The Greedy Basis Equals the Theta Basis A Rank Two Haiku

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1 The Greedy Basis Equals the Theta Basis A Rank Two Haiku Man Wai Cheung (UCSD), Mark Gross (Cambridge), Greg Muller (Michigan), Gregg Musiker (University of Minnesota) *, Dylan Rupel (Notre Dame), Salvatore Stella (Roma La Sapienza), and Harold Williams (University of Texas) AMS Central Spring Sectional, Combinatoial Ideals and Applications April 17, musiker/haiku.pdf CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

2 Outline 1 Introduction to Cluster Algebras Greedy Bases 3 Theta Bases in Rank 4 Sketch of their Equivalence Thank you for support from NSF Grant DMS , the University of Cambridge, Northeastern Univeristy, and North Carolina State University, and the 014 AMS Mathematics Research Community on Cluster Algebras in Snowbird, UT. musiker/haiku.pdf CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, 016 / 7

3 Introduction to Cluster Algebras In the late 1990 s: Fomin and Zelevinsky were studying total positivity and canonical bases of algebraic groups. They noticed recurring combinatorial and algebraic structures. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

4 Introduction to Cluster Algebras In the late 1990 s: Fomin and Zelevinsky were studying total positivity and canonical bases of algebraic groups. They noticed recurring combinatorial and algebraic structures. Let them to define cluster algebras, which have now been linked to quiver representations, Poisson geometry Teichmüller theory, tilting theory, mathematical physics, discrete integrable systems, string theory, and many other topics. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

5 Introduction to Cluster Algebras In the late 1990 s: Fomin and Zelevinsky were studying total positivity and canonical bases of algebraic groups. They noticed recurring combinatorial and algebraic structures. Let them to define cluster algebras, which have now been linked to quiver representations, Poisson geometry Teichmüller theory, tilting theory, mathematical physics, discrete integrable systems, string theory, and many other topics. Cluster algebras are a certain class of commutative rings which have a distinguished set of generators that are grouped into overlapping subsets, called clusters, each having the same cardinality. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

6 What is a Cluster Algebra? Definition (Sergey Fomin and Andrei Zelevinsky 001) A cluster algebra A (of geometric type) is a subalgebra of k(x 1,..., x n, x n+1,..., x n+m ) constructed cluster by cluster by certain exchange relations. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

7 What is a Cluster Algebra? Definition (Sergey Fomin and Andrei Zelevinsky 001) A cluster algebra A (of geometric type) is a subalgebra of k(x 1,..., x n, x n+1,..., x n+m ) constructed cluster by cluster by certain exchange relations. Generators: Specify an initial finite set of them, a Cluster, {x 1, x,..., x n+m }. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

8 What is a Cluster Algebra? Definition (Sergey Fomin and Andrei Zelevinsky 001) A cluster algebra A (of geometric type) is a subalgebra of k(x 1,..., x n, x n+1,..., x n+m ) constructed cluster by cluster by certain exchange relations. Generators: Specify an initial finite set of them, a Cluster, {x 1, x,..., x n+m }. Construct the rest via Binomial Exchange Relations: x α x α = x d+ i γ i + x d i γ i. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

9 What is a Cluster Algebra? Definition (Sergey Fomin and Andrei Zelevinsky 001) A cluster algebra A (of geometric type) is a subalgebra of k(x 1,..., x n, x n+1,..., x n+m ) constructed cluster by cluster by certain exchange relations. Generators: Specify an initial finite set of them, a Cluster, {x 1, x,..., x n+m }. Construct the rest via Binomial Exchange Relations: x α x α = x d+ i γ i + x d i γ i. The set of all such generators are known as Cluster Variables, and the initial pattern of exchange relations (described as a valued quiver, i.e. a directed graph, or as a skew-symmetrizable matrix) determines the Seed. Relations: Induced by the Binomial Exchange Relations. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

10 Cluster Algebras, Ideally Point of view of Cluster Algebras III (Berenstein-Fomin-Zelevinsky): A is generated by x 1, x,..., x n, x 1, x,..., x n where the standard monomials in this alphabet (i.e. x i and x i forbidden from being in the same monomial) are a Z-basis for A. The polynomials x i x i x d+ j γ j generate the ideal I of relations. x d j γ j Form a Gröbner basis for I assuming a term order where the x i s are higher degree than the x i s. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

11 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

12 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a quiver with two vertices and b arrows from v 1 v.) µ 1 (B) = µ (B) = B and x 1 x 1 = x c + 1, x x = 1 + x b 1. Thus the cluster variables in this case are { xn 1 b + 1 if n is odd {x n : n Z} satisfying x n x n = xn 1 c + 1 if n is even. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

13 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a quiver with two vertices and b arrows from v 1 v.) µ 1 (B) = µ (B) = B and x 1 x 1 = x c + 1, x x = 1 + x b 1. Thus the cluster variables in this case are { xn 1 b + 1 if n is odd {x n : n Z} satisfying x n x n = xn 1 c + 1 if n is even. Example (b = c = 1): x 3 = x + 1 x 1. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

14 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a quiver with two vertices and b arrows from v 1 v.) µ 1 (B) = µ (B) = B and x 1 x 1 = x c + 1, x x = 1 + x b 1. Thus the cluster variables in this case are { xn 1 b + 1 if n is odd {x n : n Z} satisfying x n x n = xn 1 c + 1 if n is even. Example (b = c = 1): x 3 = x + 1 x 1. x 4 = x x = CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

15 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a quiver with two vertices and b arrows from v 1 v.) µ 1 (B) = µ (B) = B and x 1 x 1 = x c + 1, x x = 1 + x b 1. Thus the cluster variables in this case are { xn 1 b + 1 if n is odd {x n : n Z} satisfying x n x n = xn 1 c + 1 if n is even. Example (b = c = 1): x 3 = x + 1 x 1. x 4 = x x = x +1 x x = CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

16 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a quiver with two vertices and b arrows from v 1 v.) µ 1 (B) = µ (B) = B and x 1 x 1 = x c + 1, x x = 1 + x b 1. Thus the cluster variables in this case are { xn 1 b + 1 if n is odd {x n : n Z} satisfying x n x n = xn 1 c + 1 if n is even. Example (b = c = 1): x 3 = x + 1 x 1. x 4 = x x = x +1 x x = x 1 + x + 1 x 1 x. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

17 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a quiver with two vertices and b arrows from v 1 v.) µ 1 (B) = µ (B) = B and x 1 x 1 = x c + 1, x x = 1 + x b 1. Thus the cluster variables in this case are { xn 1 b + 1 if n is odd {x n : n Z} satisfying x n x n = xn 1 c + 1 if n is even. Example (b = c = 1): x 5 = x x 3 = x 3 = x + 1 x 1. x 4 = x x = x +1 x x = x 1 + x + 1 x 1 x. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

18 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a quiver with two vertices and b arrows from v 1 v.) µ 1 (B) = µ (B) = B and x 1 x 1 = x c + 1, x x = 1 + x b 1. Thus the cluster variables in this case are { xn 1 b + 1 if n is odd {x n : n Z} satisfying x n x n = xn 1 c + 1 if n is even. Example (b = c = 1): x 5 = x x 3 = x 3 = x + 1 x 1. x 4 = x x = x 1 +x +1 x 1 x + 1 (x + 1)/x 1 = x +1 x x = x 1 + x + 1 x 1 x. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

19 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a quiver with two vertices and b arrows from v 1 v.) µ 1 (B) = µ (B) = B and x 1 x 1 = x c + 1, x x = 1 + x b 1. Thus the cluster variables in this case are { xn 1 b + 1 if n is odd {x n : n Z} satisfying x n x n = xn 1 c + 1 if n is even. Example (b = c = 1): (Finite Type, of Type A ) x 5 = x x 3 = x 3 = x + 1 x 1. x 4 = x x = x +1 x x = x 1 + x + 1 x 1 x. x 1 +x +1 x 1 x + 1 = x 1(x 1 + x x 1 x ) = (x + 1)/x 1 x 1 x (x + 1) CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

20 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a quiver with two vertices and b arrows from v 1 v.) µ 1 (B) = µ (B) = B and x 1 x 1 = x c + 1, x x = 1 + x b 1. Thus the cluster variables in this case are { xn 1 b + 1 if n is odd {x n : n Z} satisfying x n x n = xn 1 c + 1 if n is even. Example (b = c = 1): (Finite Type, of Type A ) x 5 = x x 3 = x 3 = x + 1 x 1. x 4 = x x = x +1 x x = x 1 + x + 1 x 1 x. x 1 +x +1 x 1 x + 1 = x 1(x 1 + x x 1 x ) = x (x + 1)/x 1 x 1 x (x + 1) x CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

21 Example: Rank Cluster Algebras [ ] 0 b Let B =, b, c Z c 0 >0. ({x 1, x }, B) is a seed for a cluster algebra A(b, c) of rank. (E.g. when b = c, B = B(Q) where Q is a quiver with two vertices and b arrows from v 1 v.) µ 1 (B) = µ (B) = B and x 1 x 1 = x c + 1, x x = 1 + x b 1. Thus the cluster variables in this case are { xn 1 b + 1 if n is odd {x n : n Z} satisfying x n x n = xn 1 c + 1 if n is even. Example (b = c = 1): x 5 = x x 3 = x 3 = x + 1 x 1. x 4 = x x = x +1 x x = x 1 + x + 1 x 1 x. x 1 +x +1 x 1 x + 1 = x 1(x 1 + x x 1 x ) = x x 6 = x 1. (x + 1)/x 1 x 1 x (x + 1) x CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

22 Example: Rank Cluster Algebras Example (b = c = ): (Affine Type, of Type Ã1) x 3 = x + 1 x 1. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

23 Example: Rank Cluster Algebras Example (b = c = ): (Affine Type, of Type Ã1) x 3 = x + 1. x 4 = x = x 4 + x x 1 x 1 x x1 x. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

24 Example: Rank Cluster Algebras Example (b = c = ): (Affine Type, of Type Ã1) x 3 = x + 1. x 4 = x = x 4 + x x 1 x 1 x x1 x. x 5 = x = x 6 + 3x 4 + 3x x x 1 + x 1 x x 3 x1 3x,... CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

25 Example: Rank Cluster Algebras Example (b = c = ): (Affine Type, of Type Ã1) x 3 = x + 1. x 4 = x = x 4 + x x 1 x 1 x x1 x. x 5 = x = x 6 + 3x 4 + 3x x x 1 + x 1 x x 3 x1 3x,... If we let x 1 = x = 1, we obtain {x 3, x 4, x 5, x 6 } = {, 5, 13, 34}. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

26 Example: Rank Cluster Algebras Example (b = c = ): (Affine Type, of Type Ã1) x 3 = x + 1. x 4 = x = x 4 + x x 1 x 1 x x1 x. x 5 = x = x 6 + 3x 4 + 3x x x 1 + x 1 x x 3 x1 3x,... If we let x 1 = x = 1, we obtain {x 3, x 4, x 5, x 6 } = {, 5, 13, 34}. The next number in the sequence is x 7 = = CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

27 Example: Rank Cluster Algebras Example (b = c = ): (Affine Type, of Type Ã1) x 3 = x + 1. x 4 = x = x 4 + x x 1 x 1 x x1 x. x 5 = x = x 6 + 3x 4 + 3x x x 1 + x 1 x x 3 x1 3x,... If we let x 1 = x = 1, we obtain {x 3, x 4, x 5, x 6 } = {, 5, 13, 34}. The next number in the sequence is x 7 = = = CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

28 Example: Rank Cluster Algebras Example (b = c = ): (Affine Type, of Type Ã1) x 3 = x + 1. x 4 = x = x 4 + x x 1 x 1 x x1 x. x 5 = x = x 6 + 3x 4 + 3x x x 1 + x 1 x x 3 x1 3x,... If we let x 1 = x = 1, we obtain {x 3, x 4, x 5, x 6 } = {, 5, 13, 34}. The next number in the sequence is x 7 = = = 89, an integer! CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

29 Laurent Phenomenon and Positivity Theorem (Fomin-Zelevinsky 00) For any cluster algebra, all cluster variables, i.e. the x n s are Laurent polynomials (i.e. denominator is a monomial) in the initial cluster {x 1, x }. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

30 Laurent Phenomenon and Positivity Theorem (Fomin-Zelevinsky 00) For any cluster algebra, all cluster variables, i.e. the x n s are Laurent polynomials (i.e. denominator is a monomial) in the initial cluster {x 1, x }. Theorem (Lee-Schiffler 013) The Laurent exapnsions of cluster variables involve exclusively positive integer coefficients. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

31 Laurent Phenomenon and Positivity Theorem (Fomin-Zelevinsky 00) For any cluster algebra, all cluster variables, i.e. the x n s are Laurent polynomials (i.e. denominator is a monomial) in the initial cluster {x 1, x }. Theorem (Lee-Schiffler 013) The Laurent exapnsions of cluster variables involve exclusively positive integer coefficients. Example (b=c=3): x n x n = x r n If we let x 1 = x = 1 and r = 3, then {x n } n 1 = 1, 1,, 9, 365, , , ,... CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

32 Laurent Phenomenon and Positivity Theorem (Fomin-Zelevinsky 00) For any cluster algebra, all cluster variables, i.e. the x n s are Laurent polynomials (i.e. denominator is a monomial) in the initial cluster {x 1, x }. Theorem (Lee-Schiffler 013) The Laurent exapnsions of cluster variables involve exclusively positive integer coefficients. Example (b=c=3): x n x n = x r n If we let x 1 = x = 1 and r = 3, then {x n } n 1 = 1, 1,, 9, 365, , , ,... Lee-Schiffler provided a combinatorial interpretaion for x n s for any r 3 in terms of colored subpaths of Dyck paths. Lee-Li-Zelevinsky obtained more general combinatorial interpretation (for any A(b, c) and more than cluster variables) in terms of compatible pairs. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

33 Finite versus Infinite Type Let A(b, c) be the subalgebra of Q(x 1, x ) generated by {x n : n Z} with x n s as above. Remark: A(1, 1), A(1, ), and A(1, 3) are of finite type, i.e. {x n } is a finite set. However, for bc 4, we get all x n s are distinct. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

34 Finite versus Infinite Type Let A(b, c) be the subalgebra of Q(x 1, x ) generated by {x n : n Z} with x n s as above. Remark: A(1, 1), A(1, ), and A(1, 3) are of finite type, i.e. {x n } is a finite set. However, for bc 4, we get all x n s are distinct. Example: For A(1, 1), the cluster variables are x 1, x, x 3 = x + 1 x 1, x 4 = x + x x 1 x, x 5 = x x Example: For A(1, ), the cluster variables are x 1, x, x 3 = x + 1 x 1, x 4 = x + x 1 + 1, x 5 = x 1 + x 1 + x + 1 x 1 x x 1 x, x 6 = x x CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

35 Motivated by Positivity Goal: Construct a vector-space basis for A(b, c), i.e. {β i : i I} s.t. (a) β i β j = k c kβ k with c k 0. (i.e. positive structure constants) CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

36 Motivated by Positivity Goal: Construct a vector-space basis for A(b, c), i.e. {β i : i I} s.t. (a) β i β j = k c kβ k with c k 0. (i.e. positive structure constants) (b) Cluster monomials appear as basis elements. (i.e. β = xk dx k+1 e for k Z and d, e 0) CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

37 Motivated by Positivity Goal: Construct a vector-space basis for A(b, c), i.e. {β i : i I} s.t. (a) β i β j = k c kβ k with c k 0. (i.e. positive structure constants) (b) Cluster monomials appear as basis elements. (i.e. β = xk dx k+1 e for k Z and d, e 0) (c) Each β i is an indecomposable positive element. (The Positive Cone: Let A + A(b, c) be the subset of elements that can be written as a positive Laurent polynomial in every cluster {x k, x k+1 }. We want β i A + \ {0} cannot be written as a sum of two nonzero elements of A +.) CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

38 Example: A(, ), i.e. Affine Type, of Type Ã1 x 3 = x + 1, x 4 = x4 + x x 1 x 1 x1 x, x 5 = x6 + 3x4 + 3x x4 1 + x 1 + x 1 x x1 3,... x If we let x 1 = x = 1, then {x 3, x 4, x 5, x 6, x 7,... } = {, 5, 13, 34, 89,... }. Let z = x 1 +x +1 x 1 x = x 0 x 3 x 1 x, which is not a sum of cluster monomials. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

39 Example: A(, ), i.e. Affine Type, of Type Ã1 x 3 = x + 1, x 4 = x4 + x x 1 x 1 x1 x, x 5 = x6 + 3x4 + 3x x4 1 + x 1 + x 1 x x1 3,... x If we let x 1 = x = 1, then {x 3, x 4, x 5, x 6, x 7,... } = {, 5, 13, 34, 89,... }. Let z = x 1 +x +1 x 1 x = x 0 x 3 x 1 x, which is not a sum of cluster monomials. Claim: z = x k +x k+1 +1 x k x k+1 for all k Z. Thus z A +. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

40 Example: A(, ), i.e. Affine Type, of Type Ã1 x 3 = x + 1, x 4 = x4 + x x 1 x 1 x1 x, x 5 = x6 + 3x4 + 3x x4 1 + x 1 + x 1 x x1 3,... x If we let x 1 = x = 1, then {x 3, x 4, x 5, x 6, x 7,... } = {, 5, 13, 34, 89,... }. Let z = x 1 +x +1 x 1 x = x 0 x 3 x 1 x, which is not a sum of cluster monomials. Claim: z = x k +x k+1 +1 x k x k+1 for all k Z. Thus z A +. We define a basis for A(, ) consisting of (for k Z and d k, e k+1 0) {x d k k x e k+1 k+1 } {zk : k 1} where z 1 := z, z := z, and z k := z z k 1 z k for k 3 satsifying the desired properties. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

41 Reminder of Desired Properties Goal: Construct a vector-space basis for A(b, c), i.e. {β i : i I} s.t. (a) β i β j = k c kβ k with c k 0. (i.e. positive structure constants) (b) Cluster monomials appear as basis elements. (i.e. β = xk dx k+1 e for k Z and d, e 0) (c) Each β i is an indecomposable positive element. (The Positive Cone: Let A + A(b, c) be the subset of elements that can be written as a positive Laurent polynomial in every cluster {x k, x k+1 }. We want β i A + \ {0} cannot be written as a sum of two nonzero elements of A +.) CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

42 Greedy Basis Defined Definition Laurent series x is said to be pointed at (a 1, a ) Z if x = 1 x a 1 1 x a p,q 0 c(p, q)x bp 1 x cq with c(p, q) Z and normalized so that c(0, 0) = 1. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

43 Greedy Basis Defined Definition Laurent series x is said to be pointed at (a 1, a ) Z if x = 1 x a 1 1 x a p,q 0 c(p, q)x bp 1 x cq with c(p, q) Z and normalized so that c(0, 0) = 1. Definition Laurent series x is said to be greedy at (a 1, a ) Z if it is pointed at (a 1, a ), and for all (p, q) Z 0 \ {(0, 0}, p ( ) c(p, q) = max ( 1) k 1 a cq + k 1 q ( ) c(p k, q), ( 1) k 1 a1 bp + k 1 c(p, q k). k k k=1 k=1 This yields a unique element we denote as x[a 1, a ]. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

44 Pointed and Greedy Elements and their Terms Illustrated C C B O D1 O B A B O B A O (1) a 1, a 0 () a 1 0 < a (3) a 0 < a 1 (4) 0 < ba a 1 C O C O C D O B A B A B A (5) 0 < ca 1 a (6) 0 < a 1 < ba, 0 < a < ca 1, (a1, a) : non-imaginary root (6) 0 < a 1 < ba, 0 < a < ca 1, (a1, a) : imaginary root CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

45 Theorem (Lee-Li-Zelevinsky (01)) (1) The greedy element x[a 1, a ] is contained in cluster algebra A(b, c). CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

46 Theorem (Lee-Li-Zelevinsky (01)) (1) The greedy element x[a 1, a ] is contained in cluster algebra A(b, c). () The recurrence for c(p, q) terminates and we in fact get a finite sum, i.e. a Laurent polynomial rather than a Laurent series. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

47 Theorem (Lee-Li-Zelevinsky (01)) (1) The greedy element x[a 1, a ] is contained in cluster algebra A(b, c). () The recurrence for c(p, q) terminates and we in fact get a finite sum, i.e. a Laurent polynomial rather than a Laurent series. (3) All of the c(p, q) s are nonnegative integers. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

48 Theorem (Lee-Li-Zelevinsky (01)) (1) The greedy element x[a 1, a ] is contained in cluster algebra A(b, c). () The recurrence for c(p, q) terminates and we in fact get a finite sum, i.e. a Laurent polynomial rather than a Laurent series. (3) All of the c(p, q) s are nonnegative integers. (4) All x[a 1, a ] are indecomposable positive elements of A + A(b, c). CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

49 Theorem (Lee-Li-Zelevinsky (01)) (1) The greedy element x[a 1, a ] is contained in cluster algebra A(b, c). () The recurrence for c(p, q) terminates and we in fact get a finite sum, i.e. a Laurent polynomial rather than a Laurent series. (3) All of the c(p, q) s are nonnegative integers. (4) All x[a 1, a ] are indecomposable positive elements of A + A(b, c). (5) Collection {x[a 1, a ] : (a 1, a ) Z } is a vector-space basis for A(b, c). CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

50 Theorem (Lee-Li-Zelevinsky (01)) (1) The greedy element x[a 1, a ] is contained in cluster algebra A(b, c). () The recurrence for c(p, q) terminates and we in fact get a finite sum, i.e. a Laurent polynomial rather than a Laurent series. (3) All of the c(p, q) s are nonnegative integers. (4) All x[a 1, a ] are indecomposable positive elements of A + A(b, c). (5) Collection {x[a 1, a ] : (a 1, a ) Z } is a vector-space basis for A(b, c). (6) Further, this basis has positive integer structure constants. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

51 Theorem (Lee-Li-Zelevinsky (01)) (1) The greedy element x[a 1, a ] is contained in cluster algebra A(b, c). () The recurrence for c(p, q) terminates and we in fact get a finite sum, i.e. a Laurent polynomial rather than a Laurent series. (3) All of the c(p, q) s are nonnegative integers. (4) All x[a 1, a ] are indecomposable positive elements of A + A(b, c). (5) Collection {x[a 1, a ] : (a 1, a ) Z } is a vector-space basis for A(b, c). (6) Further, this basis has positive integer structure constants. (7) The greedy basis contains all the cluster monomials. In particular, x[ d, e] = x1 dx e if d, e 0 and other cluster monomials correspond to lattice points outside the third quadrant. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

52 Theorem (Lee-Li-Zelevinsky (01)) (1) The greedy element x[a 1, a ] is contained in cluster algebra A(b, c). () The recurrence for c(p, q) terminates and we in fact get a finite sum, i.e. a Laurent polynomial rather than a Laurent series. (3) All of the c(p, q) s are nonnegative integers. (4) All x[a 1, a ] are indecomposable positive elements of A + A(b, c). (5) Collection {x[a 1, a ] : (a 1, a ) Z } is a vector-space basis for A(b, c). (6) Further, this basis has positive integer structure constants. (7) The greedy basis contains all the cluster monomials. In particular, x[ d, e] = x1 dx e if d, e 0 and other cluster monomials correspond to lattice points outside the third quadrant. (8) For a 1, a 0, there are combiantorial formulas for x[a 1, a ] as x[a 1, a ] = 1 x a 1 1 x a (S 1,S ) compatible pairs in D a 1 a x b S 1 x c S 1. In particular, x[a 1, a ] can be written as a Laurent polynomial with positive integer coefficients in any cluster. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

53 Example: A(, ), i.e. Affine Type, of Type Ã1 Revisited x 3 = x + 1, x 4 = x4 + x x 1 x 1 x1 x, x 5 = x6 + 3x4 + 3x x4 1 + x 1 + x 1 x x1 3,... x Let z = x 1 +x +1 x 1 x = x 0 x 3 x 1 x, which is not a sum of cluster monomials. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

54 Example: A(, ), i.e. Affine Type, of Type Ã1 Revisited x 3 = x + 1, x 4 = x4 + x x 1 x 1 x1 x, x 5 = x6 + 3x4 + 3x x4 1 + x 1 + x 1 x x1 3,... x Let z = x 1 +x +1 x 1 x = x 0 x 3 x 1 x, which is not a sum of cluster monomials. We define a basis for A(, ) consisting of (for k Z and d k, e k+1 0) {x d k k x e k+1 k+1 } {zk : k 1} where z 1 := z, z := z, and z k := z z k 1 z k for k 3 z k = x[k, k] and all other lattice points correspond to cluster monomials. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

55 And Now For Something Completely Different Based on mirror symmetry, tropical geometry, holomorphic discs, symplectic geometry, etc., Gross-Hacking-Keel-Kontsevich defined scattering diagrams, broken lines, and theta functions. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

56 And Now For Something Completely Different Based on mirror symmetry, tropical geometry, holomorphic discs, symplectic geometry, etc., Gross-Hacking-Keel-Kontsevich defined scattering diagrams, broken lines, and theta functions. We focus on the rank two version of their theory: Given the cluster algebra A(b, c), we build a scattering diagram D(b, c). Here, D(1, 1), D(1, ), and D(, ) are shown: CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

57 And Now For Something Completely Different Based on mirror symmetry, tropical geometry, holomorphic discs, symplectic geometry, etc., Gross-Hacking-Keel-Kontsevich defined scattering diagrams, broken lines, and theta functions. We focus on the rank two version of their theory: Given the cluster algebra A(b, c), we build a scattering diagram D(b, c). Here, D(1, 1), D(1, ), and D(, ) are shown: 1 + x 1 + x 1 + x 1 + x x1 1 + x1 1 + x 1 1 x 1 + x 1 x 1 + x 1 x 1 + x1 4 x 1 + x1 6 x 4 ( ) 1 + x1 x x1 4 x 6 ( ) = 1 (1 x 1 x )4. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

58 And Now For Something Completely Different A Scattering Diagram D(b, c) when bc 5: The Badlands! A cone where every wall of rational slope appears. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

59 Broken Lines Definition Given a choice of A(b, c), a constructed scattering diagram D(b, c), a point q any wall, and an exponent m = (m 1, m ), we construct ϑ q, m as follows: 1) Start with a line of initial slope m /m 1. Any parallel translate will do. We don t assume this fraction is written in reduced form and instead consider for example the vector m to have larger momentum than m. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

60 Broken Lines Definition Given a choice of A(b, c), a constructed scattering diagram D(b, c), a point q any wall, and an exponent m = (m 1, m ), we construct ϑ q, m as follows: 1) Start with a line of initial slope m /m 1. Any parallel translate will do. We don t assume this fraction is written in reduced form and instead consider for example the vector m to have larger momentum than m. ) When the line hits a wall of the scattering diagram, it can either pass through it or bend/refract a certain amount. There are discrete choices for how much it bends depending on its momentum and angle of intersection. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

61 Broken Lines Definition Given a choice of A(b, c), a constructed scattering diagram D(b, c), a point q any wall, and an exponent m = (m 1, m ), we construct ϑ q, m as follows: 1) Start with a line of initial slope m /m 1. Any parallel translate will do. We don t assume this fraction is written in reduced form and instead consider for example the vector m to have larger momentum than m. ) When the line hits a wall of the scattering diagram, it can either pass through it or bend/refract a certain amount. There are discrete choices for how much it bends depending on its momentum and angle of intersection. More precisely, if d = (d 1, d ) is a primitive normal vector to the wall in question, then let U = m d = m 1 d 1 + m d. Then the line can bend by adding 0, d, d,..., or U d to m. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

62 Broken Lines and Theta Functions 3) After choosing a sequence of such bends at each intersected wall, we consider the broken lines γ that end at the point q. 4) For each broken line γ, we obtain a Laurent monomial x(γ) = x e 1 1 x e where e = (e 1, e ) is the final momentum/exponent of the broken line as it reaches q. ϑ q, m = all broken lines γ starting with exponent m and ending at the point q x(γ) CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

63 Example for A(, ) 1 + x 1 + x 1 x 1 1 x 1 x 1 1 x 1 x 1 1 x q x 1 x 1 γ 1 x 1 x 1 γ 3 x 1 x x 1 x 4 γ 1 + x 4 1 x CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

64 Example for A(, ) 1 + x 1 + x 1 x 1 1 x 1 x 1 1 x 1 x 1 1 x q x 1 x 1 γ 1 x 1 x 1 γ 3 x 1 x x 1 x 4 γ 1 + x 4 1 x ϑ = x q, (1, 1) 1x 1 + x1 1 x 1 + x1 1 x = x 1 +x +1 x 1 x = z A(, ) CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

65 Theorem (Cheung-Gross-Muller-M-Rupel-Stella-Williams) The greedy basis element x[a 1, a ] A(b, c) equals the theta function (basis element) ϑ q, T ( a 1, a ) when i) q is in the first quadrant generically ii) drawing broken lines using the scattering diagram D(b, c) iii) T is a piece-wise linear transformation that simply pushes the d-vector fan into the g-vector fan by pushing rays clockwise. 1 + x 1 + x 1 + x1 1 + x x 1 x 1 + x 1 x x 1 x 1 + x 1 x CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, 016 / 7

66 Sketch of Proof Suppose a broken line γ passes through the point (q 1, q ) with exponent/momentum m = (m 1, m ) as it does. Define the angular momentum at this point to be q m 1 q 1 m. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

67 Sketch of Proof Suppose a broken line γ passes through the point (q 1, q ) with exponent/momentum m = (m 1, m ) as it does. Define the angular momentum at this point to be q m 1 q 1 m. Claim: (1) Even as broken line γ bends at a wall, its angular momentum is constant. () For any broken line γ which has positive angular momentum, any bends in the third quadrant will lead to a decrease in slope. The opposite statement holds when γ has negative angulary momentum. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

68 Sketch of Proof Suppose a broken line γ passes through the point (q 1, q ) with exponent/momentum m = (m 1, m ) as it does. Define the angular momentum at this point to be q m 1 q 1 m. Claim: (1) Even as broken line γ bends at a wall, its angular momentum is constant. () For any broken line γ which has positive angular momentum, any bends in the third quadrant will lead to a decrease in slope. The opposite statement holds when γ has negative angulary momentum. Consequently, we get on bounds on what the exponent of the Laurent monomial x(γ) = x e 1 1 x e can be. CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

69 Sketch of Proof Allows us to build a Newton polygon (more precisely a half-open quadrilateral) of the support of ϑ q, T ( a 1, a ). C C B O D1 O B A B O B A O (1) a 1, a 0 () a 1 0 < a (3) a 0 < a 1 (4) 0 < ba a 1 C O C O C D O B A B A B A (5) 0 < ca 1 a (6) 0 < a 1 < ba, 0 < a < ca 1, (a1, a) : non-imaginary root (6) 0 < a 1 < ba, 0 < a < ca 1, (a1, a) : imaginary root CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

70 Sketch of Proof O = (0, 0), A = ( a 1 + ba, a ), B = ( a 1, a ), C = ( a 1, a + ca 1 ), D 1 = ( a 1 + ba, ca 1 (bc + 1)a ), D = (ba (bc + 1)a 1, a + ca 1 ). Then the support region of ϑ q, T ( a 1, a ) is C C B O D1 O B A B O B A O (1) a 1, a 0 () a 1 0 < a (3) a 0 < a 1 (4) 0 < ba a 1 C O C O C D O B A B A B A (5) 0 < ca 1 a (6) 0 < a 1 < ba, 0 < a < ca 1, (a1, a) : non-imaginary root (6) 0 < a 1 < ba, 0 < a < ca 1, (a1, a) : imaginary root CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

71 Sketch of Proof Scholium (based on Lee-Li-Zelevinsky s proof) Any Laurent polynomial with a support contained in one of these half-open quadrilaterals and with coefficient 1 on x a 1 1 x a must be the greedy basis element x[a 1, a ]. C C B O D1 O B A B O B A O (1) a 1, a 0 () a 1 0 < a (3) a 0 < a 1 (4) 0 < ba a 1 C O C O C D O B A B A B A (5) 0 < ca 1 a (6) 0 < a 1 < ba, 0 < a < ca 1, (a1, a) : non-imaginary root (6) 0 < a 1 < ba, 0 < a < ca 1, (a1, a) : imaginary root CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7

72 Thank You for Listening Open Question 1: Find a combiantorial bijection between broken lines and compatible subsets. Open Question : Reverse-engineer greedy bases in higher rank cluster algebras using theta functions. Slides at musiker/haiku.pdf The Greedy Basis Equals the Theta Basis: A Rank Two Haiku (with Man Wai Cheung, Mark Gross, Greg Muller, Dylan Rupel, Salvatore Stella, and Harold Williams) CGMMRSW (Univ. Minnesota, et. al.) The Greedy Basis Equals the Theta Basis April 17, / 7