Mutations of non-integer quivers: finite mutation type. Anna Felikson

Mutations of non-integer quivers: finite mutation type Anna Felikson Durham University (joint works with Pavel Tumarkin and Philipp Lampe) Jerusalem, December 0, 08

. Mutations of non-integer quivers: B = {b ij } a skew-symmetric matrix with b ij R. Mutate B by usual mutation rule: b ij = { bij, if i = k or j = k b ij + ( b ik b kj + b ik b kj ), otherwise

. Mutations of non-integer quivers: B = {b ij } a skew-symmetric matrix with b ij R. Mutate B by usual mutation rule: b ij = { bij, if i = k or j = k b ij + ( b ik b kj + b ik b kj ), otherwise Why: Philipp Lampe, On the approximate periodicity of sequences attached to Why: noncrystallographic root systems, To appear in Experimental Mathematics (08). Why: Integer finite type contains types A, B, C, D, E, F... - but not H 3, H 4! Why: Geometric realization of acyclic mutation classes by partial reflections Why: allow non-integer values.

. Mutations of non-integer quivers: Mutation µ k of a non-integer quiver: as ) reverse all arrows incident to k; as ) for every path i p k q j with p, q > 0 apply p k q µ k p k q r r = pq r

. Mutations of non-integer quivers: Mutation µ k of a non-integer quiver: as ) reverse all arrows incident to k; as ) for every path i p k q j with p, q > 0 apply p k q µ k p k q r r = pq r Example: B 3 k µ k k

. In Rank 3: { acyclic mutation class } { Geometric realisation by partial reflections }

. In Rank 3: { acyclic mutation class } { Geometric realisation Q = (b ij ) M = ( bij b ij by partial reflections } ) = v i, v j (v,..., v n ) - basis of quadratic space V of same signature as M.

. In Rank 3: { acyclic mutation class } { Geometric realisation Q = (b ij ) M = ( bij b ij by partial reflections } ) = v i, v j (v,..., v n ) - basis of quadratic space V of same signature as M. Given v V with v, v =, consider reflection r v (u) = u u, v v. Let G = s,..., s n where s i = r vi. G acts discretely in a cone C V with fundamental domain F = n i= Π i, where Π i = {u V u, v i < 0}.

. In Rank 3: { acyclic mutation class } { Geometric realisation by partial reflections } Acyclic quiver Q reflection group G = s,..., s n with chosen generating reflections Mutation Partial reflection v i v i, v k v k, µ k (v i ) = v k, v i, if k i in Q if i = k otherwise Theorem. (Barot, Geiss, Zelevinsky 06; Seven ) For integer quivers (but also for real ones in rank 3): The values v i, v j change under mutations in the same way as the weights of the arrows in Q.

. In Rank 3: { acyclic mutation class } { Geometric realisation by partial reflections }. In Rank 3: { cyclic mutation class } { Geometric realisation by partial π-rotations }

. In Rank 3: { acyclic mutation class } { Geometric realisation by partial reflections }. In Rank 3: { cyclic mutation class } { Geometric realisation by partial π-rotations } Theorem [FT 6]. Any mutation-finite rank 3 quiver is mutation-equivalent to one of Markov quiver: Finite type quivers: A 3 Affine quivers: n n B 3 4 H 3 H 3 H 3 (Here, a label k m stays for the weight b ij = cos kπ m.)

. In Rank 3: All mut. finite (but Markov) mutation classes. In Rank 3: have geometric realisation by reflections.

. In Rank 3: All mut. finite (but Markov) mutation classes. In Rank 3: have geometric realisation by reflections. Properties: aaaa Realisation in E affine type aaaa Realisation in S finite type

. In Rank 3: All mut. finite (but Markov) mutation classes. In Rank 3: have geometric realisation by reflections. Properties: aaaa Realisation in E affine type aaaa Realisation in S finite type aaaa All weights are cos πm d, m, d Z, m d. aaaa If there is d > then Q is affine. aaaa If Q is acyclic then the corresponding triangle is acute-angled asdasdsadsadsadsadsadsadsadsadsadsadsadasdsad (or has obtuse angles). asdasdsadsadsadsa (in E : m d + m d + m 3 d 3 = ).

3. In Rank n: Want: To use rank 3 classification to get general classification. Expectation: aa Finite type should come from finite root systems aa Acyclic quivers should come form acute-angled simplices aa (in the corresponding reflection group) How to proceed: aa d 4: Q is a symmetrisation of an integer diagram aaaaaaaaaaaaaaaaaaaaa orbifolds and F -type quivers aa d : computer search gives fin. many examples aaaaaaaaaaaaaaaaaaaaa in ranks 3,4,,6 and nothing else. aa d > : easy to check that nothing in rank ; aaaaaaaaaaaaaaaaaaaaa in rank 4: there are 3 series of answers. All of them geometric!

3. In Rank n: Want: To use rank 3 classification to get general classification. Expectation: aa Finite type should come from finite root systems aa Acyclic quivers should come from acute-angled simplices aa (in the corresponding reflection group) How to proceed: aa d 4: Q is a symmetrisation of an integer diagram aaaaaaaaaaaaaaaaaaaaa orbifolds and F -type quivers aa d : computer search gives fin. many examples aaaaaaaaaaaaaaaaaaaaa in ranks 3,4,,6 and nothing else. aa d > : easy to check that nothing in rank ; aaaaaaaaaaaaaaaaaaaaa in rank 4: there are 3 series of answers. All of them geometric!

3. In Rank n: Answer: Thm. [FT 8] Mut.-fin. G,n or orbifold or as in Table: Finite type Affine type Extended affine type rank 3 rank 4 rank rank 6 4 F 4 H 3 H 4 H 3 n n H 3 G,n n n H 4 H 4 H 4 H 4 H n n n n n n n n n n+ n n+ n+ n+ n n+ H 3 4 F4 3 G (,+),n G (, ),n G (, ),n+ H4 H (,) 3 4 4 4 4 F (,+) 4 F (, ) 4 H (,) 4

3. In Rank n: Expected: Acyclic quivers are coming form acute-angled simplices

3. In Rank n: Expected: Acyclic quivers are coming form acute-angled simplices aaa YES! - all acyclic quivers correspond to acute angled simplices aaa YES! - there many be many of them in one mutation class. aaa But not all acute-angled simplices (decorated with acyclic quivers) aaa But not all a induce mut.-finite mutation classes. aaaaaaaaaaaaaaaaaa Why??? - No idea... aaaaaaaaaaaaaaaaaaaaaa i.e. geometricity does not follow a-priory

4. Exchange graph: Rank 3 - finite type Example: H 3 (mutation class and exchange graph ) (cf. generalised associahedron in Fomin - Reading)

4. Exchange graph: Rank 3 - finite type Exchange graphs for H 3 and H 3 are graphs on a torus aaaaa (with two acyclic belts each): N H G M M N G H D C B A F E C D B A F E H 3 H 3 Two different acyclic representatives in each of H 3 and H 3 :

4. Exchange graph: Rank 3 - affine type from here: joint with Philipp Lampe

4. Exchange graph: Rank 3 - affine type Initial acyclic seed from here: joint with Philipp Lampe

4. Exchange graph: Rank 3 - affine type Initial acyclic seed from here: joint with Philipp Lampe An acyclic belt

4. Exchange graph: Rank 3 - affine type Initial acyclic seed from here: joint with Philipp Lampe An acyclic belt Belt (or billiard) line passes through two feet of altitudes cf. Fagnano s problem: billiard tranjectory in triangle.

4. Exchange graph: Rank 3 - affine type Away from acyclic belt: two feet of altitudes are on the belt line. from here: joint with Philipp Lampe Belt (or billiard) line passes through two feet of altitudes cf. Fagnano s problem: billiard tranjectory in triangle.

4. Exchange graph: Rank 3 - affine type from here: joint with Philipp Lampe Away from acyclic belt: two feet of altitudes are on the belt line. If there are shifts, they are parallel to the belt line Belt (or billiard) line passes through two feet of altitudes cf. Fagnano s problem: billiard tranjectory in triangle.

4. Exchange graph: Rank 3 - affine type from here: joint with Philipp Lampe Invariant under mutation: T ( ) = a i sin(a j ) sin(a k ) If there are shifts, they are parallel to the belt line H H H 3

4. Exchange graph: Rank 3 - affine type from here: joint with Philipp Lampe Invariant under mutation: T ( ) = a i sin(a j ) sin(a k ) Triangles with same angles are congruent If there are shifts, they are parallel to the belt line Need: to find all shifts! H H H 3

4. Exchange graph: Rank 3 - affine type There are more shifts: each infinite region induces a shift: from here: joint with Philipp Lampe a α

4. Exchange graph: Rank 3 - affine type There are more shifts: each infinite region induces a shift: from here: joint with Philipp Lampe If a if the finite size, α the angle then T ( ) = a sin α So, a = T ( ) sin α a α

4. Exchange graph: Rank 3 - affine type from here: joint with Philipp Lampe Theorem. [FL 08] Let Q be an affine type rank 3 mutation-finite quiver. Then the exchange graph of Q grows polynomially and is quasi-isometric to some lattice L. rk Z (L) = { ϕ(d), for some d Z; ϕ(d), otherwise. Here, ϕ(d) = #{k {,,..., d} gcd(k, d) = } aaaaaaaa is the Euler s totient function.

4. Exchange graph: Rank 3 - affine type from here: joint with Philipp Lampe Example: exchange graph for d =

4. Exchange graph: Rank 3 - affine type from here: joint with Philipp Lampe Remark: Similar belt line (and similar geometry) takes place when Q has a geometric presentation by reflections on S or H.

. Remarks on definition of mutation We defined: Mutation Partial reflection v i v i, v k v k, if k i in Q µ k (v i ) = v k, if i = k v i, otherwise Not an involution!

. Remarks on definition of mutation We defined: Mutation Partial reflection v i v i, v k v k, if k i in Q µ k (v i ) = v k, if i = k v i, otherwise Not an involution! Define: if v k is positive: µ k (v i ) = v i v i, v k v k, v k, v i, if k i in Q if i = k otherwise if v k is negative: µ k (v i ) = v i v i, v k v k, v k, v i, if k i in Q if i = k otherwise What to mean by positive / negative?

. Remarks on definition of mutation What to mean by positive / negative? In rank : aaa Period Period 7 admissible positions forbidden positions of the reference point

. Remarks on definition of mutation What to mean by positive / negative? In rank 3: Reference point is in admissible position, In rank 3: if it is in admissible position In rank 3: for every rank subquiver in every cluster. Theorem [FL 8] For every rank 3 finite type quiver, asd () there are geometric realisations asd () with the reference point in an admissible position; asd () all such realisations result in the same exchange graph; asd (3) in all such realisations, the reference point belongs asd (3) to some acute-angled (i.e. acyclic) domain;

. Remarks on definition of mutation What to mean by positive / negative? Theorem [FL 8] For every rank 3 affine quiver, asd there exists a unique admissible position of the reference point: asd it is the limit point at the end of the belt line. aasdasdasdasdsa

Non-integer quivers:geometry and mutation-finiteness Non-integer quivers:geometry and mutation-finiteness Introduction B = {b ij } a skew-symmetric matrix with b ij R. Mutate { B by usual mutation rule: b ij = b ij, if i = k or j = k b ij + ( b ik b kj + b ik b kj ), otherwise B defines a non-integer quiver (with arrows of real weights b ij = b ji ). Mutation µ k of a non-integer quiver: ) reverse all arrows incident to k; ) for every path i p k q j with p, q > 0 apply: k k Example: B 3 p r k q µ k p µ k q r = pq r k Anna Felikson Pavel Tumarkin Question: when a real quiver Q is mutation-finite? Example: H 3 (mutation class and exchange graph ) θ θ θ θ θ θ where θ = cos π (cf. generalised associahedron in [FR]) Quivers of rank 3 Theorem [FT]. Any mutation-finite rank 3 quiver is mutation-equivalent to one of Markov quiver: Finite type quivers: A 3 Affine quivers: n n B 3 4 H 3 H 3 H 3 Here, a label k m stays for the weight b ij = cos kπ m. All of these (but Markov) are mutation-acyclic. Geometric realisation by reflections (GR) GR of a quiver of rank n: vectors v,...,v n in a quadratic vector space V s.t. (v i, v i ) = and (v i, v j ) = b ij. Mutation = partial reflection: µ k (v i ) = v i, v i, if b ki 0, i k if i = k v i (v i, v k )v k, if b ki < 0 Mutation class has a GR if GR of quivers commute with mutations. Theorem [FT,FT]. Mutation class of any real acyclic quiver with b ij i,j admits a GR. Finite mutation type: classification Theorem [FT3]. A mut.-fin. non-int. quiver of rank n > 3 is either of orbifold type, or mut.-equiv. to one of F 4, F 4, F (,+) 4, F (, ) 4 or n+ n+ n n+ n n n+ n+ n n n n n n n n n n Exchange graphs for H 3 and H 3 are graphs on a torus (with two acyclic belts each): C B H 3 H D A B F C E D A 3 M G N H M N When GR exists, we define (geometric) Y -seeds (n-tuples of vectors in V ) and exchange graphs. Example: exchange graph of H = Here, a label k m stays for the weight b ij = cos kπ m. E F Each of the mutation classes H 3 and H 3 has two different acyclic representatives: H 3 : H 3 : H G Theorem [FL]. Let Q be an affine type rank 3 mut.-fin. quiver. Then exchange graph of Q grows polynomially and is quasi-isometric to a lattice of some dimension. References [FL] A. Felikson, Ph. Lampe, Exchange graphs for non-integer affine quivers with 3 vertices, in preparation. [FT] A. Felikson, P. Tumarkin, Geometry of mutation classes of rank 3 quivers, arxiv:609.0888, [FT] A. Felikson, P. Tumarkin, Acyclic cluster algebras, reflection groups and curves on a punctured disc, arxiv:709.0360. [FT3] A. Felikson, P. Tumarkin, Non-integer quivers of finite mutation type, in preparation. [FR] S. Fomin, N. Reading, Root systems and generalized associahedra, Geometric combinatorics, 63-3, IAS/Park City Math. Ser., 3, Amer. Math. Soc., Providence, RI, 007. [L] Ph. Lampe, On the approximate periodicity of sequences attached to noncrystallographic root systems, To appear in Experimental Mathematics (08), arxiv:607.043.