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Mutation-linear algebra and universal geometric cluster algebras Nathan Reading NC State University Mutation-linear (“μ-linear”) algebra Universal geometric cluster algebras The mutation fan Universal geometric cluster algebras from surfaces

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  • Mutation-linear algebra anduniversal geometric cluster algebras

    Nathan Reading

    NC State University

    Mutation-linear (“µ-linear”) algebra

    Universal geometric cluster algebras

    The mutation fan

    Universal geometric cluster algebras from surfaces

  • Mutation maps ηBk

    Let B be a skew-symmetrizable 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 piecewise-linear homeomorphisms ofRn. Their inverses are also mutation maps.

    1. Mutation-linear (“µ-linear”) algebra 1

    .

  • B-coherent 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 B-coherent 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 B-coherent relation is a linear relationamong vectors in Rn all agreeing in sign.

    1. Mutation-linear (“µ-linear”) algebra 2

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  • 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 R-basis 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

    B-coherent linear relation.

    (ii) Independence: If S is a finite subset of I and∑

    i∈S cibi is aB-coherent linear relation, then ci = 0 for all i ∈ S .

    Example: For any R, {±1} ⊂ R1 is an R-basis for [0].

    Theorem

    Every skew-symmetrizable B has an R-basis.

    Proof: Zorn’s lemma, same argument that shows that every vectorspace has a (Hamel) basis. The proof is non-constructive.

    1. Mutation-linear (“µ-linear”) algebra 3

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  • Why do µ-linear algebra?

    • Universal geometric cluster algebras

    • The mutation fan

    1. Mutation-linear (“µ-linear”) algebra 4

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  • 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

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  • 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

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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 R-linear 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 R-linear 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 R-linear 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 R-linear 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

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  • Previous results (for B bipartite, finite type):

    Fomin-Zelevinsky, 2006:Coefficient rows of universal extended exchange matrix are indexedby almost positive co-roots β∨. The row for β∨ has i th entryε(i)[β∨ : α∨i ] for i = 1, . . . , n.

    Reading-Speyer, 2007:The linear map taking a positive root αi to −ε(i)ωi maps almostpositive roots into rays of (bipartite) Cambrian fan.

    Yang-Zelevinsky, 2008:Rays of the Cambrian fan are spanned by vectors whosefundamental weight coordinates are g-vectors of cluster variables.

    Putting this all together (including sign change and “ ∨ ”):Universal extended exchange matrix for B has coefficient rowsgiven by g-vectors 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 µ-linear algebra

    Theorem (R., 2012)

    A collection (bi : i ∈ I ) are universal coefficients for B over R ifand only if they are an R-basis for B.

    (That is, to make a universal extended exchange matrix, extend Bwith coefficient rows forming an R-basis 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 R-linear 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 R-linear 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 hand-waving:

    B B

    2. Universal geometric cluster algebras 12

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  • The mutation fan

    One “easy” way to get B-coherent 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.

    B-classes: equivalence classes of ≡B .

    B-cones: closures of B-classes.

    Facts:B-classes are cones.B-cones are closed cones.Each map ηBk is linear on each B-cone.

    3. The mutation fan 13

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  • The mutation fan (continued)

    Mutation fan for B:The collection FB of all B-cones and all faces of B-cones.

    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

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  • Example: B =[

    0 2 −2−2 0 2

    2 −2 0

    ](Markov quiver)

    3. The mutation fan 15

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  • Positive bases

    An R-basis for B is positive if, for every a = (a1, . . . , an) ∈ Rn, theunique B-coherent linear relation a−

    ∑i∈I cibi has all ci ≥ 0.

    In many examples, positive R-bases exist, but not always, and itdepends on R.

    There is at most one positive R-basis for B, up to scaling eachbasis element by a positive unit in R.

    Theorem (R., 2012)

    If a positive R-basis 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 “R-part” ofFB (e.g. the “rational part”) of FB .

    3. The mutation fan 16

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  • g-Vectors enter the picture

    Conjecture

    The nonnegative orthant (R≥0)n is a B-cone. (Equivalently,principal coefficients are sign-coherent.)

    Theorem (R., 2011)

    Assume the above conjecture holds for every exchange matrix thatis mutation-equivalent to B or to −B. Then the g-vector fan forBT is a subfan of FB .

    Proof uses a Nakanishi-Zelevinsky result.

    One interpretation of this theorem: The mutation fan is the naturalgeneralization of the g-vector 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 semi-invariants of quivers, orindices of “unreachable” objects in CQ,W (Plamondon’s talk).

    3. The mutation fan 17

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  • Rank 2 finite type

    FB is finite and rational. Taking a smallest integer vector in eachray, we get a positive R-basis for B for any R. (These are theg-vectors 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

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  • Rank 2 affine type

    FB is infinite but still rational. Again a positive R-basis for B forany R. (These are the g-vectors 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

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  • 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 non-negative span of

    [√

    6, −√

    6−√

    2]

    and

    [3(√

    6 +√

    2), −2√

    6].

    A positive R-basis for B if R = R:Take a nonzero vector in each ray of FB .

    A Q-basis 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

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  • Rank 2 wild type (continued). . . the case R = Z

    B =[

    0 2−3 0

    ], C is shaded cone.

    A Z-basis for B:• The smallest integerpoint in each rationalray, and

    • Two integer pointsin C whose Z-linearspan contains allinteger points in C .

    The two red pointswork because they area Z-basis for Z2.

    This is not a positivebasis.

    3. The mutation fan 21

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  • Rank 2 wild type (continued). . . the case R = Z

    B =[

    0 2−3 0

    ], C is shaded cone.

    A Z-basis for B:• The smallest integerpoint in each rationalray, and

    • Two integer pointsin C whose Z-linearspan contains allinteger points in C .The two red pointswork because they area Z-basis for Z2.

    This is not a positivebasis.

    3. The mutation fan 21

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  • Finite and affine type

    As mentioned earlier: In finite type, g-vectors 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 g-vectors in affine type,we should be able to prove the following conjecture:

    Conjecture

    For B of affine type, a positive R-basis for B (over any R) consistsof the g-vectors for BT , plus one additional integer vector in theboundary of the Tits cone.

    This is not surprising in light of Sherman-Zelevinsky 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. Z-basis or Q-basis 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

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  • 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- orQ-basis 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 once-punctured 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

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  • 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

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  • 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 once-punctured torus.In preparation.

    Mutation-linear (``-linear'') algebraUniversal geometric cluster algebrasThe mutation fanUniversal geometric cluster algebras from surfaces