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Mutationlinear algebra anduniversal geometric cluster algebras
Nathan Reading
NC State University
Mutationlinear (“µlinear”) algebra
Universal geometric cluster algebras
The mutation fan
Universal geometric cluster algebras from surfaces

Mutation maps ηBk
Let B be a skewsymmetrizable exchange matrix.
Given a = (a1, . . . , an) ∈ Rn, let B̃ be B, extended with onecoefficient row a.
For a sequence k = kq, kq−1, . . . , k1, define ηBk (a) to be the
coefficient row of µk(B̃). Concretely, for k = k :
ηBk (a) = (a′1, . . . , a
′n), where
a′j =
−ak if j = k ;aj + akbkj if j 6= k , ak ≥ 0 and bkj ≥ 0;aj − akbkj if j 6= k , ak ≤ 0 and bkj ≤ 0;aj otherwise.
The mutation maps ηBk are piecewiselinear homeomorphisms ofRn. Their inverses are also mutation maps.
1. Mutationlinear (“µlinear”) algebra 1
.

Bcoherent linear relations
Let S be a finite set, let (vi : i ∈ S) be vectors in Rn and let(ci : i ∈ S) be real numbers.
The formal expression∑
i∈S civi is a Bcoherent linear relation if∑i∈S
ciηBk (vi ) = 0, and (1)∑
i∈Scimin(η
Bk (vi ), 0) = 0 (2)
hold for every finite sequence k = kq, . . . , k1.
In particular,∑
i∈S civi is a linear relation in the usual sense.
Example: For B = [0], a Bcoherent relation is a linear relationamong vectors in Rn all agreeing in sign.
1. Mutationlinear (“µlinear”) algebra 2
.

Basis for B
Let R be Z or a field between Q and R. Usually Z, Q, or R.
The vectors (bi : i ∈ I ) in Rn are an Rbasis for B if and only ifthe following two conditions hold.
(i) Spanning: For all a ∈ Rn, there exists a finite subset S of Iand coefficients ci in R such that a−
∑i∈S cibi is a
Bcoherent linear relation.
(ii) Independence: If S is a finite subset of I and∑
i∈S cibi is aBcoherent linear relation, then ci = 0 for all i ∈ S .
Example: For any R, {±1} ⊂ R1 is an Rbasis for [0].
Theorem
Every skewsymmetrizable B has an Rbasis.
Proof: Zorn’s lemma, same argument that shows that every vectorspace has a (Hamel) basis. The proof is nonconstructive.
1. Mutationlinear (“µlinear”) algebra 3
.

Why do µlinear algebra?
• Universal geometric cluster algebras
• The mutation fan
1. Mutationlinear (“µlinear”) algebra 4
.

Tropical Semifield (broadly defined)
I : an indexing set (no requirement on cardinality).
(ui : i ∈ I ): formal symbols (tropical variables).
Tropical semifield Trop(ui : i ∈ I ):Elements are products
∏i∈I u
aii with ai ∈ R.
Multiplication as usualTrop(ui : i ∈ I ) is the module R I , written multiplicatively.
Auxiliary addition∏
i∈I uaii ⊕
∏i∈I u
bii =
∏i∈I u
min(ai ,bi )i .
Topology on R: discrete.
Topology on Trop(ui : i ∈ I ): product topology as R I .I finite: discrete.I countable: FPS.
2. Universal geometric cluster algebras 5
.

Cluster algebra of geometric type (broadly defined)
Extended exchange matrix: a collection of rows indexed by [n] ∪ I .
In matrix notation, B̃ = [bij ].
Rows of B̃ indexed by [n] are the matrix B.
Other rows are coefficient rows. These are vectors in Rn.
Coefficients: yj =∏
i∈I ubiji ∈ Trop(ui : i ∈ I ) for each j ∈ [n].
Cluster algebra of geometric type:
A(x, B̃) = A(x,B, {y1, . . . , yn}).
(The usual definition: take I finite and R = Z.)
2. Universal geometric cluster algebras 6
.

Coefficient specialization
B̃ and B̃ ′: extended exchange matrices of rank n.
For each sequence k and j ∈ [n], write yk,j for the j th coefficientdefined by µk(B̃) and y
′k,j for the j
th coefficient defined by µk(B̃′).
A ring homomorphism ϕ : A(x, B̃)→ A(x′, B̃ ′) is a coefficientspecialization if
(i) the exchange matrices B and B ′ coincide;
(ii) ϕ(xj) = x′j for all j ∈ [n] (i.e. initial cluster variables coincide);
(iii) ϕ restricts to a continuous Rlinear map on tropical semifields.
(iv) ϕ(yk,j) = y′k,j and ϕ(yk,j ⊕ 1) = y ′k,j ⊕ 1 for each k and each j .
Continuity refers to the product topology on tropical semifields.
An Rlinear map is just a multiplicative homomorphism ofsemifields when R = Z or Q.
2. Universal geometric cluster algebras 7
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Coefficient specialization
B̃ and B̃ ′: extended exchange matrices of rank n.
For each sequence k and j ∈ [n], write yk,j for the j th coefficientdefined by µk(B̃) and y
′k,j for the j
th coefficient defined by µk(B̃′).
A ring homomorphism ϕ : A(x, B̃)→ A(x′, B̃ ′) is a coefficientspecialization if
(i) the exchange matrices B and B ′ coincide;
(ii) ϕ(xj) = x′j for all j ∈ [n] (i.e. initial cluster variables coincide);
(iii) ϕ restricts to a continuous Rlinear map on tropical semifields.
(iv) ϕ(yk,j) = y′k,j and ϕ(yk,j ⊕ 1) = y ′k,j ⊕ 1 for each k and each j .
Continuity refers to the product topology on tropical semifields.
An Rlinear map is just a multiplicative homomorphism ofsemifields when R = Z or Q.
2. Universal geometric cluster algebras 7
.

Universal geometric cluster algebras
Fix an exchange matrix B.
Category: geometric cluster algebras with exchange matrix B;coefficient specializations. (Depends on R.)
Universal geometric cluster algebra: universal object in category.
Isomorphism type of a geometric cluster algebra depends only onthe extended exchange matrix B̃. So we are really looking foruniversal geometric exchange matrices.
We’ll use the term universal geometric coefficients for thecoefficient rows of a universal geometric exchange matrix.
2. Universal geometric cluster algebras 8
.

Previous results (for B bipartite, finite type):
FominZelevinsky, 2006:Coefficient rows of universal extended exchange matrix are indexedby almost positive coroots β∨. The row for β∨ has i th entryε(i)[β∨ : α∨i ] for i = 1, . . . , n.
ReadingSpeyer, 2007:The linear map taking a positive root αi to −ε(i)ωi maps almostpositive roots into rays of (bipartite) Cambrian fan.
YangZelevinsky, 2008:Rays of the Cambrian fan are spanned by vectors whosefundamental weight coordinates are gvectors of cluster variables.
Putting this all together (including sign change and “ ∨ ”):Universal extended exchange matrix for B has coefficient rowsgiven by gvectors for BT .
(Works for any R or for a much larger category)
2. Universal geometric cluster algebras 9
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Universal geometric coefficients and µlinear algebra
Theorem (R., 2012)
A collection (bi : i ∈ I ) are universal coefficients for B over R ifand only if they are an Rbasis for B.
(That is, to make a universal extended exchange matrix, extend Bwith coefficient rows forming an Rbasis for B.)
Example:[
01−1
]is universal for any R.
2. Universal geometric cluster algebras 10
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Why universal coefficients correspond to bases
Recall: a coefficient specialization is a continuous Rlinear mapbetween tropical semifields taking coefficients to coefficientseverywhere in the Y pattern.
Proposition.Let Trop(ui : i ∈ I ) and Trop(vj : j ∈ J) be tropical semifields andfix a family (pij : i ∈ I , j ∈ J) of elements of R. Then the followingare equivalent.
(i) ∃ continuous Rlinear map ϕ from Trop(ui : i ∈ I ) toTrop(vj : j ∈ J), with ϕ(ui ) =
∏j∈J v
pijj for all i ∈ I .
(ii) For each j ∈ J, there are only finitely many indices i ∈ I suchthat pij is nonzero.
When these conditions hold, the unique map ϕ is
ϕ(∏i∈I
uaii
)=∏j∈J
v∑
i∈I pijajj .
2. Universal geometric cluster algebras 11
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Why universal coefficients correspond to bases (continued)
A picture suitable for handwaving:
B B
2. Universal geometric cluster algebras 12
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The mutation fan
One “easy” way to get Bcoherent linear relations: Find vectors inthe same domain of linearity of all mutation maps and make alinear relation among them.
Define an equivalence relation ≡B on Rn by setting
a1 ≡B a2 ⇐⇒ sgn(ηBk (a1)) = sgn(ηBk (a2)) ∀k.
sgn(a) is the vector of signs (−1, 0,+1) of the entries of a.
Bclasses: equivalence classes of ≡B .
Bcones: closures of Bclasses.
Facts:Bclasses are cones.Bcones are closed cones.Each map ηBk is linear on each Bcone.
3. The mutation fan 13
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The mutation fan (continued)
Mutation fan for B:The collection FB of all Bcones and all faces of Bcones.
Theorem (R., 2011)
FB is a complete fan.
Example: For B = [0]:µ1 is negation and FB = {{0},R≥0,R≤0}.
3. The mutation fan 14
.

Example: B =[
0 2 −2−2 0 2
2 −2 0
](Markov quiver)
3. The mutation fan 15
.

Positive bases
An Rbasis for B is positive if, for every a = (a1, . . . , an) ∈ Rn, theunique Bcoherent linear relation a−
∑i∈I cibi has all ci ≥ 0.
In many examples, positive Rbases exist, but not always, and itdepends on R.
There is at most one positive Rbasis for B, up to scaling eachbasis element by a positive unit in R.
Theorem (R., 2012)
If a positive Rbasis for B exists, then FB is simplicial. The basisconsists of exactly one vector in each ray of FB .
For R 6= R, there is a similar statement in terms of the “Rpart” ofFB (e.g. the “rational part”) of FB .
3. The mutation fan 16
.

gVectors enter the picture
Conjecture
The nonnegative orthant (R≥0)n is a Bcone. (Equivalently,principal coefficients are signcoherent.)
Theorem (R., 2011)
Assume the above conjecture holds for every exchange matrix thatis mutationequivalent to B or to −B. Then the gvector fan forBT is a subfan of FB .
Proof uses a NakanishiZelevinsky result.
One interpretation of this theorem: The mutation fan is the naturalgeneralization of the gvector fan of a cluster algebra of finite type.This leads to lots of speculative ideas, e.g. about “nice” additivebases for cluster algebras or virtual semiinvariants of quivers, orindices of “unreachable” objects in CQ,W (Plamondon’s talk).
3. The mutation fan 17
.

Rank 2 finite type
FB is finite and rational. Taking a smallest integer vector in eachray, we get a positive Rbasis for B for any R. (These are thegvectors for BT .)
0 1−1 01 00 1−1 00 −11 −1
0 1−2 01 00 1−1 00 −11 −12 −1
0 1−3 01 00 1−1 00 −11 −13 −22 −13 −1
3. The mutation fan 18
.

Rank 2 affine type
FB is infinite but still rational. Again a positive Rbasis for B forany R. (These are the gvectors for BT with an additional limitingvector.)
0 1−4 0−1 01 −13 −2· ·· ·· ·0 −14 −38 −5· ·· ·· ·0 14 −18 −3· ·· ·· ·1 03 −15 −2· ·· ·· ·2 −1
3. The mutation fan 19
.

Rank 2 wild type
For B =[
0 2−3 0
], the fan FB is infinite and algebraic, not rational.
All cones are rational except theshaded cone, which is the nonnegative span of
[√
6, −√
6−√
2]
and
[3(√
6 +√
2), −2√
6].
A positive Rbasis for B if R = R:Take a nonzero vector in each ray of FB .
A Qbasis for B:Take a rational vector in each ray except the irrational rays.Then take any two linearly independent rational vectors in theshaded cone. This is not a positive basis.
3. The mutation fan 20
.

Rank 2 wild type (continued). . . the case R = Z
B =[
0 2−3 0
], C is shaded cone.
A Zbasis for B:• The smallest integerpoint in each rationalray, and
• Two integer pointsin C whose Zlinearspan contains allinteger points in C .
The two red pointswork because they area Zbasis for Z2.
This is not a positivebasis.
3. The mutation fan 21
.

Rank 2 wild type (continued). . . the case R = Z
B =[
0 2−3 0
], C is shaded cone.
A Zbasis for B:• The smallest integerpoint in each rationalray, and
• Two integer pointsin C whose Zlinearspan contains allinteger points in C .The two red pointswork because they area Zbasis for Z2.
This is not a positivebasis.
3. The mutation fan 21
.

Finite and affine type
As mentioned earlier: In finite type, gvectors for BT are a basisfor B.
This can be proved directly and uniformly using the geometry ofroot systems.
Similar arguments can be made in affine type. Using results of apaper in progress (joint with Speyer) on gvectors in affine type,we should be able to prove the following conjecture:
Conjecture
For B of affine type, a positive Rbasis for B (over any R) consistsof the gvectors for BT , plus one additional integer vector in theboundary of the Tits cone.
This is not surprising in light of ShermanZelevinsky and in light ofthe surfaces case. (In fact, I have a proof in the surfaces case.)
3. The mutation fan 22
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3. The mutation fan 23
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Universal geometric cluster algebras from surfaces
Allowable curves: Almost, but not quite, the set of curves thatappear in (unbounded measured) laminations.
Compatibility of allowable curves: Almost but not quite that theydon’t intersect. (One kind of intersection is allowed.)
What should happen:
1. Zbasis or Qbasis for B(T ) consists of shear coordinates ofallowable curves.
2. (Rational part of) the mutation fan consists of cones spannedby shear coordinates of pairwise compatible sets of allowablecurves.
What happens depends on two key properties of the surface:
The Curve Separation Property and the Null Tangle Property.
(Properties are proved in some cases. No counterexamples.)4. Universal geometric cluster algebras from surfaces 24
.

The Null Tangle Property
A tangle is a finite weighted collection Ξ of distinct allowablecurves, with no requirement of compatibility. The shearcoordinates of Ξ are the weighted sum of the shear coordinates ofits curves. A null tangle has shear coordinates zero with respect toevery tagged triangulation.∗
The Null Tangle Property: A null tangle has all weights zero.
Theorem (R., 2012)
The shear coordinates of allowable curves are a (positive) Z orQbasis for B(T ) if and only if the Null Tangle Property holds.
Theorem (R., 2012)
The Null Tangle Property holds for surfaces with polynomialgrowth and for the oncepunctured torus.
4. Universal geometric cluster algebras from surfaces 25
.

The Curve Separation Property
The Curve Separation Property: If λ and ν are incompatibleallowable curves, then there exists a tagged triangulation∗ T and atagged arc γ ∈ T such that the shear coordinates of λ and ν withrespect to T have strictly opposite signs in the entry indexed by γ.
Null Tangle Property =⇒ Curve Separation Property
Theorem (R., 2012)
The rational part of the mutation fan consists of cones spanned byshear coordinates of pairwise compatible sets of allowable curves ifand only if the Curve Separation Property holds.
Theorem (R., 2012)
The Curve Separation Property holds except possibly for surfacesof genus > 1 with no boundary components and exactly onepuncture.
4. Universal geometric cluster algebras from surfaces 26
.

Proving the properties
Curve Separation Property: For each incompatible λ and ν,construct a tagged triangulation T . Many cases.
For example, if λ connects two distinct boundary segments:
λ
1
2
3
4
Similarly, in any null tangle, a curve λ connecting two distinctboundary segments occurs with weight zero.
4. Universal geometric cluster algebras from surfaces 27
.

Thanks for listening.
References:
Universal geometric cluster algebras. arXiv:1209.3987
Universal geometric cluster algebras from surfaces.arXiv:1209.4095
Universal geometric coefficients for the oncepunctured torus.In preparation.
Mutationlinear (``linear'') algebraUniversal geometric cluster algebrasThe mutation fanUniversal geometric cluster algebras from surfaces