friezes and triangulations
Transcript of friezes and triangulations
Infinite friezes and triangulationsof annuli
joint Baur - Jacobsen - Kulkarni - Todorov
FD Seminar,
25 February 2021
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 : =
[email protected].,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