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Title: Cluster Algebras from Triangulations of Surfaces
Some References for this talk:
Cluster Algebras from Surfaces: Sergey Fomin, Michael Shapiro, Dylan Thurston (2006), http://arxiv.org/abs/math/0608367
Sequences: Gregg Musiker (2002), http://www.math.umn.edu/~musiker/uthesis.pdf
T-paths formula: Ralf Schiffler, Hugh Thomas (2007-2008): http://arxiv.org/abs/0712.4131 and http://arxiv.org/abs/0809.2593
T-paths formula for punctured surfaces: Emily Gunawan (2014): http://arxiv.org/pdf/1409.3610v2.pdf
Outline:
0. Warm-up
1. Triangulations
2. Cluster algebra from surfaces
3. Closed-form formula (T-paths)
4. Atomic bases
Warm-up
=======
a) Recurrence x(n) = [ x(n-1) + 1 ] / x(n-2)
Set x1 = x2 = 1
x3 = (x2 + 1)/x1 = (1 + 1)/1 = 2
x4 = (2+1)/1 = 3
x5 = (3+1)/2 = 2
x6 = (2+1)/3 = 1 = x1
x7 = (1+1)/2 = 1 = x2
This sequence is 5-periodic
b) Recurrence x(n) = [ x(n-1)^2 + 1 ] / x(n-2)
Set x1 = x2 = 1
x3 = (x2^2 + 1)/x1 = (1^2 + 1)/1 = 2
x4 = (2^2 + 1)/1 = 5
x5 = (5^2 + 1)/2 = 26/2 = 13
x6 = (13^2 + 1)/5 = 170/5 = 34
x7 = (34^2 + 1)/13 = 1157/13 = 89
x8 = (89^2 + 1)/34 = 7922/34 = 233
Integer sequence 1,1,2,5,34,89,233,…
Do you recognize this sequence?
1. Triangulations
==============
Let (S,M) be an orientable Riemann surface S (possibly with empty boundary) with marked points M on the boundary or in the interior (called punctures).
An ideal triangulation (triangulation for short) cuts up (S,M) into pieces:
o
/ \
/ \
o_____o (3 vertices, 3 edges),
_o__
/ \
( \
\ __ |
\ ( ) |
\_\/_/
o (2 vertices, 3 edges),
__________
/ \
( )
| __ __ |
\ \ )( / /
\__\||/__/
o (1 vertex, 3 edges), and
_____
( )
\ o /
\|/
o (the self-folded triangle with 2 vertices, and only 2 edges).
================================
Example: a pentagon triangulation
1 /\ 2
/ \
================================
Example: an annulus triangulation (with one marked point on each boundary).
inside
__
|\1|
2 |_\| 2
outside
||
________
/ \
/ |
| ____ |
| / )1 |
| | D_/ |
| \ | |
| \ | 2 /
\__\|____/
outside
================================
Define: a mutation/flip:
__
|\ | < - >
|_\|
__
| /|
|/_|
=================================
Examples:
Draw pentagon flips at 1,2,3,4,5.
Draw annulus flips at 1,2,3, . . .
__
|\1|
2 |_\| 2
outside
|
| 1
\|/
__
|\2|
3 |_\| 3
outside
|
| 2
\|/
__
|\3|
4 |_\| 4
outside
|
| 3
\|/
__
|\4|
5 |_\| 5
outside
|
| 4
\|/
====================
Define Ptolemy Rule:
d
__
a|\k| < - >
|_\| c
b
d
__
a|j/|
|/_| c
b
kj = ac + bd
j = (ac + bd)/j
Set the weight of the boundary edge to 1
Example pentagon:
=================
x3=(x2+1)/x1
x4=(x3+1)/x2
x5=(x4+1)/x3
x6=(x5+1)/x4
x7=(x6+1)/x5
x4=(x3+1)/x2
(x2+1)/x1 + 1
=_____________
x2
x2 + 1 + x1
=_____________
x1 x2
x5=(x4+1)/x3
(x2 + 1 + x1)/(x1 x2) + 1
= _________________________
(x2+1)/x1
x2 + 1 + x1 + x1x2 x1
= __________________ _____
x2 + 1 x1 x2
(x2 + 1)(1 + x1) 1
= __________________ ____
x2 + 1 x2
= (x1 + 1)/ x2
x6=(x5+1)/x4
(x1+1)/x2 + 1
= ___________________
(x2 + 1 + x1)/x1x2
x1 + 1 + x2 x1 x2
= _____________ _____
x2 + 1 + x1 x2
= x1
x7=(x6+1)/x5
x1 + 1
= ____________
(x1 + 1)/ x2
= x2
Example annulus(1,1):
=====================
x3 = (x2^2+1)/x1
x4 = (x3^2+1)/x2
[(x2^2+1) /x1]^2 + 1
= ____________________
x2
[x2^4 + 2 x2 + 1 + x1^2]
= ________________________
x1^2 x2
2. Cluster Algebras from surfaces
=================================
Def:
- Fix a triangulation T on (S,M)
- Put weights x1,. . ., xn on the internal diagonals (arcs)
- Compute all the _cluster variables_ by applying flips
__
|\ | < - >
|_\|
__
| /|
|/_|
on every edge repeatedly.
__ {x1,x2,x3} ___ {x1,x2’,x3} ___ ___
| | |
| | |
___ {x1’,x2,x3} ___
|
- The _cluster algebra_ A(S,M) of type (S,M) is the subring of Q(x1, . . ., xn) generated by all the cluster variables
Example:
- A(pentagon) is generated by {x1, x2,(x2 + 1)/x1, (x1 + 1)/x2, (x1 + x2 + 1)/x1 x2}
- A(annulus 1,1) is generated by {x1, x2,(x2^2 + 1)/x1, (x2^4 + 2 x2^2 + 1 + x1^2)/(x1^2 x2), . . .}
Note: In slightly greater generality, instead of triangulations, the cluster algebra data can be encoded in a skew-symmetric matrix or a quiver (a directed graph with no loop or two-cycles).
triangulation matrix quiver
==============================
1/\2 0 1 1 -> 2
/ \ -1 0
______________________________
inside
__
|\1| 0 -2 2 => 1
2 |_\| 2 2 0
outside
______________________________
Laurent Phenomenon: Every cluster variable is a Laurent polynomial in the variables of x1, . . ., xn
Positivity: with positive coefficients
3. Closed-form formula (T-paths)
================================
- Fix an initial ideal triangulation T.
- Let gamma be a non-initial arc (i.e. in our examples, gamma crosses T). Choose an orientation of gamma.
- Let tau_1, tau_2, . . ., tau_d be the arcs crossed by gamma, in order.
- Let triangle_0, triangle_1, . . ., triangle_d be the arcs crossed by gamma, in order.
e.g. let gamma be arc 4 in the pentagon, and let gamma be the arc which starts from the outer boundary, crosses 1, 2, and 1, ending up in the inner boundary.
- Def: A (T,gamma)path w=(w1,w2, . . ., w_l) is an odd-length path along edges of T such that:
1. w is homotopic to gamma
2. the even step crosses gamma, obeying the order of tau_1, . . ., tau_d.
3. No backtrack
4. More subtle homotopy requirements
T-path formula:
x(gamma) = the sum over all T-paths w (weights of even steps of w) / (weights of odd steps of w)
(for personal note)
a / \e
b|___|d
c
2 e
o___o___o
a| |c |d
o___o___o
b 1
=============
b e 1
= ___
x1 x1
o___o o
|
o___o o
=============
a c e 1
= ____
x1 x2 x1x2
o o o
| | |
o o o
=============
a e 1
= ____
x2 x2
o o___o
|
o o___o
============== +
x4 = 1/x1 + 1/x1x2 + 1/x2 = (x2 + 1 + x1)/ (x1 x2)
Example annulus:
Initial triangulaton:
inside
___ ___
|\ |\ |
2| \1|2\1| 2
|__\|__\|
outside
^
|\ |\ |
| \ | \ |
| \| \|
x2 x2 x2 x2^3
= ______
x1 x1 x1^2
====================
____ ^
\ \ |
\ \ |
__\ \|
Out In x2 x2
= ______
x1 x1 x1^2
==================
___>
|\ \
| \ \
| \___\
x2 Out In x2
= _____
x1 x1 x1
==================
____ ____>
\ | \
\ | \
__\ |___\
Out In Out In 1
= ________
x1 x2 x1 x1^2 x2
==========================
____>
|
|
___|
Out In 1
= ____
x2 x2
==================+
7. Atomic Bases
===============
Def: Let A be a (coefficient-free) cluster algebra
-The semiring A+ = {positive elements of A}
{the element which can be written as a Laurent polynomial with positive coefficients}
- An indecomposable positive element is a positive element which is not the sum of two positive elements.
- Let B = {the set of indecomposable positive elements}
- B forms a basis :if and only if B is the atomic basis of A
Not all cluster algebras have this property, e.g. the one arising from the matrix
0 -3
3 0
Conjecture for most (S,M):
==========================
Let A(S,M) be a cluster algebra arising from a marked surface (S,M).
Def: A cluster monomial is a product of compatible cluster variables (i.e. cluster variables from arcs that do not cross).
Note: A cluster monomial corresponds to a partial triangulation where multiplicity is counted.
For example, xa ab^4 ac^2 corresponds to the partial triangulation
\ |b /
a\ | /c
\|/
Def: A bracelet is a (non-contractible) loop which wraps itself finitely many times.
Conjecture: The atomic basis for A(S,M) is a collection of corresponding to partial triangulations and bracelets.
True for a disk with (n+3) points (type An),
a once-punctured disk with n boundary points (type Dn),
and an annulus with (n1,n2) points (type affine A(n1,n2)).
Type affine D(n-1) corresponds to twice-punctured disk with n-3 boundary points.