Unitary friezes and frieze vectors Emily Gunawan and Ralf Schiffler - - PowerPoint PPT Presentation

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Unitary friezes and frieze vectors Emily Gunawan and Ralf Schiffler - - PowerPoint PPT Presentation

Jun 11, 2023 193 likes •433 views

Unitary friezes and frieze vectors Emily Gunawan and Ralf Schiffler University of Connecticut The 31st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC) University of Ljubljana, Slovenia 5 July 2019 Emily

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Unitary friezes and frieze vectors

Emily Gunawan and Ralf Schiffler University of Connecticut The 31st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC) University of Ljubljana, Slovenia 5 July 2019 Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 1 / 23
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Frieze

In architecture, a frieze is an image that repeats itself along one direction. Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 2 / 23
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Conway and Coxeter, 1970s

Definition A Conway – Coxeter frieze pattern is an array of positive integers such that: 1 it is bounded above and below by a row of 1s 2 every diamond b a d c satisfies the diamond rule ad − bc = 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 2 2 2 1 4 1 2 2 2 3 3 1 3 3 1 3 3 1 3 3 1 2 2 1 4 1 2 2 2 1 4 1 2 1 1 1 1 1 1 1 1 1 1 1 1 Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 3 / 23
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Conway and Coxeter, 1970s

Theorem A Conway – Coxeter frieze pattern with n nontrivial rows ← → a triangulation of an (n + 3)-gon. 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 2 2 2 1 4 1 2 2 2 3 3 1 3 3 1 3 3 1 3 3 1 2 2 1 4 1 2 2 2 1 4 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 2 2 2 Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 4 / 23
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Fomin and Zelevinsky, 2001

Start with a quiver (directed graph) Q on n vertices with no loops and no 2-cycles. Example: type Ap,q An acyclic quiver Q is of type Ap,q if and
  • nly if
its underlying graph is a circular graph with n = p + q vertices, the quiver Q has p counterclockwise arrows and q clockwise arrows For example, this is a quiver of type A1,2 → Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 5 / 23
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  • €¥
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  • Shapiro
  • thurston
, 2006 ) a b
  • k ¢
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  • curve
between marked points ( d M2 A triangulation is a MK
  • Maximal
Collection
  • f
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  • crossing
arcs "lab

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A flip Mk replaces ( d
  • ¢
' a b a b Ptolemy He k with k '
  • k '=
adtbc µ = × ,
  • 1×3
c d c d k Xz Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 6 / 23
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SLIDE 7
  • a
cluster { Xn , Xz ,X3} X , × } a ↳

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K §
  • (
d Mz mutation at Xz MK
  • a
new cluster {14,141×2×3} "lab

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  • Ptolemy
He ¥
  • k '=
adtbc # =×nt×3 K Xz Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 7 / 23
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SLIDE 8
  • ¥E¥'
n .

Ej

.

.ae#i

: F M x'=×Iye
  • f

|Xz

XI = 11¥
  • repeat
this mutation MZ process to produce all clusters
  • A
X3 x "=×¥I

t.to#xi=xitI

Qp¥⇒

"*

"

Iaea

" Xz ' = × ' + × } Xz Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 8 / 23
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SLIDE 9
  • ¥E¥'
n .

Ej

.

.ae#i

: F M x'=×Iye
  • f

|Xz

XI = 11¥
  • repeat
this mutation MZ process to produce all clusters
  • A
X3 x "=×¥I

t.to#xi=xitI

Qp¥⇒

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Iaea

" Xz ' = × ' + × } Xz Def (Fomin – Zelevinsky, 2001) { cluster variables } =
  • all clusters x
{ elements of x} The cluster algebra A(Q) is the Z-algebra of Q(x1, . . . , xn) generated by all cluster variables. Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 9 / 23
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  • X
, ×3 A If j follows : 1 3 a b

€€¥)

counterclockwise

Tzk

µ

, along a k triangle
  • (
d Mz I i then MK
  • draw
j←i

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'¥7 ' ( d y,

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, p 2 Ptolemy the ¥
  • k '=
adtbc # =×nt×3 K Xz Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 10 / 23
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Friezes

Definition Let Q be a quiver and A(Q) the cluster algebra from Q. A frieze of type Q is a ring homomorphism F : A(Q) → R We say that F is positive integral if R = Z and F maps every cluster variable to a positive integer Examples: The identity frieze Id : A(Q) → A(Q). A frieze F : A(Q) → Z defined by fixing a cluster x and sending each cluster variable in x to 1. Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 11 / 23
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Friezes examples

1 1 1 1 x3 x1x3+1+x2 x2x3 x2+1 x1 x1 x2 x1x3+1 x2 x2 2 +2x2+1+x1x3 x1x2x3 x2 x1 x1x3+1+x2 x1x2 x2+1 x3 x3 1 1 1 1 Figure: The identity frieze Id : A(Q) → A(Q) for the type A3 quiver Q = 1 → 2 ← 3. 1 1 1 1 1 3 2 1 1 2 5 1 1 3 2 1 1 1 1 1 Figure: Setting x1=x2=x3=1 produces a Conway – Coxeter frieze pattern. Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 12 / 23
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Unitary friezes

Definition We say that a frieze F is unitary if there exists a cluster x in A(Q) such that F maps every cluster variable in x to 1. Proposition 1 (G – Schiffler) Let F be a positive unitary integral frieze, i.e., there is a cluster x such that F(u) = 1 for all u ∈ x. Then x is unique. Sketch of Proof: If u is a cluster variable not in a cluster x, then the Laurent expansion of u in x has two or more terms. Remark All positive integral friezes of type A are unitary (due to Conway and Coxeter), but there are non-unitary positive integral friezes of type D, D, E, and E (due to Fontaine and Plamondon). Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 13 / 23
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Friezes of type Ap,q

Theorem 2 (G – Schiffler) All positive integral friezes of type Ap,q are unitary. Example: There are the two friezes of type A1,2, up to translation. . . . . . . 11 26 41 2 3 7 1 1 1 7 3 2 41 26 11 362 153 97 2131 1351 571 18817 7953 5042 Figure: An A1,2 frieze obtained by specializing the cluster variables of an acyclic seed to 1. The peripheral arcs have frieze values 2 and 3. . . . . . . 5 18 13 3 2 7 1 2 1 7 2 3 13 18 5 123 34 47 233 322 89 2207 610 843 Figure: An A1,2 frieze obtained by specializing the cluster variables of a non-acyclic seed to 1. The peripheral arcs have frieze values 1 and 5. Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 14 / 23
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Friezes of type Ap,q

Algorithm for finding the cluster where each cluster variable has frieze value 1: Let F be a positive integral frieze of type Ap,q. Pick any acyclic cluster x0 := {x1, . . . , xn}. If not all cluster variables of x0 have weight 1, we mutate x0 at xk with maximal frieze value. Let x′ k := µk(xk) Then: F(x′ k) < F(xk) Furthermore, if the vertex k is not a sink/source, then F(x′ k) = 1 If not every cluster variable in x1 := {x′ k} ∪ x0\{xk} has weight 1, repeat this procedure, and so on. Since F is positive integral, this process must stop. Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 15 / 23
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Friezes of type Ap,q

Every acyclic shape, for example,
  • and
  • tells us the frieze
values of a cluster. Example (A possible step in the algorithm) . . . . . . 11 26 41 2 3 7 1 1 1 7 3 2 41 26 11 362 153 97 2131 1351 571 18817 7953 5042 Figure: An A1,2 frieze obtained by specializing the cluster variables of an acyclic seed to 1. The peripheral arcs have frieze values 2 and 3. Mutating at the position with frieze value 11 produces a new frieze value 3×7+1 11 = 2 < 11. Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 16 / 23
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Friezes of type Ap,q

Every acyclic shape, for example,
  • and
  • tells us the frieze
values of a cluster. Example (A possible step in the algorithm) . . . . . . 5 18 13 3 2 7 1 2 1 7 2 3 13 18 5 123 34 47 233 322 89 2207 610 843 Figure: An A1,2 frieze obtained by specializing the cluster variables of a non-acyclic seed to 1. The peripheral arcs have frieze values 1 and 5. Mutating at the position with frieze value 18 (which is not a sink/source) produces a new frieze value 5+13 18 = 1. Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 17 / 23
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Frieze vectors

Definition Fix a cluster x = (x1, . . . , xn). A vector (a1, . . . , an) ∈ Zn >0 can be used to define a frieze F : A(Q) → Q by defining F(xi) = ai for all i = 1, . . . , n. We say that (a1, . . . , an) is a positive integral frieze vector relative to x if F maps every cluster variable to a positive integer. If (a1, . . . , an) determines a unitary frieze, we say that (a1, . . . , an) is a unitary frieze vector. . . . . . . 89 233 13 34 2 5 1 1 5 2 34 13 233 89 1597 610 The slices display the frieze vectors . . . , (233, 89), (34, 13), (5, 2), (1, 1), (2, 5), (13, 34), (89, 233), (610, 1597), . . . relative to a cluster with the quiver 1 ⇒ 2. Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 18 / 23
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Frieze vectors algorithm

Proposition 3 A vector (a1, . . . , an) ∈ Zn is a frieze vector relative to an acyclic Q iff ak divides
  • k→j in Q
xj +
  • k←j in Q
xj for all k = 1, . . . , n. Example A vector (a1, a2, a3) ∈ Z3 >0 is a positive frieze vector relative to 1 → 2 ← 3 iff a2 + 1 a1 , a1a3 + 1 a2 , a2 + 1 a3 are integers.

k

1

fit EH EH

4 11

3 3 i

1

3,2 25,2N 3

I

43,1 113,2 kill 1,2 l 1,112

Lil

Ya Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 19 / 23
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Frieze vectors

Theorem 4 (G – Schiffler) Fix A(Q) and fix an arbitrary cluster x = (x1, . . . , xn). Then there is a bijection between clusters in A(Q) and unitary frieze vectors relative to x. Sketch of Proof: Define φ : { clusters in A(Q)} → { unitary frieze vectors } x′ = {x′ 1, . . . , x′ n} → φ(x′) = F(x) = (a1, . . . , an) where F is the frieze defined by specializing the cluster variables in x′ to 1. Then φ is a bijection. Injectivity follows from Proposition 1. Surjectivity follows from the construction of φ. Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 20 / 23
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Friezahedron (work in progress)

In type An, Dn, and E6, it is known that there are finitely many positive integral frieze vectors. Take the convex hull of these points in Rn. sage: V = [ [1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 2, 3], [1, 3, 2], [2, 1, 1], [2, 1, 2], [2, 3, 1], [2, 3, 4], [2, 5, 2], [3, 2, 1], [3, 2, 3], [3, 5, 3], [4, 3, 2] ] sage: P = Polyhedron(V)

9

3,213

III II

l

14 i 3,2 25,2N 3

I

I

43,1 113,2 Ht 112,1 41,2

Lil

Yin Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 21 / 23
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References

[CC73] [FZ02] [FST08] [FP16] [GS] [GS19]
  • J. H. Conway and H. S. M. Coxeter.
Triangulated polygons and frieze patterns.
  • Math. Gaz., 57:87–94, 175–183, 1973.
  • B. Fontaine and P.-G. Plamondon.
Counting friezes in type Dn.
  • J. Algebraic Combin., 44(2):433–445, 2016.
  • S. Fomin, M. Shapiro, and D. Thurston.
Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math., 201(1):83–146, 2008.
  • S. Fomin and A. Zelevinsky.
Cluster algebras. I. Foundations.
  • J. Amer. Math. Soc., 15(2):497–529, 2002.
  • E. Gunawan and R. Schiffler.
Frieze vectors and unitary friezes. arxiv 1806.00940, 2018.
  • E. Gunawan and R. Schiffler.
Unitary friezes and frieze vectors. S´
  • em. Lothar. Combin., 82B:Art. 74, 12pp, 2019.
Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 22 / 23
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Comments and questions

Thank you! Hvala!

Emily Gunawan & Ralf Schiffler (UConn) Unitary friezes & frieze vectors FPSAC 19 23 / 23