Algebraic PL-Invariants and Cluster Algebras
Alastair Darby
The 43rd Symposium on Transformation Groups
Fudan University
Joint work with Zhi Lü

November 18, 2016

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Table of contents

1

PL-Manifolds

2

Cluster Algebras

3

Triangulated Surfaces

4

Higher Dimensional Manifolds

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

PL-Manifolds
Suppose that K1 and K2 are simplicial complexes.
A PL-map ϕ : K1 → K2 is a simplicial map from a subdivision of
K1 to a subdivision of K2 . So K1 and K2 are PL-homeomorphic iff
there exists a simplicial complex isomorphic to a subdivision of theboth of them.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

PL-Manifolds
Suppose that K1 and K2 are simplicial complexes.
A PL-map ϕ : K1 → K2 is a simplicial map from a subdivision of
K1 to a subdivision of K2 . So K1 and K2 are PL-homeomorphic iff
there exists a simplicial complex isomorphic to a subdivision of theboth of them.
A PL-sphere is a triangulated sphere which is PL-homeomorphic tothe boundary of a simplex.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

PL-Manifolds
Suppose that K1 and K2 are simplicial complexes.
A PL-map ϕ : K1 → K2 is a simplicial map from a subdivision of
K1 to a subdivision of K2 . So K1 and K2 are PL-homeomorphic iff
there exists a simplicial complex isomorphic to a subdivision of theboth of them.
A PL-sphere is a triangulated sphere which is PL-homeomorphic tothe boundary of a simplex.
In dim ≤ 3, any triangulated sphere is PL.
In dim = 4, the question is open.
In dim ≥ 5, there exist non–PL-sphere triangulations.
Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

PL-Manifolds
All manifolds are assumed to be connected, closed and oriented.
Let α ∈ K . Then lkK α = {α0 ∈ K | α ∪ α0 ∈ K , α ∩ α0 = ∅}.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

PL-Manifolds
All manifolds are assumed to be connected, closed and oriented.
Let α ∈ K . Then lkK α = {α0 ∈ K | α ∪ α0 ∈ K , α ∩ α0 = ∅}.
Definition
A PL-manifold of dim n is a simplicial complex K of dim n suchthat lkK α is a PL-sphere of dim n − |α|, for all non-empty α ∈ K .

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

PL-Manifolds
All manifolds are assumed to be connected, closed and oriented.
Let α ∈ K . Then lkK α = {α0 ∈ K | α ∪ α0 ∈ K , α ∩ α0 = ∅}.
Definition
A PL-manifold of dim n is a simplicial complex K of dim n suchthat lkK α is a PL-sphere of dim n − |α|, for all non-empty α ∈ K .
In dim ≤ 3, Smooth = PL = Top
In dim = 4, Smooth = PL ⊂ Top
In dim ≥ 4, Smooth ⊂ PL ⊂ Top

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

PL-Spheres

A PL-sphere is not the same as a PL-manifold homeomorphic to asphere but to the ‘standard’ sphere, i.e. the PL-structure is givenby the boundary of a simplex. These notions coincide indimensions other than 4.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

PL-Spheres

A PL-sphere is not the same as a PL-manifold homeomorphic to asphere but to the ‘standard’ sphere, i.e. the PL-structure is givenby the boundary of a simplex. These notions coincide indimensions other than 4.

Problem
Is a PL-structure on S 4 unique?

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Bistellar Moves
Let K be a triangulated manifold of dim n.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Bistellar Moves
Let K be a triangulated manifold of dim n. Assume α ∈ K is an(n − j)-face such that lkK α = ∂β, for some β ∈/ K.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Bistellar Moves
Let K be a triangulated manifold of dim n. Assume α ∈ K is an(n − j)-face such that lkK α = ∂β, for some β ∈/ K .Thenbmα K = (K r (α ∗ ∂β)) ∪ (∂α ∗ β)
is called a bistellar j-move on the bistellar pair (α, β).

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Bistellar Moves
Let K be a triangulated manifold of dim n. Assume α ∈ K is an(n − j)-face such that lkK α = ∂β, for some β ∈/ K .Thenbmα K = (K r (α ∗ ∂β)) ∪ (∂α ∗ β)
is called a bistellar j-move on the bistellar pair (α, β).
A 0-move adds a vertex and an n-move deletes one. All othermoves don’t change the number of vertices.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Bistellar Moves
Let K be a triangulated manifold of dim n. Assume α ∈ K is an(n − j)-face such that lkK α = ∂β, for some β ∈/ K .Thenbmα K = (K r (α ∗ ∂β)) ∪ (∂α ∗ β)
is called a bistellar j-move on the bistellar pair (α, β).
A 0-move adds a vertex and an n-move deletes one. All othermoves don’t change the number of vertices.
Let f = (f0 , f1 , . . . , fn ) be the f -vector of K . ThenLemma
f (bmα K ) = f (K ) ⇔ n = 2j.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Bistellar Moves

Two simplicial complexes are bistellarly equivalent if one can betransformed into another by a finite sequence of bistellar moves.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Bistellar Moves

Two simplicial complexes are bistellarly equivalent if one can betransformed into another by a finite sequence of bistellar moves.
Theorem (Pachner ’87)
Two PL-manifolds are bistellarly equivalent if and only if they are
PL-homeomorphic.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Cluster Algebras
Introduced by Fomin and Zelevinsky in the early 2000s as an
attempt to create an algebraic framework for dual canonical bases
and total positivity in semisimple groups.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Cluster Algebras
Introduced by Fomin and Zelevinsky in the early 2000s as an
attempt to create an algebraic framework for dual canonical bases
and total positivity in semisimple groups.
A cluster algebra A of rank n is a commutative ring (with unit andno zero divisors) equipped with:

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Cluster Algebras
Introduced by Fomin and Zelevinsky in the early 2000s as an
attempt to create an algebraic framework for dual canonical bases
and total positivity in semisimple groups.
A cluster algebra A of rank n is a commutative ring (with unit andno zero divisors) equipped with:a distinguished family of generators X called cluster variables;

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Cluster Algebras
Introduced by Fomin and Zelevinsky in the early 2000s as an
attempt to create an algebraic framework for dual canonical bases
and total positivity in semisimple groups.
A cluster algebra A of rank n is a commutative ring (with unit andno zero divisors) equipped with:a distinguished family of generators X called cluster variables;such that X is grouped into overlapping subsets called clusters
that all have cardinality n.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Cluster Algebras
Introduced by Fomin and Zelevinsky in the early 2000s as an
attempt to create an algebraic framework for dual canonical bases
and total positivity in semisimple groups.
A cluster algebra A of rank n is a commutative ring (with unit andno zero divisors) equipped with:a distinguished family of generators X called cluster variables;such that X is grouped into overlapping subsets called clusters
that all have cardinality n.
x1 = {x11 , . . . , xn1 },
X =

[

xi

x2 = {x12 , . . . , xn2 }, . . .
non-disjoint union

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Exchange Property
The clusters have the following exchange property:
For every cluster x and x ∈ x, there exists another cluster x0 andx 0 ∈ x0 such thatx0 = (x r {x}) ∪ {x 0 },Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Exchange Property
The clusters have the following exchange property:
For every cluster x and x ∈ x, there exists another cluster x0 andx 0 ∈ x0 such thatx0 = (x r {x}) ∪ {x 0 },where x and x 0 are related by the exchange relation
xx 0 = M + M 0 ,where M, M 0 are monomials without common divisors in thevariables x ∩ x0 = x r {x}.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Exchange Property
The clusters have the following exchange property:
For every cluster x and x ∈ x, there exists another cluster x0 andx 0 ∈ x0 such thatx0 = (x r {x}) ∪ {x 0 },where x and x 0 are related by the exchange relation
xx 0 = M + M 0 ,where M, M 0 are monomials without common divisors in thevariables x ∩ x0 = x r {x}.
Furthermore, any two clusters can be obtained from each other by
a sequence of exchanges.
Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank 1
Let A = C[SL2 ] = C[a, b, c, d]/(ad − bc − 1) be the coordiantering, where we write an element of SL2 as
a b, with ad − bc = 1.
c dAlastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank 1
Let A = C[SL2 ] = C[a, b, c, d]/(ad − bc − 1) be the coordiantering, where we write an element of SL2 as
a b, with ad − bc = 1.
c dConsider a, d as cluster variables and b, c as scalers. Then we justhave two clusters {a}, {d} and A is the algebra over C[b, c]
generated by cluster variables a, d subject to the exchange relationad = 1 + bc,

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank 1
Let A = C[SL2 ] = C[a, b, c, d]/(ad − bc − 1) be the coordiantering, where we write an element of SL2 as
a b, with ad − bc = 1.
c dConsider a, d as cluster variables and b, c as scalers. Then we justhave two clusters {a}, {d} and A is the algebra over C[b, c]
generated by cluster variables a, d subject to the exchange relationad = 1 + bc,
which we can write as the ring C[b, c][a, a−1 ] of Laurent
polynomials.
Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank 2Cluster algebras of rank 2, A(b, c), depend only two positive
integers b, c ∈ Z>0 .

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank 2Cluster algebras of rank 2, A(b, c), depend only two positive
integers b, c ∈ Z>0 .
The cluster variables are elements xm , m ∈ Z, defined recursively
by the exchange relations:(b + 1, if m is odd;xm
xm−1 xm+1 =c + 1, if m is even.
xm

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank 2Cluster algebras of rank 2, A(b, c), depend only two positive
integers b, c ∈ Z>0 .
The cluster variables are elements xm , m ∈ Z, defined recursively
by the exchange relations:(b + 1, if m is odd;xm
xm−1 xm+1 =c + 1, if m is even.
xm

Each xm is then a rational function of x1 and x2 .

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank 2
So A(b, c) is the subring generated by xm , m ∈ Z, inside the fieldof rational functions
Q(x1 , x2 ) := {f (x1 , x2 )| f , g are polynomials such that g 6= 0}.
g (x1 .x2 )Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank 2
So A(b, c) is the subring generated by xm , m ∈ Z, inside the fieldof rational functions
Q(x1 , x2 ) := {f (x1 , x2 )| f , g are polynomials such that g 6= 0}.
g (x1 .x2 )The clusters are pairs {xm , xm+1 }, m ∈ Z.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank 2
So A(b, c) is the subring generated by xm , m ∈ Z, inside the fieldof rational functions
Q(x1 , x2 ) := {f (x1 , x2 )| f , g are polynomials such that g 6= 0}.
g (x1 .x2 )The clusters are pairs {xm , xm+1 }, m ∈ Z.
Starting with {x1 , x2 } we can reach any other cluster by a series of
exchanges· · · ←→ {x0 , x1 } ←→ {x1 , x2 } ←→ {x2 , x3 } ←→ · · ·Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nFor rank n the construction is similar.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nFor rank n the construction is similar.
Each cluster x = {x1 , . . . , xn } is a collection of (algebraicallyindependent) elements of some ambient field.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nFor rank n the construction is similar.
Each cluster x = {x1 , . . . , xn } is a collection of (algebraicallyindependent) elements of some ambient field.
Each cluster variable xk can be exchanged from x to form a newcluster x0 = (x r {xk }) ∪ {xk0 },Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nFor rank n the construction is similar.
Each cluster x = {x1 , . . . , xn } is a collection of (algebraicallyindependent) elements of some ambient field.
Each cluster variable xk can be exchanged from x to form a newcluster x0 = (x r {xk }) ∪ {xk0 }, where xk and xk0 are related by anexchange relation
xk xk0 = M + M 0 ,where M, M 0 are disjoint monomials in the variables x ∩ x0 .

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nThe exponents in M and M 0 are encoded in an (n × n)−integer
matrix B = (bij ) (usually skew-symmetric) called the exchange
matrix.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nThe exponents in M and M 0 are encoded in an (n × n)−integer
matrix B = (bij ) (usually skew-symmetric) called the exchange
matrix.
The exchange relations look likeY [b ]
Y [−b ]
xk xk0 =xi ik + +xi ik + ,iiwhere [b]+ = max{b, 0}.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nThe exponents in M and M 0 are encoded in an (n × n)−integer
matrix B = (bij ) (usually skew-symmetric) called the exchange
matrix.
The exchange relations look likeY [b ]
Y [−b ]
xk xk0 =xi ik + +xi ik + ,iiwhere [b]+ = max{b, 0}.
A pair (x, B) is called a seed, which can be realised as a quiver.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nFor each index k we can extend x 7→ x0 to(x, B) 7−→ (x0 , B 0 )Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nFor each index k we can extend x 7→ x0 to(x, B) 7−→ (x0 , B 0 )called a seed mutation in direction k, where we use the matrix
mutation µk : B → B 0 = (bij0 ) given by(−bij ,
i = k or j = k;bij0 =
bij + [bik ]+ [bkj ]+ − [−bik ]+ [−bkj ]+ , otherwise.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nFor each index k we can extend x 7→ x0 to(x, B) 7−→ (x0 , B 0 )called a seed mutation in direction k, where we use the matrix
mutation µk : B → B 0 = (bij0 ) given by(−bij ,
i = k or j = k;bij0 =
bij + [bik ]+ [bkj ]+ − [−bik ]+ [−bkj ]+ , otherwise.

Note that µk (µk (B)) = B.
Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nDefinition
The cluster algebra is then defined as the subring of Q(x1 , . . . , xn )generated by all cluster variables,

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nDefinition
The cluster algebra is then defined as the subring of Q(x1 , . . . , xn )generated by all cluster variables, i.e. the union of all clusters
obtained from the initial cluster x by iterating seeds in all
directions.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nDefinition
The cluster algebra is then defined as the subring of Q(x1 , . . . , xn )generated by all cluster variables, i.e. the union of all clusters
obtained from the initial cluster x by iterating seeds in all
directions.
A cluster algebra is said to be of finite type if it has only a finite
number of seeds.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Rank nDefinition
The cluster algebra is then defined as the subring of Q(x1 , . . . , xn )generated by all cluster variables, i.e. the union of all clusters
obtained from the initial cluster x by iterating seeds in all
directions.
A cluster algebra is said to be of finite type if it has only a finite
number of seeds.
Theorem (Fomin, Zelevinsky 2003)
Cluster algebras of finite type can be classified in terms of the
Dynkin diagrams of finite-dimensional simple Lie algebras.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

The Laurent Phenomenon

One of the main consequences of the definition of matrix
mutations.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

The Laurent Phenomenon

One of the main consequences of the definition of matrix
mutations.
Every cluster variable (a rational function in the elements of agiven cluster) is a Laurent polynomial with integer coefficients.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

The Laurent Phenomenon

One of the main consequences of the definition of matrix
mutations.
Every cluster variable (a rational function in the elements of agiven cluster) is a Laurent polynomial with integer coefficients.
Surprising since every cluster variable appears as thedenominator of the expression used for producing a new one.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

The Laurent Phenomenon

One of the main consequences of the definition of matrix
mutations.
Every cluster variable (a rational function in the elements of agiven cluster) is a Laurent polynomial with integer coefficients.
Surprising since every cluster variable appears as thedenominator of the expression used for producing a new one.
Conjecture: All coefficients in these Laurent polynomials are
positive.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
Suppose we have a pair (S, M) where S is a closed oriented
connected surface and M is a set of marked points on S with
|M| = m. We want to consider all triangulations K of S with mvertices as the marked points.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
Suppose we have a pair (S, M) where S is a closed oriented
connected surface and M is a set of marked points on S with
|M| = m. We want to consider all triangulations K of S with mvertices as the marked points.
Consider the category of PL-surfaces K with a fixed number ofvertices.
Then
K∼= K 0 ⇐⇒ related by a finite sequence of bistellar 1-moves.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
Suppose we have a pair (S, M) where S is a closed oriented
connected surface and M is a set of marked points on S with
|M| = m. We want to consider all triangulations K of S with mvertices as the marked points.
Consider the category of PL-surfaces K with a fixed number ofvertices.
Then
K∼= K 0 ⇐⇒ related by a finite sequence of bistellar 1-moves.
Given (S, M) we can form a cluster algebra A(S, M).
Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
The set of cluster variables X is the set of potential edges of K .
Every triangulation K will haven = 6g + 3m − 6edges, where g is the genus of S. These will form our clusters.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
The set of cluster variables X is the set of potential edges of K .
Every triangulation K will haven = 6g + 3m − 6edges, where g is the genus of S. These will form our clusters.
For every triangulation K we form the (n × n)-exchange matrix B
as follows:
B(K ) = B =

X

B ∆,

∆∈S

where ∆ is a 2-simplex of S.
Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
Using the following matrix mutation B 0 = µk (B) in direction k:(−bij ,
if i = k of j = k;bij0 =
bij + [bik ]+ [bkj ]+ − [−bik ]+ [−bkj ]+ , otherwise,
we obtain the following theorem:

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
Using the following matrix mutation B 0 = µk (B) in direction k:(−bij ,
if i = k of j = k;bij0 =
bij + [bik ]+ [bkj ]+ − [−bik ]+ [−bkj ]+ , otherwise,
we obtain the following theorem:
Theorem (Fomin, Shapiro, Thurston ’08)
Suppose K 0 is obtained from K by a 1-move on the edge labelledk. Then µk (B(K )) = B(K 0 ).
Therefore, A(K ) ∼= A(K 0 ) as cluster algebras of rank 6g + 3m − 6.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
For each surface S we can now form a family of cluster algebras
Am , where m corresponds to the number of vertices in thetriangulations of S.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
For each surface S we can now form a family of cluster algebras
Am , where m corresponds to the number of vertices in thetriangulations of S.
We have a canonical inclusion Am ,→ Am+1 induced by performinga zero move on a 2-simplex.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
For each surface S we can now form a family of cluster algebras
Am , where m corresponds to the number of vertices in thetriangulations of S.
We have a canonical inclusion Am ,→ Am+1 induced by performinga zero move on a 2-simplex.
This forms a direct system and we can define
A = lim Am .
m

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Triangulated Surfaces
For each surface S we can now form a family of cluster algebras
Am , where m corresponds to the number of vertices in thetriangulations of S.
We have a canonical inclusion Am ,→ Am+1 induced by performinga zero move on a 2-simplex.
This forms a direct system and we can define
A = lim Am .
m

So to compare two triangulations with a different number ofvertices we can embed their cluster algebras into ones of higher
rank.
Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Pair Ordering
We now generalise the exchange matrix B to higher-dimensional
triangulated manifolds. Consider a simplicial complex K on thevertex set [m].

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Pair Ordering
We now generalise the exchange matrix B to higher-dimensional
triangulated manifolds. Consider a simplicial complex K on thevertex set [m]. Using the ordering 1 < · · · < m on the vertices we
impose the lexographical ordering < on 2[m] , e.g. when m = 3 wehave
∅ < {1} < {2} < {3} < {1, 2} < {1, 3} < {2, 3} < {1, 2, 3}.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Pair Ordering
We now generalise the exchange matrix B to higher-dimensional
triangulated manifolds. Consider a simplicial complex K on thevertex set [m]. Using the ordering 1 < · · · < m on the vertices we
impose the lexographical ordering < on 2[m] , e.g. when m = 3 wehave
∅ < {1} < {2} < {3} < {1, 2} < {1, 3} < {2, 3} < {1, 2, 3}.
Let α = (a0 , a1 , . . . , an ) be an oriented n-simplex. Then thestandard boundary operator is defined as∂(a0 , a1 , . . . , an ) =nX(−1)j+1 (a0 , . . . , aˆj , . . . , an ).

j=0

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Pair Ordering
We generalise this as follows: for any 0 ≤ k ≤ n defineX∂ (k) (a0 , a1 , . . . , an ) =(−1)j1 +···+jk +1 (. . . , aˆj1 , . . . , aˆjk , . . . ).
1≤j1 <···<jk ≤nObviously, we have ∂ (1) = ∂.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Pair Ordering
We generalise this as follows: for any 0 ≤ k ≤ n defineX∂ (k) (a0 , a1 , . . . , an ) =(−1)j1 +···+jk +1 (. . . , aˆj1 , . . . , aˆjk , . . . ).
1≤j1 <···<jk ≤nObviously, we have ∂ (1) = ∂.
We now define a pair ordering ≺ on all (n − 1)-faces of a simplex
α = (a0 , a1 , . . . , an ).

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Pair Ordering
We generalise this as follows: for any 0 ≤ k ≤ n defineX∂ (k) (a0 , a1 , . . . , an ) =(−1)j1 +···+jk +1 (. . . , aˆj1 , . . . , aˆjk , . . . ).
1≤j1 <···<jk ≤nObviously, we have ∂ (1) = ∂.
We now define a pair ordering ≺ on all (n − 1)-faces of a simplex
α = (a0 , a1 , . . . , an ).
Take two facets f and g of α such that f < g . Definef \ g = (f ∪ g ) \ (f ∩ g ) = α \ (f ∩ g ).
Note that dim f \ g = 1. Consider the coefficient cfg of f \ g in
∂ (n−1) α.
Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Pair Ordering
Set
f ≺g

if cfg =+1

g ≺f

if cfg =-1.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Pair Ordering
Set
f ≺g

if cfg =+1

g ≺f

if cfg =-1.

ExampleTake a = (1, 2, 3). Then the one faces have the lexographicalordering (12) < (13) < (23). By considering∂(1, 2, 3) = (23) − (13) + (12)
we obtain the pair ordering for all one faces as follows:(12) ≺ (13);(23) ≺ (13);Alastair Darby The 43rd Symposium on Transformation Groups(13) ≺ (23).

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Exchange Matrix
Consider a triangulated manifold K of dimension n. Let F denotethe set of (n − 1)-faces of K .

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Exchange Matrix
Consider a triangulated manifold K of dimension n. Let F denotethe set of (n − 1)-faces of K . For every n-simplex α in K we defineα)the skew-symmetric matrix B α (K ) = (bfg
f ,g ∈F of size fn−1 (K ) bysetting+1, if f ≺ g ;α
bfg = −1, if g ≺ f ;0,otherwise,for f , g ⊂ α.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Exchange Matrix
Consider a triangulated manifold K of dimension n. Let F denotethe set of (n − 1)-faces of K . For every n-simplex α in K we defineα)the skew-symmetric matrix B α (K ) = (bfg
f ,g ∈F of size fn−1 (K ) bysetting+1, if f ≺ g ;α
bfg = −1, if g ≺ f ;0,otherwise,for f , g ⊂ α.
Then we define
B(K ) =X

B αK .

α∈F

Note that B(K ) is skew-symmetric with entries belonging to{−1, 0, +1}.
Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Matrix Mutations
We now restrict to the case of 2j = n, i.e. bisteller n-moves on amanifold of dimension 2n.
Question
How does B(K ) change into B(bmα K )? We only need to consider
the local structure around α.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Matrix Mutations
We now restrict to the case of 2j = n, i.e. bisteller n-moves on amanifold of dimension 2n.
Question
How does B(K ) change into B(bmα K )? We only need to consider
the local structure around α.
WLOG assume that α = (1, . . . , j + 1) and β = (j + 2, . . . , 2j + 2).
Set
Fi = (−1)i (1, . . . , j + 1, j + 2, . . . , 2j \+ 3 − i, . . . , 2j + 2),
for i = 1, . . . , j + 1.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Matrix Mutations
We now restrict to the case of 2j = n, i.e. bisteller n-moves on amanifold of dimension 2n.
Question
How does B(K ) change into B(bmα K )? We only need to consider
the local structure around α.
WLOG assume that α = (1, . . . , j + 1) and β = (j + 2, . . . , 2j + 2).
Set
Fi = (−1)i (1, . . . , j + 1, j + 2, . . . , 2j \+ 3 − i, . . . , 2j + 2),
TS
for i = 1, . . . , j + 1. Note α = Fi . Let Λ(F ) = Fi .

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Matrix MutationsThen
bmα Λ(F ) = Λ(H),where
ˆ . . . , j + 1, . . . , 2j + 2)
Hi = (−1)i (1, . . . , i,
TS
Tfor i = 1, . . . , j + 1, β = Hi and Λ(H) = Hi . Note β = Hi .

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Matrix MutationsThen
bmα Λ(F ) = Λ(H),where
ˆ . . . , j + 1, . . . , 2j + 2)
Hi = (−1)i (1, . . . , i,
TS
Tfor i = 1, . . . , j + 1, β = Hi and Λ(H) = Hi . Note β = Hi .
Lemma
bmα
For any n-face γ not in Λ(F ) we have that bfg = bfg, for
f , g ∈ F(γ). Furthermore,XX
B γ (K ) =B γ (bmα K ).
γ∈F (K )\Fγ∈F (bmα K )\H

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Matrix Mutations
Let σ be a permutation on [2j + 2] such that σ(α) = β. Then σ
induces a combinatorial equivalence between Λ(F ) and Λ(H).

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Matrix Mutations
Let σ be a permutation on [2j + 2] such that σ(α) = β. Then σ
induces a combinatorial equivalence between Λ(F ) and Λ(H).
We consider the particular involution
1· · · j + 1 j + 2 · · · 2j + 2
σ=.
2j + 2 · · · j + 2 j + 1 · · ·1

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Matrix Mutations
Let σ be a permutation on [2j + 2] such that σ(α) = β. Then σ
induces a combinatorial equivalence between Λ(F ) and Λ(H).
We consider the particular involution
1· · · j + 1 j + 2 · · · 2j + 2
σ=.
2j + 2 · · · j + 2 j + 1 · · ·1
We define a matrix mutation µσ (B) = B 0 with respect to σ bysetting(bfg ,if f , g ∈/ Λ(F );0bfg =−bσ(f )σ(g ) , if f , g ∈ Λ(F ).

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Matrix Mutations
Let σ be a permutation on [2j + 2] such that σ(α) = β. Then σ
induces a combinatorial equivalence between Λ(F ) and Λ(H).
We consider the particular involution
1· · · j + 1 j + 2 · · · 2j + 2
σ=.
2j + 2 · · · j + 2 j + 1 · · ·1
We define a matrix mutation µσ (B) = B 0 with respect to σ bysetting(bfg ,if f , g ∈/ Λ(F );0bfg =−bσ(f )σ(g ) , if f , g ∈ Λ(F ).
Propositionµσ (B(K )) = B(bmα (K )).
Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Cluster Algebra
Suppose we have a triangulated manifold K . A bistellar cluster
χ(K ) = {xf | f ∈ F(K )}is a set of abstract variables associated to K . Denote by Q(χ(K ))the field of rational functions over χ(K ).

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Cluster Algebra
Suppose we have a triangulated manifold K . A bistellar cluster
χ(K ) = {xf | f ∈ F(K )}is a set of abstract variables associated to K . Denote by Q(χ(K ))the field of rational functions over χ(K ).
Associated to K we get a bistellar seed (χ(K ), B(K )), where B(K )is the exchange matrix associated to K .

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Cluster Algebra
Suppose we have a triangulated manifold K . A bistellar cluster
χ(K ) = {xf | f ∈ F(K )}is a set of abstract variables associated to K . Denote by Q(χ(K ))the field of rational functions over χ(K ).
Associated to K we get a bistellar seed (χ(K ), B(K )), where B(K )is the exchange matrix associated to K .
We then form a cluster algebra A(K ) by performing bistellar seedmutations in all possible directions.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Cluster Algebra
The bistellar exchange relations are of the form
Y [b g ]Y [−bfg ]+xf xσ(f ) =xg f + +xg.
g ∈Λ(F )Alastair Darby The 43rd Symposium on Transformation Groupsg ∈Λ(F )Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Cluster Algebra
The bistellar exchange relations are of the form
Y [b g ]Y [−bfg ]+xf xσ(f ) =xg f + +xg.
g ∈Λ(F )g ∈Λ(F )We then have that
Theorem
The associated cluster algebra A(K ) to a PL-manifold is a
PL-invariant.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Higher Dimensional Manifolds
This theory also works for bistellar moves that are nothalf-dimensional.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Higher Dimensional Manifolds
This theory also works for bistellar moves that are nothalf-dimensional.
One problem is that, these moves change the number ofcodimension 1 faces of K .

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Higher Dimensional Manifolds
This theory also works for bistellar moves that are nothalf-dimensional.
One problem is that, these moves change the number ofcodimension 1 faces of K .
Our solution to these problems is to define an algebra similar
to a cluster algebra with some differences.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Higher Dimensional Manifolds
This theory also works for bistellar moves that are nothalf-dimensional.
One problem is that, these moves change the number ofcodimension 1 faces of K .
Our solution to these problems is to define an algebra similar
to a cluster algebra with some differences.
We have defined generalised matrix mutations for performing
bistellar moves on arbitrary dimensional faces.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Higher Dimensional Manifolds
This theory also works for bistellar moves that are nothalf-dimensional.
One problem is that, these moves change the number ofcodimension 1 faces of K .
Our solution to these problems is to define an algebra similar
to a cluster algebra with some differences.
We have defined generalised matrix mutations for performing
bistellar moves on arbitrary dimensional faces.
So now our goal is to fit this into some algebraic framework
similar to cluster algebras to give (possibly complete)invariants of PL-manifolds.

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras

PL-Manifolds
Cluster Algebras
Triangulated Surfaces
Higher Dimensional Manifolds

Thank you!

Alastair Darby The 43rd Symposium on Transformation Groups

Algebraic PL-Invariants and Cluster Algebras