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Mutation-Linear Algebra and Universal Geometric Cluster Algebras

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Mutation-Linear Algebra and Universal Geometric Cluster Algebras Mutation-linear algebra and universal geometric cluster algebras Nathan Reading NC State University Mutation-linear (\µ-linear") algebra Universal geometric cluster algebras The mutation fan Universal geometric cluster algebras from surfaces B Mutation maps ηk Let B be a skew-symmetrizable exchange matrix. n Given a = (a1;:::; an) 2 R , let Be be B, extended with one coefficient row a. B For a sequence k = kq; kq−1;:::; k1, define ηk (a) to be the coefficient row of µk(Be). Concretely, for k = k: B 0 0 ηk (a) = (a1;:::; an), where 8 −a if j = k; > k <> a + a b if j 6= k, a ≥ 0 and b ≥ 0; a0 = j k kj k kj j a − a b if j 6= k, a ≤ 0 and b ≤ 0; > j k kj k kj : aj otherwise. B The mutation maps ηk are piecewise-linear homeomorphisms of n R . Their inverses are also mutation maps. 1. Mutation-linear (\µ-linear") algebra 1 . B-coherent linear relations n Let S be a finite set, let (vi : i 2 S) be vectors in R and let (ci : i 2 S) be real numbers. P The formal expression i2S ci vi is a B-coherent linear relation if X B ci ηk (vi ) = 0; and (1) i2S X B ci min(ηk (vi ); 0) = 0 (2) i2S hold for every finite sequence k = kq;:::; k1. P In particular, i2S ci vi is a linear relation in the usual sense. Example: For B = [0], a B-coherent relation is a linear relation n among vectors in R all agreeing in sign. 1. Mutation-linear (\µ-linear") algebra 2 . Basis for B Let R be Z or a field between Q and R. Usually Z, Q, or R. n The vectors (bi : i 2 I ) in R are an R-basis for B if and only if the following two conditions hold. (i) Spanning: For all a 2 Rn, there exists a finite subset S of I P and coefficients ci in R such that a − i2S ci bi is a B-coherent linear relation. P (ii) Independence: If S is a finite subset of I and i2S ci bi is a B-coherent linear relation, then ci = 0 for all i 2 S. Example: For any R, {±1g ⊂ R1 is an R-basis for [0]. Theorem Every skew-symmetrizable B has an R-basis. Proof: Zorn's lemma, same argument that shows that every vector space has a (Hamel) basis. The proof is non-constructive. 1. Mutation-linear (\µ-linear") algebra 3 . Why do µ-linear algebra? • Universal geometric cluster algebras • The mutation fan 1. Mutation-linear (\µ-linear") algebra 4 . Tropical Semifield (broadly defined) I : an indexing set (no requirement on cardinality). (ui : i 2 I ): formal symbols (tropical variables). Tropical semifield Trop(ui : i 2 I ): Q ai Elements are products i2I ui with ai 2 R. Multiplication as usual I Trop(ui : i 2 I ) is the module R , written multiplicatively. Q ai Q bi Q min(ai ;bi ) Auxiliary addition i2I ui ⊕ i2I ui = i2I ui . Topology on R: discrete. I Topology on Trop(ui : i 2 I ): product topology as R . I finite: discrete. I countable: FPS. 2. Universal geometric cluster algebras 5 . Cluster algebra of geometric type (broadly defined) Extended exchange matrix: a collection of rows indexed by [n] [ I . In matrix notation, Be = [bij ]. Rows of Be indexed by [n] are the matrix B. Other rows are coefficient rows. These are vectors in Rn. Q bij Coefficients: yj = i2I ui 2 Trop(ui : i 2 I ) for each j 2 [n]. Cluster algebra of geometric type: A(x; Be) = A(x; B; fy1;:::; yng): (The usual definition: take I finite and R = Z.) 2. Universal geometric cluster algebras 6 . Continuity refers to the product topology on tropical semifields. An R-linear map is just a multiplicative homomorphism of semifields when R = Z or Q. Coefficient specialization Be and Be 0: extended exchange matrices of rank n. th For each sequence k and j 2 [n], write yk;j for the j coefficient 0 th 0 defined by µk(Be) and yk;j for the j coefficient defined by µk(Be ). A ring homomorphism ' : A(x; Be) !A(x0; Be 0) is a coefficient specialization if (i) the exchange matrices B and B0 coincide; 0 (ii) '(xj ) = xj for all j 2 [n] (i.e. initial cluster variables coincide); (iii) ' restricts to a continuous R-linear map on tropical semifields. 0 0 (iv) '(yk;j ) = yk;j and '(yk;j ⊕ 1) = yk;j ⊕ 1 for each k and each j. 2. Universal geometric cluster algebras 7 . Coefficient specialization Be and Be 0: extended exchange matrices of rank n. th For each sequence k and j 2 [n], write yk;j for the j coefficient 0 th 0 defined by µk(Be) and yk;j for the j coefficient defined by µk(Be ). A ring homomorphism ' : A(x; Be) !A(x0; Be 0) is a coefficient specialization if (i) the exchange matrices B and B0 coincide; 0 (ii) '(xj ) = xj for all j 2 [n] (i.e. initial cluster variables coincide); (iii) ' restricts to a continuous R-linear map on tropical semifields. 0 0 (iv) '(yk;j ) = yk;j and '(yk;j ⊕ 1) = yk;j ⊕ 1 for each k and each j. Continuity refers to the product topology on tropical semifields. An R-linear map is just a multiplicative homomorphism of semifields when R = Z or Q. 2. Universal geometric cluster algebras 7 . Universal geometric cluster algebras Fix an exchange matrix B. Category: geometric cluster algebras with exchange matrix B; coefficient specializations. (Depends on R.) Universal geometric cluster algebra: universal object in category. Isomorphism type of a geometric cluster algebra depends only on the extended exchange matrix Be. So we are really looking for universal geometric exchange matrices. We'll use the term universal geometric coefficients for the coefficient rows of a universal geometric exchange matrix. 2. Universal geometric cluster algebras 8 . Previous results (for B bipartite, finite type): Fomin-Zelevinsky, 2006: Coefficient rows of universal extended exchange matrix are indexed by almost positive co-roots β_. The row for β_ has i th entry _ _ "(i)[β : αi ] for i = 1;:::; n. Reading-Speyer, 2007: The linear map taking a positive root αi to −"(i)!i maps almost positive roots into rays of (bipartite) Cambrian fan. Yang-Zelevinsky, 2008: Rays of the Cambrian fan are spanned by vectors whose fundamental weight coordinates are g-vectors of cluster variables. Putting this all together (including sign change and \ _ "): Universal extended exchange matrix for B has coefficient rows given by g-vectors for BT . (Works for any R or for a much larger category) 2. Universal geometric cluster algebras 9 . Universal geometric coefficients and µ-linear algebra Theorem (R., 2012) A collection (bi : i 2 I ) are universal coefficients for B over R if and only if they are an R-basis for B. (That is, to make a universal extended exchange matrix, extend B with coefficient rows forming an R-basis for B.) h 0 i Example: 1 is universal for any R. −1 2. Universal geometric cluster algebras 10 . Why universal coefficients correspond to bases Recall: a coefficient specialization is a continuous R-linear map between tropical semifields taking coefficients to coefficients everywhere in the Y -pattern. Proposition. Let Trop(ui : i 2 I ) and Trop(vj : j 2 J) be tropical semifields and fix a family (pij : i 2 I ; j 2 J) of elements of R. Then the following are equivalent. (i) 9 continuous R-linear map ' from Trop(ui : i 2 I ) to Q pij Trop(vj : j 2 J), with '(ui ) = j2J vj for all i 2 I . (ii) For each j 2 J, there are only finitely many indices i 2 I such that pij is nonzero. When these conditions hold, the unique map ' is P Y ai Y i2I pij aj ' ui = vj : i2I j2J 2. Universal geometric cluster algebras 11 . Why universal coefficients correspond to bases (continued) A picture suitable for hand-waving: B B 2. Universal geometric cluster algebras 12 . The mutation fan One \easy" way to get B-coherent linear relations: Find vectors in the same domain of linearity of all mutation maps and make a linear relation among them. B n Define an equivalence relation ≡ on R by setting B B B a1 ≡ a2 () sgn(ηk (a1)) = sgn(ηk (a2)) 8k: sgn(a) is the vector of signs (−1; 0; +1) of the entries of a. B-classes: equivalence classes of ≡B . B-cones: closures of B-classes. Facts: B-classes are cones. B-cones are closed cones. B Each map ηk is linear on each B-cone. 3. The mutation fan 13 . The mutation fan (continued) Mutation fan for B: The collection FB of all B-cones and all faces of B-cones. Theorem (R., 2011) FB is a complete fan. Example: For B = [0]: µ1 is negation and FB = ff0g; R≥0; R≤0g. 3. The mutation fan 14 . h 0 2 −2 i Example: B = −2 0 2 (Markov quiver) 2 −2 0 3. The mutation fan 15 . Positive bases n An R-basis for B is positive if, for every a = (a1;:::; an) 2 R , the P unique B-coherent linear relation a − i2I ci bi has all ci ≥ 0. In many examples, positive R-bases exist, but not always, and it depends on R. There is at most one positive R-basis for B, up to scaling each basis element by a positive unit in R.
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