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

• tal

positivity, combinatorics, T eichmüller theory, representation theory, knot theory, Lie algebras, … Defjned via combinatorial data: Quivers and mutations.

Quivers

Quiver: fjnite directed graph (for us:) no loops or 2-cycles parallel arrows allowed. 1

✗

1 2

✗

1 2

✓

1 3 4 2 acyclic 1 3 4 2 with cycle(s)

Quiver mutations I

Mutation of a quiver 𝑅 at vertex 𝑗. 1 2 3

• 1. For arrows 𝑘 → 𝑗 → 𝑙, add arrows

𝑘 → 𝑙. 1 2 3

• 2. Flip all arrows incident

with 𝑗. 1 2 3 = 2 1 3

• 3. Remove 2-cycles.

✓

Parallel mutation of seed: In {𝑦1, … , 𝑦𝑜} replace 𝑦𝑗 by 𝑦′

𝑗:

𝑦𝑗𝑦′

𝑗 = ∏ 𝑘→𝑗

𝑦𝑘 + ∏

𝑗→𝑘

𝑦𝑘. {𝑦1, 𝑦2, 𝑦3} ⇝ {𝑦1, 𝑦1+𝑦3

𝑦2

, 𝑦3}

Quiver mutations II

2 1 3 {𝑦1, 𝑦1+𝑦3

𝑦2

, 𝑦3}

• 1. For arrows 𝑘 → 𝑗 → 𝑙, add arrows

𝑘 → 𝑙. 2 1 3

• 2. Flip all arrows incident with 𝑗.

2 1 3

• 3. Remove 2-cycles.

2 1 3 ⇝ 1 2 3 𝑦1𝑦′

1 = 𝑦′ 2 + 𝑦3, so new seed { 𝑦1+(1+𝑦2)𝑦3 𝑦1𝑦2

, 𝑦1+𝑦3

𝑦2

, 𝑦3}.

Cluster algebras

Let 𝑅 be a quiver on vertices {1, … , 𝑜} and {𝑦1, … , 𝑦𝑜} 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(𝑦1, … , 𝑦𝑜) generated by all cluster variables. ℤ[𝑦1, … , 𝑦𝑜] ⊂ 𝐵 ⊂ ℤ(𝑦1, … , 𝑦𝑜).

Cluster algebras

Example (𝐵3) 𝐵3: 1 2 3 𝐵(𝐵3) = ℤ[𝑦1, 𝑦2, 𝑦3, 1 + 𝑦2 𝑦1 , 𝑦1 + 𝑦3 𝑦2 , 1 + 𝑦2 𝑦3 , 𝑦1 + (1 + 𝑦2)𝑦3 𝑦1𝑦2 , (1 + 𝑦2)𝑦1 + 𝑦3 𝑦2𝑦3 , (1 + 𝑦2)(𝑦1 + 𝑦3) 𝑦1𝑦2𝑦3 ]

Laurent phenomenon

Theorem (Fomin, Zelevinsky 2002) Denominators of cluster variables are monomials, hence 𝐵 ⊆ ℤ[𝑦±1

1 , … , 𝑦±1 𝑜 ].

Finite type classifjcation

Theorem (Fomin, Zelevinsky 2003) Cluster algebras of fjnite type (=having fjnitely many cluster variables) are classifjed by Dynkin diagrams. 𝐵𝑜 𝐶𝑜 𝐷𝑜 𝐸𝑜 𝐹6 𝐹7 𝐹8 𝐺4 𝐻2

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

𝑔𝑗 ∈ ℤ[𝑦1, … , 𝑦𝑜] with 𝑦𝑗𝑦′

𝑗 = 𝑔𝑗 are irreducible and

pairwise distinct. Example If 𝑔𝑗 = 𝑕1 ⋯ 𝑕𝑙, then 𝑦𝑗𝑦′

𝑗 = 𝑕1 ⋯ 𝑕𝑙.

If 𝑔𝑗 = 𝑔𝑘 for 𝑗 ≠ 𝑘, then 𝑦𝑗𝑦′

𝑗 = 𝑦𝑘𝑦′ 𝑘.

Theorem (Lampe, 2012, 2014) Classifjcation of factoriality for simply-laced Dynkin types (𝐵𝑜, 𝐸𝑜, 𝐹𝑜).

Acyclic cluster algebras

𝑦′

𝑗... obtained from initial seed {𝑦1, … , 𝑦𝑜} by mutation at 𝑗:

𝑦𝑗𝑦′

𝑗 = 𝑔𝑗

with 𝑔𝑗 = ∏

𝑘→𝑗

𝑦𝑘 + ∏

𝑗→𝑘

𝑦𝑘 exchange polynomials. Theorem (Berenstein, Fomin, Zelevinsky 2006; Muller 2014) Let 𝑅 be acyclic. Then 𝐵 = ℤ[𝑦1, 𝑦′

1, … , 𝑦𝑜, 𝑦′ 𝑜] ≅ ℤ[𝑌1, 𝑌′ 1, … , 𝑌𝑜, 𝑌′ 𝑜]/(𝑌𝑗𝑌′ 𝑗 −𝑔𝑗).

𝐵 is fjnitely generated, noetherian, integrally closed. Corollary (Locally) acyclic cluster algebras are Krull domains.

Krull domains

Theorem Let 𝐵 be a Krull domain with divisor class group 𝐻 = 𝒟(𝐵) and 𝐻0 = { [𝔮] ∶ 𝔮 divisorial [=height-1] prime } ⊆ 𝐻. Then there exists a transfer homomorphism 𝜒∶ (𝐵 ∖ {0}, ⋅) → ℬ(𝐻0), with ℬ(𝐻0) the monoid of zero-sum sequences over 𝐻0. Corollary

1 𝐵 is factorial (= a UFD) if and only if 𝐻 is trivial. 2 Factorization theory of 𝐵 determined by 𝐻 and 𝐻0.

First Main Result

Theorem (Garcia Elsener, Lampe, S., 2017) Let 𝐵 = 𝐵(𝑅) be a Krull domain (e.g., 𝑅 acyclic), and {𝑦1, … , 𝑦𝑜} a seed. Then 𝐻 = 𝒟(𝐵) ≅ ℤ𝑠 for some 𝑠 ≥ 0, and every class contains infjnitely many prime divisors (𝐻0 = 𝐻). 𝐵 is factorial if and only if 𝑠 = 0. 𝑠 = 𝑢 − 𝑜 with 𝑢 the number of height-1 primes containing

• ne of 𝑦1, … , 𝑦𝑜.

Consequences

Corollary For 𝑅 acyclic1, 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.

1without isolated vertices

Goal Get explicit description of the rank 𝑠 of 𝒟(𝐵) ≅ ℤ𝑠, 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 𝑐𝑗𝑘 = #{arrows 𝑗 → 𝑘} − #{arrows 𝑘 → 𝑗}. a vector 𝑒 ∈ ℤ𝑜 with 𝑒𝑗 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 𝑔𝑗, 𝑔𝑘 have a common factor. 2 there exist odd 𝑑𝑗, 𝑑𝑘 ∈ ℤ: 𝑑𝑘𝑐∗𝑗 = 𝑑𝑗𝑐∗𝑘. 3 v2(𝑒𝑗) = v2(𝑒𝑘) and 𝑐∗𝑗/𝑒𝑗 = ±𝑐∗𝑘/𝑒𝑘.

Partnership is an equivalence relation on [1, 𝑜]: Partner sets.

Example

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

Main result for acyclic quivers

For a partner set 𝑊 ⊆ [1, 𝑜] and 𝑒 ≥ 1, let c(𝑊, 𝑒) = #{𝑗 ∈ 𝑊 ∣ 𝑒 divides 𝑒𝑗}. (Recall: 𝑒𝑗 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) ≅ ℤ𝑠 with 𝑠 = ∑

𝑊 a partner set

𝑠𝑊, where 𝑠𝑊 = ∑

𝑒≥1 𝑒 odd

(2c(𝑊,𝑒) − 1) − #𝑊.

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 𝐵𝑜 is factorial if 𝑜 ≠ 3, and 𝒟(𝐵3) ≅ ℤ. T ype 𝐶𝑜 is factorial if 𝑜 ≠ 3, and 𝒟(𝐶3) ≅ ℤ. T ype 𝐷𝑜 is factorial. T ype 𝐸𝑜 has 𝒟(𝐸𝑜) ≅ ℤ for 𝑜 > 4, and 𝒟(𝐸4) ≅ ℤ4. T ypes 𝐹6, 𝐹7, and 𝐹8 are factorial. T ype 𝐺4 is factorial. T ype 𝐻2 has 𝒟(𝐻2) ≅ ℤ.

Summary

For cluster algebras that are Krull domains, the class group is always of form ℤ𝑠. For acyclic cluster algebras, 𝑠 can be expressed directly in terms of the quiver and is trivial to compute. Similar results hold

• ver 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?