Combinatorics of X -variables in finite type cluster algebras

Transcription

1 Combinatorics of X -variables in finite type cluster algebras arxiv: Slides available at Melissa Sherman-Bennett, UC Berkeley AMS Fall Central Sectional Meeting 2018 M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

2 Big Picture (Fock Goncharov (2009)) Cluster varieties in 2 flavors: A-varieties (with A-variables) and X -varieties (with X -variables) A-variables cluster variables X -variables coefficients There is a duality between A-varieties and X -varieties (GHKK (2018)), but on the algebraic side much less is understood about X -variables X -variables appear naturally in total positivity and in scattering amplitudes in N = 4 Super Yang-Mills theory (GGSVV (2014)) M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

3 Definitions Let F = Q(t 1,..., t n ). An X -seed Σ in F is a pair (x, B), where x = (x 1,..., x n ) with x i F and B = (b ij ) is a skew-symmetrizable n n integer matrix. Mutation at k {1,..., n}: where x j = (x, B) µ k (x, B ) if j = k x j (x k + 1) b kj if b kj 0 x j (x 1 k + 1) b kj if b kj > 0 x 1 j and B is obtained from B by matrix mutation at k. M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

4 Definitions Let F = Q(t 1,..., t n ). An A-seed Σ in F is a pair (a, B), where a = (a 1,..., a n ) consists of algebraically independent elements of F and B = (b ij ) is a skew-symmetrizable n n integer matrix. Mutation at k {1,..., n}: where a j = a 1 k a j ( (a, B) µ k (a, B ) a b ik i + ) a b ik i b ik >0 b ik <0 and B is obtained from B by matrix mutation at k. if j = k if j k M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

5 Seed Patterns T n : n-regular tree with edges labeled with 1,..., n so each vertex sees each label. A seed pattern S is a collection of seeds {Σ t } t Tn such that if t k t in T n, then Σ t = µ k (Σ t ). An A-seed pattern is finite type if it contains finitely many seeds. M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

6 Motivation Theorem (Fomin Zelevinsky (2003)) An A-seed pattern S(a, B) is finite type if and only if B is mutation equivalent to a matrix whose Cartan companion is a finite type Cartan matrix. Further, there is a bijection between A-variables and almost positive roots (positive or negative simple) in the corresponding root system. Combinatorics of finite type A-seed patterns of classical type are encoded in tagged triangulations of certain marked surfaces (Fomin Shapiro Thurston (2008)). M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

7 Motivation Theorem (Fomin Zelevinsky (2003)) An A-seed pattern S(a, B) is finite type if and only if B is mutation equivalent to a matrix whose Cartan companion is a finite type Cartan matrix. Further, there is a bijection between A-variables and almost positive roots (positive or negative simple) in the corresponding root system. Combinatorics of finite type A-seed patterns of classical type are encoded in tagged triangulations of certain marked surfaces (Fomin Shapiro Thurston (2008)). Question Let S(x, B) be an X -seed pattern of classical type. What can we say about its combinatorics? M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

8 The Answer Theorem (S.B. (2018)) Let S be an X -seed pattern of classical type and let P be the corresponding marked surface. Then there is a bijection between the quadrilaterals (with a choice of diagonal) appearing in triangulations of P and the X -variables of S. M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

9 The Answer Theorem (S.B. (2018)) Let S be an X -seed pattern of classical type and let P be the corresponding marked surface. Then there is a bijection between the quadrilaterals (with a choice of diagonal) appearing in triangulations of P and the X -variables of S. Classical types: Type A n B n, C n D n X (S) 2 ( ) n n(n + 1)(n ) 3 n(n 1)(n2 + 4n 6) Exceptional types: Type E 6 E 7 E 8 F 4 G 2 X (S) M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

10 A-seed patterns of classical types S a A-seed pattern of type A n (D n ), P an (n + 3)-gon (punctured n-gon). {A-variables of S} {arcs of tagged triangulations of P} {seeds Σ in S} {triangulations T of P}. If Σ corresponds to T, the A-variables of Σ correspond to the arcs of T and the exchange matrix of Σ can be obtained from T. Mutating Σ at k corresponds to flipping the k th arc in T. There is an analogous story for types B n, C n involving triangulations preserved by a particular group action. M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

11 A surjection... Let S be an X -seed pattern of classical type, and P be the appropriate marked surface. Think of seeds (x, B) as triangulations T of P with arcs labeled by X -variables (B is the signed adjacency matrix of T ). Mutating/flipping an arc γ may change the labels of the arcs adjacent to γ. If we mutate away from the quadrilateral of an arc γ, the label of γ does not change. x j = if j = k x j (x k + 1) b kj if b kj 0 x j (x 1 k + 1) b kj if b kj > 0 x 1 j M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

12 A surjection... Let S be an X -seed pattern of classical type, and P be the appropriate marked surface. Think of seeds (x, B) as triangulations T of P with arcs labeled by X -variables (B is the signed adjacency matrix of T ). Mutating/flipping an arc γ may change the labels of the arcs adjacent to γ. If we mutate away from the quadrilateral of an arc γ, the label of γ does not change. M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

13 A surjection... Fact (Fomin Shapiro Thurston (2008)) Let T, T be two tagged triangulations of a marked surface which both contain arcs τ 1,..., τ s. Then T can be obtained from T by a sequence of arc flips avoiding τ 1,..., τ s. So α : {q {γ} q a quadrilateral with diagonal γ in P} X (S) is well-defined and surjective. M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

14 ...which is injective. Proposition The X -variables associated to distinct quadrilaterals are distinct. Method of proof: Consider a particular A-seed pattern R of each type (can be found in e.g. Intro. to cluster algebras); A-variables are rational functions on a vector space V. Look at a related X -seed pattern ˆR; X -variables are rational functions of A-variables in R. For any pair of X -variables labeling diagonals of different quadrilaterals, verify that they are different functions on V. M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

15 Corollaries and a Conjecture Type A n B n, C n D n X 2 ( ) n n(n + 1)(n ) 3 n(n 1)(n2 + 4n 6) X pc n(n + 1) 2n 2 2n(n 1) Type E 6 E 7 E 8 F 4 G 2 X X pc Note: X pc is the number of X -variables when we replace + with tropical plus in the X -variable mutation formulas. The values follow from results of (Speyer Thomas (2013)). M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

16 Corollaries and a Conjecture Corollary Let S be an X -seed pattern of type Z n. The X -variables in S are in bijection with ordered pairs of exchangeable A-variables in an A-seed pattern of type Z n. The X -variables in S are in bijection with ordered pairs of almost-positive roots with compatibility degree 1 in the root system of type Z n. Conjecture Let T be a tagged triangulation of a marked surface (S, M), B the signed adjacency matrix of T, and S := S(x, B). Then the following map is a bijection: α : {q {γ} q a quadrilateral with diagonal γ in (S, M)} X (S). M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12

17 References V. Fock, A. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009) S. Fomin, M. Shapiro, D. Thurston, Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (1) (2008) S. Fomin, A. Zelevinsky, Cluster algebras II. Finite type classification, Invent. Math. 154 (1) (2003) J. Golden, A. Goncharov, M. Spradlin, C. Vergu, A. Volovich, Motivic amplitudes and cluster coordinates, Journal of High Energy Physics 2014 (1). M. Gross, P. Hacking, S. Keel, M. Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2) (2018) D. Speyer, H. Thomas, Acyclic cluster algebras revisited, in: Algebras, quivers and representations, Vol. 8 of Abel Symp., Springer, Heidelberf, 2013, M. Sherman-Bennett (UC Berkeley) X -variables in finite type AMS Fall Sectionals / 12