Infinite friezes and triangulationsof annuli

joint Baur - Jacobsen - Kulkarni - Todorov

FD Seminar,

25 February 2021

Outline

r Finite frieze patterns

2 Infinite friezes

3 Representation theoretic aspects

1 Frieze Patterns Coxeter ' 71 Infinite arrays ofnumbers where

a bad ⇒ ad -be -- l

l l- . .

. of 0000010000000 - -

.

:row O 's

- - - - - l,l l l l l l , l l l y y y y row 1 's

- - -- 4 f 2 2 2 1 4 11 2 22 l 4 l - - - - i

-- - - - 31

,I 3 3 1 3 31

,a 3 3 1 33 I . . . . } width =

3

.- - - - 212 l 4 l 2 2 f l 4 1 22 2 - - - -

- n y l l l l l l l l ll l l -

- row t 's

O OO OO !- .

- .0 row O 's

I s

confer ' 71 •Finite friezes of

width m are periodic with

period dividing mx3

•Friezes are invariant under glide symmetry

conway - center'

" 1¥:÷÷÷:%:'t I :i÷7÷g÷÷ )- - - - -

d l l l l y d y y y , y y y . . . . y,quiddity

row

l" " " " " " " " " " "

" "

-- - - - p l 3 3 1 3 13 1 3 3

t 3 3 I - - - iy

2

.- - - - 212

l 4 l 2 2,2 I 4 1 22 2 - - - -

- - - - . l l l l l l d l l l l l l l - -- i 2 2

( l

"

quiddityseqn

2 Infinite friezesBaur- Parsons- Tschabold

'16 Periodic integral friezes

ii. ?"o o o o dfitfua.ae row of O's & l 's : infinite width

H l l l l,

1 a bad ⇒ ad - be =L

- . . . . 2 ' 5 4 I 2,5 4 - -

- - - - -

'9 19 3 I 9, 19 . . .

[BPT] Infinite periodic friezes come

- - - - - 4134 14 2 4134 14 - - -

from triangulations of annuli- . .

. . .1st 25 9 7 15 25 . . -

- . . - 26 1 11 16 31 261,11 16 - - •

4d go ,

- .. . . .

19,

7 55 115 19 7 - - - 14 2 41'

'

y

'

! ! !"

Y'

! n . ④• •②

•5 4 71I 2424

I, I Ig

l

l. I '

I•

'

2 !'

,

I

C annulus,T triangulation

k.

• "

kit t.nl02

'

, ? 021

→ ( Fy , Ft ) pair of infinite'

.

.

i

...

'

• s

friezesassociated with (Gati 'T)

Question How are these two friezes related ?

Definition skeletalfrieze-n.no l's in its quiddity segn gs

skeletaltriangulatiens-s.no arcs connectingmarked points on

the same boundary component Is

Note T -- T'

iff Og is a non- oriented cycle p→ .

→→ . . . ..

T

t arrows

→ Ts its skeletalProposition J triangulation of chit annulus

triangulation(Ek . E ) pair of friezes assoc . ( Cat 's T)

(s

(J! .FI) - " - ( cists ; T)

T → ( Fa , Ft )

I o f .

y•

Ts→ ( Es , Es ).

Theorem [BGJKT]skeletal quiddity skeletal

sequencefi. . 'i::Ln-

-

k + Itai - 2)

T

Theorem [BGJKT] Let Fa be an infinite frieze .

Then

i Is uniquely determines Is such that CFI, Fes) pair

ii JI givesrise to an infinite of infinite friezes Ft

such that (Fk , Ft ) pair .

Growth coefficient F infinite frieze of period n,ice . g=

Ca, as - . - an )

iii°

a°

a . . . i0,0i :-.

The grewthcoeq.net- -

- ai aim - - - . -.

aj- a aj- - -

- row ①of F

is defined by"

"

"

""i!! " ' : .

.

, ÷:"

"

"

;!"

Sf :-

- di,mid

- ai-anti -z

- . .. aiijs Airlift ditsy - -

-- ↳ '

invariant for F- - -

- - - - .di ,j- t aid , j

- - - - --

--- - - - ai

.,j - - - - . . rowj [ BPT ]

÷

Theorem BGJKT aim = IL t -Hl' FINK{ in . ,j )

pair- excludingLI : # pairs excluded from hi, . . ,j }

Sf = (Z C-htt Itak ) + Sn

IEft . . . ; n }O n odd

g."

Intending a Itierwrse

z Module theoretic interpretationharrows

chit annulus,Js triangulation → as .p→

. - - - - -

\.

X.→ . . .. .

7

→ A -- ka' cluster - tilted algebra of

t arrows

type Ankit

→The AR - quiver of S contains

rank k rank t

two non-homogeneous tubes ofrank k and rank t •M

→ M indecomposable in these tubes

Consider a specialised CC - map

SCM ) =I X. ( Gre. M )e : dim N

N E M

M rigid ⇒SCM) = # submodule of M

M non-rigid ⇒s (M) = # perfect matchings of

the snake graph assoc .

with M [a - Schroll'18]

Mi Min - - - - Mitt- e Mi.tt

'

.

'

.

'

i s(Mi ) = di'

.

'

. i .

\ ),

s'

' ' !i. Tu

' g(Mi,i+t ) = ai,iet

', in.

. Mit : for thtMii .it - t Mitt.it

- . .

t. . . Whenever o→xM→B→M→0

- : Mi

-is an AR

- sequencein meds .

then slam )sCM ) - SCB) -- l .

diamond rule£

M --Mi,itt N : = Mi ⑦ Mitt ⑦ -

- .. ⑦ Mitt

R -- Mia .it -t Nj : =

NHMjtOMjHlNjnjzi-NllMjitOMjriptOMja@MjaetJiNjnjq.n.,jk :

= N ) (Mj @ Mjhl ⑦ -- -

' ⑦ Mj.@ Mjhl )

Theorem [BGJKT]itt itt

shut - slit ) =s(N) - Zsllvj ) + Is CNN.jo ) t - - - . - t R

je,jz=iJi just '

qq.ntk.rs t even

0 t odd