Class groups of cluster algebras Daniel Smertnig University of - PowerPoint PPT Presentation

Class groups of cluster algebras Daniel Smertnig University of Waterloo with Ana Garcia Elsener (Graz, Austria) and Philipp Lampe (Kent, UK) arXiv:1712.06512

Cluster algebras Cluster algebras were introduced by Fomin and Zelevinsky in 2002. More than 600 preprints on the arXiv. Connections to many difgerent area of mathematics: T otal positivity, combinatorics, T eichmüller theory, representation theory, knot theory, Lie algebras, … Defjned via combinatorial data: Quivers and mutations.

Quivers 1 2 4 3 1 2 4 3 2 1 2 1 1 Quiver : fjnite directed graph (for us:) no loops or 2-cycles parallel arrows allowed. ✗ ✗ ✓ acyclic with cycle(s)

Quiver mutations I 1 𝑦 2 𝑦 𝑘 . 𝑗→𝑘 𝑘→𝑗 𝑦 𝑗 𝑦 ′ 3 1 2 2 , 𝑦 3 } 3 2 1 1 2 3 Mutation of a quiver 𝑅 at vertex 𝑗 . 1. For arrows 𝑘 → 𝑗 → 𝑙 , add arrows 𝑘 → 𝑙 . 2. Flip all arrows incident 3 = with 𝑗 . 3. Remove 2-cycles. ✓ Parallel mutation of seed : In {𝑦 1 , … , 𝑦 𝑜 } replace 𝑦 𝑗 by 𝑦 ′ 𝑗 : {𝑦 1 , 𝑦 2 , 𝑦 3 } ⇝ {𝑦 1 , 𝑦 1 +𝑦 3 𝑗 = ∏ 𝑦 𝑘 + ∏

Quiver mutations II 1 𝑦 2 𝑦 1 𝑦 2 𝑦 1 𝑦 ′ 3 2 1 ⇝ 3 1 2 2 3 2 𝑦 2 1 3 , 𝑦 3 } 2 1 3 , 𝑦 3 }. {𝑦 1 , 𝑦 1 +𝑦 3 1. For arrows 𝑘 → 𝑗 → 𝑙 , add arrows 𝑘 → 𝑙 . 2. Flip all arrows incident with 𝑗 . 3. Remove 2-cycles. 2 + 𝑦 3 , so new seed { 𝑦 1 +(1+𝑦 2 )𝑦 3 , 𝑦 1 +𝑦 3 1 = 𝑦 ′

ℤ[𝑦 1 , … , 𝑦 𝑜 ] ⊂ 𝐵 ⊂ ℤ(𝑦 1 , … , 𝑦 𝑜 ). Cluster algebras Let 𝑅 be a quiver on vertices {1, … , 𝑜} and {𝑦 1 , … , 𝑦 𝑜 } an initial seed. Mutation yields a (possibly infjnite) collection of seeds. Each element of a seed is a cluster variable . Defjnition The cluster algebra 𝐵 = 𝐵(𝑅) is the subalgebra of ℤ(𝑦 1 , … , 𝑦 𝑜 ) generated by all cluster variables.

Cluster algebras 𝑦 2 𝑦 1 𝑦 2 𝑦 3 , 𝑦 2 𝑦 3 𝑦 1 𝑦 2 , 𝑦 3 ] 𝑦 1 3 2 1 Example ( 𝐵 3 ) 𝐵 3 : 𝐵(𝐵 3 ) = ℤ[𝑦 1 , 𝑦 2 , 𝑦 3 , 1 + 𝑦 2 , 𝑦 1 + 𝑦 3 , 1 + 𝑦 2 𝑦 1 + (1 + 𝑦 2 )𝑦 3 , (1 + 𝑦 2 )𝑦 1 + 𝑦 3 (1 + 𝑦 2 )(𝑦 1 + 𝑦 3 )

Laurent phenomenon 𝐵 ⊆ ℤ[𝑦 ±1 Theorem (Fomin, Zelevinsky 2002) Denominators of cluster variables are monomials, hence 1 , … , 𝑦 ±1 𝑜 ].

Finite type classifjcation 𝐵 𝑜 𝐶 𝑜 𝐷 𝑜 𝐸 𝑜 𝐹 6 𝐹 7 𝐹 8 𝐺 4 𝐻 2 Theorem (Fomin, Zelevinsky 2003) Cluster algebras of fjnite type (=having fjnitely many cluster variables) are classifjed by Dynkin diagrams.

Goal Understand factorizations of elements into atoms ( irreducible elements ) in cluster algebras. 1 When is 𝐵(𝑅) factorial (a UFD )? 2 What happens if it is not?

Factorizations: earlier results Theorem (Geiß, Leclerc, Schröer, 2012) 1 Cluster variables are (pairwise non-associated) atoms. 2 If 𝐵 is factorial, all exchange polynomials 𝑔 𝑗 ∈ ℤ[𝑦 1 , … , 𝑦 𝑜 ] with 𝑦 𝑗 𝑦 ′ 𝑗 = 𝑔 𝑗 are irreducible and pairwise distinct. Example If 𝑔 𝑗 = 𝑕 1 ⋯ 𝑕 𝑙 , then 𝑦 𝑗 𝑦 ′ 𝑗 = 𝑕 1 ⋯ 𝑕 𝑙 . If 𝑔 𝑗 = 𝑔 𝑘 for 𝑗 ≠ 𝑘 , then 𝑦 𝑗 𝑦 ′ 𝑘 . 𝑗 = 𝑦 𝑘 𝑦 ′ Theorem (Lampe, 2012, 2014) Classifjcation of factoriality for simply-laced Dynkin types ( 𝐵 𝑜 , 𝐸 𝑜 , 𝐹 𝑜 ).

Acyclic cluster algebras 𝐵 = ℤ[𝑦 1 , 𝑦 ′ 𝑦 𝑗 𝑦 ′ 𝑜 ]/(𝑌 𝑗 𝑌 ′ 1 , … , 𝑌 𝑜 , 𝑌 ′ 𝑘→𝑗 𝑜 ] ≅ ℤ[𝑌 1 , 𝑌 ′ 𝑗→𝑘 𝑦 𝑘 𝑦 ′ 1 , … , 𝑦 𝑜 , 𝑦 ′ 𝑗 ... obtained from initial seed {𝑦 1 , … , 𝑦 𝑜 } by mutation at 𝑗 : with exchange polynomials. 𝑗 = 𝑔 𝑗 𝑔 𝑗 = ∏ 𝑦 𝑘 + ∏ Theorem (Berenstein, Fomin, Zelevinsky 2006; Muller 2014) Let 𝑅 be acyclic . Then 𝑗 −𝑔 𝑗 ). 𝐵 is fjnitely generated , noetherian , integrally closed . Corollary (Locally) acyclic cluster algebras are Krull domains .

𝜒∶ (𝐵 ∖ {0}, ⋅) → ℬ(𝐻 0 ), Krull domains Theorem Let 𝐵 be a Krull domain with divisor class group 𝐻 = 𝒟(𝐵) and 𝐻 0 = { [𝔮] ∶ 𝔮 divisorial [=height- 1 ] prime } ⊆ 𝐻. Then there exists a transfer homomorphism with ℬ(𝐻 0 ) the monoid of zero-sum sequences over 𝐻 0 . Corollary 1 𝐵 is factorial (= a UFD ) if and only if 𝐻 is trivial. 2 Factorization theory of 𝐵 determined by 𝐻 and 𝐻 0 .

First Main Result Theorem (Garcia Elsener, Lampe, S., 2017) Let 𝐵 = 𝐵(𝑅) be a Krull domain (e.g., 𝑅 acyclic), and {𝑦 1 , … , 𝑦 𝑜 } a seed. Then 𝐻 = 𝒟(𝐵) ≅ ℤ 𝑠 for some 𝑠 ≥ 0 , and every class contains infjnitely many prime divisors ( 𝐻 0 = 𝐻 ). 𝐵 is factorial if and only if 𝑠 = 0 . 𝑠 = 𝑢 − 𝑜 with 𝑢 the number of height-1 primes containing one of 𝑦 1 , … , 𝑦 𝑜 .

Consequences Corollary For 𝑅 acyclic 1 , the necessary conditions of Geiß–Leclerc–Schöer are suffjcient for 𝐵(𝑅) to be factorial. Corollary Acyclic cluster algebras with (invertible) principal coeffjcients are factorial. Corollary If 𝐵 = 𝐵(𝑅) is a Krull domain but not factorial, then Kainrath’s Theorem applies: for every 𝑀 ⊆ ℤ ≥2 there exists 𝑏 ∈ 𝐵 with L (𝑏) = 𝑀 . 1 without isolated vertices

Goal Get explicit description of the rank 𝑠 of 𝒟(𝐵) ≅ ℤ 𝑠 , directly in terms of 𝑅 . Restrict to 𝑅 acyclic.

Exchange matrix For a quiver 𝑅 , defjne Signed adjacency matrix: skew-symmetric 𝑜 × 𝑜 -matrix 𝐶 = 𝐶(𝑅) , with 𝑐 𝑗𝑘 = #{ arrows 𝑗 → 𝑘} − #{ arrows 𝑘 → 𝑗}. a vector 𝑒 ∈ ℤ 𝑜 with 𝑒 𝑗 the gcd of the 𝑗 -th column of 𝐶 .

Partners Defjnition Vertices 𝑗 , 𝑘 ∈ [1, 𝑜] are partners if the following equivalent conditions hold. 1 Exchange polynomials 𝑔 𝑗 , 𝑔 𝑘 have a common factor. 2 there exist odd 𝑑 𝑗 , 𝑑 𝑘 ∈ ℤ : 𝑑 𝑘 𝑐 ∗𝑗 = 𝑑 𝑗 𝑐 ∗𝑘 . 3 v 2 (𝑒 𝑗 ) = v 2 (𝑒 𝑘 ) and 𝑐 ∗𝑗 /𝑒 𝑗 = ±𝑐 ∗𝑘 /𝑒 𝑘 . Partnership is an equivalence relation on [1, 𝑜] : Partner sets .

Example 0 1) 1, 𝑒 = (1, ⎠ ⎟ ⎟ ⎟ ⎟ ⎞ 0 −1 0 1 −1 0 1 0 ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ 𝐶 = 3 2 1 Example Partner sets: {1, 3} , {2} .

𝑠 = Main result for acyclic quivers ∑ 𝑠 𝑊 , ∑ 𝑒≥1 For a partner set 𝑊 ⊆ [1, 𝑜] and 𝑒 ≥ 1 , let c (𝑊, 𝑒) = #{𝑗 ∈ 𝑊 ∣ 𝑒 divides 𝑒 𝑗 }. (Recall: 𝑒 𝑗 is gcd of the 𝑗 -th column of adjacency matrix 𝐶 ) Theorem (Garcia Elsener, Lampe, S. 2017) Let 𝑅 be acyclic and 𝐵 = 𝐵(𝑅) . Then 𝒟(𝐵) ≅ ℤ 𝑠 with 𝑊 a partner set where (2 c (𝑊,𝑒) − 1) − #𝑊. 𝑠 𝑊 = 𝑒 odd

Corollary: fjnite type Corollary If 𝑅 is acyclic and without parallel arrows , then 𝐵(𝑅) is factorial if and only if there are no partners 𝑗 ≠ 𝑘 . Corollary For the cluster algebras of Dynkin types: T ype 𝐵 𝑜 is factorial if 𝑜 ≠ 3 , and 𝒟(𝐵 3 ) ≅ ℤ . T ype 𝐶 𝑜 is factorial if 𝑜 ≠ 3 , and 𝒟(𝐶 3 ) ≅ ℤ . T ype 𝐷 𝑜 is factorial. T ype 𝐸 𝑜 has 𝒟(𝐸 𝑜 ) ≅ ℤ for 𝑜 > 4 , and 𝒟(𝐸 4 ) ≅ ℤ 4 . T ypes 𝐹 6 , 𝐹 7 , and 𝐹 8 are factorial. T ype 𝐺 4 is factorial. T ype 𝐻 2 has 𝒟(𝐻 2 ) ≅ ℤ .

Summary For cluster algebras that are Krull domains, the class group is always of form ℤ 𝑠 . For acyclic cluster algebras, 𝑠 can be expressed directly in terms of the quiver and is trivial to compute . Similar results hold over fjelds of characteristic 0 as ground ring, and for skew-symmetrizable cluster algebras with (invertible) frozen variables. Open questions How to determine 𝑠 in the locally acyclic case? When is 𝐵(𝑅) a Krull domain? [completely] integrally closed?