 9 Therefore, yields ∂ ∂ ∂ ∂ Z FdðpÞ GdðpÞ dFd p dGd p 1 ∂ ∂ ∂ ∂ ω − x22 x22 dp x22 dp x22 ð x12½ f D1ð½ f ÞÞdt = = 0; x12 ∂ ∂ dF ∂p dG ∂p Eρ FdðpÞ GdðpÞ d d ! ∂ ∂ ∂ ∂ X X X x23 x23 dp x23 dp x23 π 8 = pffiffiffiffiffiffiffiffiffiffiffiffi ω a ρk − α a ρk − ka ρk x x ki ki ki 3 ki ~ 11 13 k;i k;i k;i and hence ord J = ord J > d, a contradiction. □ X Corollary 10. 2πρ − The function x12 is not a cluster variable in the cluster + pffiffiffiffiffiffiffiffiffiffiffiffi ka ρk 1 = 0: x x ki algebra ACGð3Þ. k;i 11 13 Proof: Assume that there exists a cluster containing x12. This cluster contains also three stable variables s , s , s and five cluster Equivalently, 1 2 3 variables v ; v ; v ; v ; v ∈ C½Mat such that fx ; v g = ω x v . 1 2 3 4 5 3 12 j j 12 j π 2 Choose x11, x12, x13, x21, x22, x23, s1, s2, s3, as local coordinates in an pffiffiffiffiffiffiffiffiffiffiffiffi a ρk ω − α − k = 0; open subset of C9 and write all cluster variables as elements of M~ . x x ki ki 3 11 13 Because the cluster variables belonging to the same cluster are or algebraically independent, the rank of the 2 × 5 matrix
6of8 | www.pnas.org/cgi/doi/10.1073/pnas.1315283111 Gekhtman et al. Downloaded by guest on October 3, 2021 0 1 n o ∂ ∂ ∂ ∂ ∂ i;n v1 v2 v3 v4 v5 F = det X ½ : i ∈ ½n ; j ∈ ½i − 1 SPECIAL FEATURE B C 1 ½j;n+j−i ∂x22 ∂x22 ∂x22 ∂x22 ∂x22 Jac = B C @ ∂ ∂ ∂ ∂ ∂ A v1 v2 v3 v4 v5 and ∂x23 ∂x23 ∂x23 ∂x23 ∂x23 n o = ½1;n n½j+1;j+n−i : ∈ ; ∈ : F2 det X½1;i i ½n j ½i equals 2 for generic values of parameters x11, x12, x13, x21, s1, s2, s3. However, by Lemma 5, any 2 × 2 minor of Jac vanishes identically, a contradiction. □ By Theorem 7, the family Corollary 11. n o The function x12 does not belong to the cluster ; ; − + w0 = ½i n = ½1 n i 1 : ∈ ; ∈ − algebra ACGð3Þ. F1 detðW0XW0Þ½j;n+j−i det X½i−j+1;n−j+1 i ½n j ½i 1 Proof: Note that cluster algebra ACGð3Þ has three independent T x = t i−2 · x T x = t j−2 · x compatible toric actions: 1ð ijÞ ij, 2ð ijÞ ij, and also consists of cluster variables in ACG. However, according w T3ðxijÞ = t · xij. Any cluster variable of ACGð3Þ is a polynomial in xij 0 ∪ to ref. 13, theorem 4.13, F1 F2 is one of the test families for homogeneous with respect to all three weights w1, w2, w3 deter- total positivity (it corresponds to the reduced word n − 1 ... mined by the toric actions T , T , T . Therefore, we can associate 1 2 3 1n − 1 ...2 ...n − 11 ...n − 11 ...n − 2 ...1), i.e., positivity of all with every cluster coordinate zC in a cluster C the weight vector = ; minors in this family guarantees total positivity of X.Thisshows wðzCÞ ðw1ðzCÞ w2ðzCÞÞ containing only the first two weights. Note ⊂ ≤ ; ≤ that TPCGðnÞ TPðnÞ. that the weight vectors for all nine matrix entries xij,1 i j 3, are To complete the proof, we need to construct a totally positive all distinct, implying that no cluster variable is a linear combination matrix X such that at least one of the functions forming the of matrix entries xij with more than one nontrivial coefficient. initial cluster for ACG is negative when evaluated at X.LetS = Taking that into account and recalling that the cluster algebra n δ ; − × ∈ C = 1 + ð i j 1Þi;j=1 be the n n shift matrix. For t ,defineEðtÞ contains onlyP regular functions on Mat3, we see that if any linear − tS. It is easy to see that if t > 0thenmatricesE t and E −t 1 polynomial cijxij belongs to ACGð3Þ then cij ≠ 0 if and only if xij ð Þ ð Þ = − T − −1 = itself is a cluster variable in ACGð3Þ. Because x12 is not a cluster are totally nonnegative. Let X0 ðEð 1Þ Eð 1ÞÞ and XðtÞ ∉ □ T n−1 variable by Corollary 10, we conclude that x12 ACGð3Þ. X0ðEðtÞEðtÞ Þ . The Cauchy–Binet identity implies that X0 is Remark: The same arguments imply that x does not belong to = 1 + > 12 totally nonnegative and XðtÞ X ð OðtÞÞ is totally positive if t MATHEMATICS ′ 0 the Cremmer–Gervais cluster algebra ACGð3Þ associated with 0. However, the value of the cluster variable ψ ðXÞ = x − ; x − ′ 2 n 2 n n1 SL . Note first that again x is not a cluster variable in A ð3Þ. x − ; x − ; at X is −1andso,fort positive and sufficiently 3 12 ′ CG n 1 n n 1 1 0 Indeed, if there is a cluster C in ACGð3Þ containing x12, then by small, XðtÞ ∈ TPðnÞnTPCGðnÞ. □ adding det X to C we will obtain a cluster in ACGð3Þ containing x , in a contradiction with Corollary 10. The proof of Corollary Further Directions 12 ′ 11 can be used literally to conclude that x12 ∉ ACG ð3Þ. We would like to outline briefly further directions of research. Connection to Total Positivity 1. Our main goal is to prove Conjecture 1. In addition to the results reported above, the conjecture has been verified for A comprehensive analysis of total positivity in reductive groups SL5; see ref. 14. In all of the cases studied so far, one can that was performed in ref. 13 gave a major impetus to the de- produce a quiver that is remarkably similar to the one shown velopment of the theory of cluster algebras. As it was explained in Fig. 1 above. Namely, it consists of n2 vertices placed on in ref. 5, remark 2.16, the set of totally positive elements in a grid with diagonals. Besides, vertices of the upper and the a simple Lie group coincides with the set of elements in an open lower rows, as well as vertices of the leftmost and the right- double Bruhat cell that form a positive locus with respect to the most columns, are connected with additional edges, which are standard cluster structure. This locus is formed by elements on determined by the isometry γ. which functions that form a single extended cluster (and there- 2. We conjecture that the result of Theorem 8 can be extended fore all cluster variables) take positive values. to other exotic cluster structures as well. In the case of GLn, the set TPðnÞ of totally positive elements is 3. Theorems 3 and 5 follow from similar results valid in the formed by matrices with all minors positive. It is a well-studied double. This hints at a possibility of endowing the double class of matrices that has many important applications. It is with a cluster structure associated with the Cremmer– natural to ask what can be said about the positive locus in GLn Gervais bracket. with respect to the Cremmer–Gervais cluster structure introduced above. We denote this locus by TPCGðnÞ. A careful analysis of cluster transformations that form the sequence featured in ACKNOWLEDGMENTS. Special thanks are due to Bernhard Keller, whose T quiver mutation applet proved to be an indispensable tool in our work on Theorem 7 allows us to make the first step toward understanding this project, to Thomas Lam for attracting our attention to the total positivity TPCGðnÞ. question, and to Sergei Fomin for illuminating discussions. This paper was Theorem 12. partially written during our joint stays at Mathematisches Forschungsinstitut Oberwolfach (Research in Pairs program, August 2010), at the Hausdorff – : Institute (Research in Groups program, June August 2011) and at the TPCGðnÞ TPðnÞ Mathematical Sciences Research Institute (Cluster Algebras program, Au- gust–December 2012). We are grateful to these institutions for warm hospi- Proof: It is shown in ref. 11 that in the process of applying tality and excellent working conditions. We are also grateful to our home transformations forming T to the initial cluster described in institution for support in arranging visits by collaborators. M.G. was supported in part by National Science Foundation (NSF) Grants DMS 0801204 and DMS Theorem 3, one recovers the following two families of minors as 1101462. M.S. was supported in part by NSF Grants DMS 0800671 and DMS cluster variables in ACG: 1101369. A.V. was supported in part by Israel Science Foundation Grant 162/12.
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