Download LECTURE 2 1. Motivation and plans Why might one study

LECTURE 2
1. Motivation and plans
Why might one study cluster algebras? We could split up the reasons
into “intrinsic” and “extrinsic” reasons. Historically, it is the extrinsic
reasons that came first.
It turns out that many interesting commutative rings are actually
cluster algebras (i.e., there is some surface or B-matrix such that the
cluster algebra associated to it, is isomorphic to the commutative ring
of interest). And, in fact, one needn’t restrict to commutative rings
here, as there are non-commutative (“quantum”) generalizations of
cluster algebras (see [BZ]) which mean that we can talk about quantum
co-ordinate rings (see [GLS]) as posessing a quantum cluster algebras
structure. Thus, one could hope to use the cluster algebra structure to
study these rings. Foci are total positivity (given a family of functions
on a space, which we want to know are positive on some region of the
space, how few of the functions do we need to test to detect that all
of them will be positive) (see [F]) and linear bases of cluster algebras
(in particular Lusztig’s canonical and semi-canonical bases, which are
defined in a rather inexplicit way – see [GLS]).
Another, more recent motivation for the study of cluster algebras,
is their connections to physics. The link to N = 2 supersymmetric
quantum field theories has been studied in [CV, GMN] and other work
by those authors. Coming at the physics from a slightly different angle
(and as a mathematician), there is Gregg Musiker [LM+]. There is
also a connection between cluster algebras and scattering amplitudes
for N = 4 super Yang-Mills [A+].
I am, for the most part, going to ignore the extrinsic motivation
for cluster algebras in this course. This is partly because of my lack of
expertise, and also because, by definition, extrinsic motivations presuppose some extra knowledge outside cluster algebras, and I am trying
not to add any such requirements.
The intrinsic motivation for studying cluster algebras is that they
have a subtle and fascinating structure with links to other interesting
mathematical topics. The following are three topics which I plan to
treat in this course, after two initial sections, one on cluster algebras
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arising from surfaces (as described in Lecture 1), and one on general
cluster algebras.
Finite type. What cluster algebras have a finite number of clusters?
It turns out that the answer is a Dynkin diagram classification: [FZ2].
This is more or less the classification of crystallographic finite Coxeter
groups (four infinite families An , Bn , Cn , Dn , and some exceptional
types, E6 , E7 , E8 , F4 , G2 ). (Note that on the level of Coxeter groups,
Bn and Cn are identical.) Only the type An cluster algebras arise in
the way that we have described from marked surfaces – in fact, the An
cluster algebra is the cluster algebra associated to the triangulations
of an n + 3-gon. It is pretty easy to convince yourself that any other
surface has an infinite number of cluster variables, and thus an infinite
number of clusters.
Cluster variables. Once we leave the setting of cluster variables arising from surfaces, we no longer are given from the beginning a way
to index cluster variables. In the finite type case mentioned above,
the cluster variables correspond to “almost positive roots”. There is
a class of algebras (the acyclic, skew-symmetric cluster algebras) for
which there is a nice interpretation of cluster variables as associated
to exceptional representations of a quiver. We will discuss this. No
familiarity with quiver representations will be assumed.
Linear bases. As mentioned above, one of the goals of studying cluster algebras was to use the cluster algebra structure on a commutative
ring to provide a linear basis for the ring. We will discuss the problem
of finding an atomic basis for cluster algebras.
2. Cluster algebra from a polygon: well-definedness of
edge-labels
We want to establish claim (i) from Lecture 1 in the case of a polygon.
Lemma 1. Any triangulation can be reached from any other one by a
sequence of flips.
Let’s define a particular triangulation: the “fan”, in which every
vertex is connected to the vertex 1. To prove Lemma 1, it suffices to
show that every triangulation can be connected to the fan triangulation
by a sequence of flips. (Connect T to the fan, and then connect the
fan to S; the result connects T to S.) So we are reduced to showing
Lemma 2. The fan triangulation can be reached from any triangulation
by a sequence of flips.
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Proof. Given a triangulation, let’s remove one edge to obtain an empty
quadrilateral, with vertices i < j < k < `. There are two ways to fill
in the missing diagonal. Say we prefer to replace the diagonal with
the higher vertices j` by the one with lower vertices ik. Each time
we do this, the sum of the endpoints of all the diagonals goes down,
so we must eventually get stuck. Suppose we are stuck, and suppose
that consecutive edges connected to 1 are 1i and 1j. We could flip
ij to something lower, unless it is a boundary. So all the triangles
connected to 1 have boundaries on their edges opposite 1, and thus the
triangulation is the fan.
We call such a sequence of moves from a given triangulation to the
fan triangulation a “decreasing sequence of flips”. The reverse of a
decreasing sequence of flips is an increasing sequence of flips.
Remark 1. Now that we have proved that it is possible to get from
any triangulation to any other triangulation by a sequence of flips, it
is natural to ask how many flips are required. It turns out this is a
very hard problem! Sleator, Tarjan, and Thurston [STT] proved using
hyperbolic geometry (!) that for n sufficiently large, the answer is 2n −
10. However, their proof didn’t give any indication of how big n should
be. Obviously, for n too small, the answer is negative, so it can’t hold
for all n. Computer checks showed that, while it fails for n = 12, it
holds for n = 13, 14, . . . (but this soon becomes very hard to check).
After 25 years, Pournin [P] showed in 2012 using combinatorics that
the statement holds for n ≥ 13.
We can draw a graph of triangulations, and connect a pair of vertices if the corresponding triangulations are related by a single flip.
There are some distinguished cycles in this graph. Let T be a triangulation. Let e and f be two edges which admit decreasing flips. Suppose
first that they don’t border the same triangle. Then there is a cycle
T, µe T, µf µe T, µe µf µe T = µf T, T .
There are also distinguished 5-cycles. Suppose e and f are edges
which do border a common triangle. If we remove e and f , the result is a
pentagon. Then if we flip e, then f , then continue, at each step flipping
the edge we didn’t just flip, we travel through the five triangulations
of a pentagon before getting back where we started.
How do these distinguished cycles look with respect to the orientation
of their edges? See the page of diagrams. If we draw the edges so
that the “decreasing” direction is downwards, then the four-cycles look
like diamonds. T is the highest triangulation, and you can get from
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the highest triangulation to the lowest triangulation by two different
decreasing steps of length two.
Now consider a distinguished 5-cycle. It involves a pentagon in the
triangulation, whose vertices are, let’s say, p1 < p2 < · · · < p5 . T is
a highest triangulation (everyone connected to p5 ) and a lowest triangulation (everyone connected to p1 ). There are two decreasing paths
from the highest triangulation to the lowest, one of length two, and the
other of length three.
References
[A+] N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, and J. Trnka. Scattering amplitudes and the positive Grassmannian.
arXiv:1212.5606.
[BZ] A. Berenstein and A. Zelevinsky, Quantum cluster algebras. Adv. Math. 195
(2005), no. 2, 405–455.
[CV] S. Cecotti and C. Vafa. Classification of complete N = 2 supersymmetric
theories in 4 dimensions. arXiv:1103.5832.
[CS] L. Chekhov and M. Shapiro. Teichmüller spaces of Riemann surfaces with
orbifold points of arbitrary order and cluster variables. arXiv:1111.3963.
[F]
S. Fomin. Total positivity and cluster algebras. arXiv:1005.1086.
[FST] S. Fomin, M. Shapiro, and D. Thurston. Cluster algebras and triangulated
surfaces. arXiv:math/0608367.
[FZ1] S. Fomin and A. Zelevinsky. Cluster algebras I: Foundations.
arXiv:math/0104151.
[FZ2] S. Fomin and A. Zelevinsky. Cluster algebras II: Finite type classification.
arXiv:math/0208229.
[GMN] D. Gaiotto, G. Moore, and A. Neitzke. Spectral networks and snakes.
arXiv:1209.0866.
[GLS] C. Geiß, B. Leclerc, and J. Schrer. Cluster structures on quantum coordinate
rings. arXiv:1104.0531.
[GH+] M. Gross, P. Hacking, S. Keel, M. Kontsevich. Canonical bases for cluster
algebras. arXiv:1411.1394.
[LS] K. Lee and R. Schiffler. Positivity for cluster algebras. arXiv:1306.2415.
[LM+] Megan Leoni, Gregg Musiker, Seth Neel, and Paxton Turner. Aztec Castles
and the dP3 quiver. arXiv:1308.3926.
[N]
I. Nikolaev. K-theory of cluster C∗-algebras. arXiv:1512.00267.
[P]
L. Pournin. The diameter of associahedra. arXiv:1207.6296.
[STT] D. Sleator, R. Tarjan, and W. Thurston. Rotation distance, triangulations,
and hyperbolic geometry. J Am. Math. Soc. 1 (1988), 647–681.