Báo cáo toán học: "The cluster basis of Z[x1,1, . . . , x3,3]" potx

We also show that the transitionmatrices relating this cluster basis to the natural and the dual canonical bases areunitriangular and nonnegative.. In the event thatthis conjectured equa

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The cluster basis of Z[x 1,1 , , x 3,3 ]

Mark Skandera

Department of MathematicsLehigh University, Bethlehem, PA 18015

mas906@lehigh.eduSubmitted: Sep 6, 2007; Accepted: Nov 1, 2007; Published: Nov 12, 2007

Mathematics Subject Classification: 05C99, 15A15, 16W99

Abstract

We show that the set of cluster monomials for the cluster algebra of type D4contains a basis of the Z-module Z[x1,1, , x3,3] We also show that the transitionmatrices relating this cluster basis to the natural and the dual canonical bases areunitriangular and nonnegative These results support a conjecture of Fomin andZelevinsky on the equality of the cluster and dual canonical bases In the event thatthis conjectured equality is true, our results also imply an explicit factorization ofeach dual canonical basis element as a product of cluster variables

1 Introduction

The coordinate ring O(SL(n, C)) of polynomial functions in the entries of matrices inSL(n, C) may be realized as a quotient,

O(SL(n, C)) = C[x1,1, , xn,n]/(det(x) − 1),where x = (x1,1, , xn,n) is a matrix of n2 commuting variables We will call i the rowindex and j the column index of the variable xi,j

Viewing the rings O(SL(n, C)) and C[x1,1, , xn,n] as vector spaces, one often appliesthe canonical homomorphism to a particular basis of C[x1,1, , xn,n] in order to obtain

a basis of O(SL(n, C)) Some bases of C[x1,1, , xn,n] which appear in the literature arethe natural basis of monomials, the bitableau basis of D´esarm´enien, Kung and Rota [6],and the dual canonical (crystal) basis of Lusztig [23] and Kashiwara [20] Since thetransition matrices relating these bases have integer entries and determinant 1, each isalso a basis of the Z-module Z[x1,1, , xn,n] In the case n = 3, work of Berenstein, Fominand Zelevinsky [2, 12, 14, 16] suggests that certain polynomials which arise as clustermonomials in the study of cluster algebras may form a basis of Z[x1,1, , x3,3] and thatthis basis may be equal to the dual canonical basis (In fact, unpublished work of these

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authors [15] implies that these cluster monomials do form a basis of Z[x1,1, , x3,3].) Theanalogous statement for n ≥ 4 is known to be false.

After recalling the definition of cluster monomials in Section 2, we will perform ratherelementary computations in Section 3 to observe a bijective correspondence between anappropriate set of cluster monomials and 3 × 3 nonnegative integer matrices This cor-respondence will lead to our main theorems in Section 4 which show our set of clustermonomials to form a basis of Z[x1,1, , x3,3] Using the correspondence we also relatethe cluster basis by unitriangular transition matrices to the natural and dual canonicalbases, and give conjectured formulae for the irreducible factorization of dual canonicalbasis elements

2 Cluster monomials of type D4

Fomin and Zelevinsky defined a class of commutative rings called cluster algebras [12] inorder to study total positivity and dual canonical bases in semisimple algebraic groups.(See also [2], [14], [16].) This definition continued earlier work of the authors with Beren-stein [1], [3], [11] and of Lusztig [24] Further work has revealed connections betweencluster algebras and other topics such as Laurent phenomena [13], Teichm¨uller spaces [8],Poisson geometry [17] and algebraic combinatorics [10]

Each cluster algebra has a distinguished set of generators called cluster variables whichare grouped into overlapping subsets called clusters Those cluster algebras generated

by a finite set of cluster variables enjoy a classification similar to the Cartan-Killingclassification of semisimple Lie algebras [14] We shall consider clusters of the clusteralgebra of type D4, which arises in the study of total nonnegativity within SL(3, C) andGL(3, C) In particular, one may decompose G = SL(3, C) or GL(3, C) as in [11], [24]into a union of intersections of double cosets called double Bruhat cells {Gu,v| u, v ∈ S3}.Letting u and v be the longest element w0 of S3, we obtain the double Bruhat cell Gw 0 ,w 0

,whose coordinate ring O(Gw 0 ,w 0) contains Z[x1,1, , x3,3] as a subring and which has acluster algebra structure of type D4 More precisely, for G = GL(3, C), the coordinate ringO(Gw 0 ,w 0

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When writing x{i 1 , ,ik},{j 1 , ,jk} and ∆{i 1 , ,ik},{j 1 , ,jk}, we will tacitly assume set elements

to satisfy i1 < · · · < ik and j1 < · · · < jk To economize notation, we also may denote thesubmatrix and minor by xi 1 ···i k ,j 1 ···j k and ∆i 1 ···i k ,j 1 ···j k

In terms of this notation, our cluster variables are the sixteen polynomials

x1,1, x1,2, x2,1, x2,2, x2,3, x3,2, x3,3,

∆12,12, ∆12,13, ∆13,12, ∆13,13, ∆13,23, ∆23,13, ∆23,23,Imm213 =

def x1,2x2,1x3,3− x1,2x2,3x3,1− x1,3x2,1x3,2+ x1,3x2,2x3,1,Imm132 =

or are different colorings of the same diameter Three examples and the correspondingsets of cluster variables are

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sym-the simplicial complex in which vertices are cluster variables and faces are sets of clustervariables whose geometric realizations satisfy the noncrossing conditions above Theseare shown in the following table, where we have named the clusters in a manner consis-tent with the naming of the thirty-four clusters shown in [11, Fig 8] In the table, wehave partitioned the clusters into twelve blocks whose significance will become clear inObservations 3.1 - 3.12.

Cluster

name

ClustervariablesdefA x2,2, x2,3, x3,2, ∆23,23

ClustervariablesbcdG x1,2, x2,1, x2,2, ∆12,12abcG x1,1, x1,2, x2,1, ∆12,12

bdfG x1,2, x2,2, x3,2, ∆12,12bfEG x1,2, x3,2, ∆13,12, ∆12,12

abEG x1,1, x1,2, ∆13,12, ∆12,12

cdeG x2,1, x2,2, x2,3, ∆12,12

ceFG x2,1, x2,3, ∆12,13, ∆12,12

acFG x1,1, x2,1, ∆12,13, ∆12,12bfCE x1,2, x3,2, ∆13,23, ∆13,12

gDEF x3,3, ∆13,13, ∆13,12, ∆12,13

aDEF x1,1, ∆13,13, ∆13,12, ∆12,13

We define five more polynomials to be frozen variables,

x1,3, ∆12,23, ∆123,123, ∆23,12, x3,1,and define the union of these with any cluster to be an extended cluster We define acluster monomial to be a product of nonnegative powers of cluster variables, and integerpowers of frozen variables, all belonging to the same extended cluster We denote by Mthe subset of cluster monomials in which exponents of frozen variables are nonnegative,i.e., the subset of cluster monomials belonging to Z[x1,1, , x3,3]

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In contrast to [11, Fig 8], we reserve the letters a, , g, A, , G for use as exponents

of cluster variables rather than using these to denote the cluster variables themselves Wewill thus express each cluster monomial having no frozen factors as

23,23∆B 23,13∆C 13,23∆D 13,13∆E 13,12∆F 12,13∆G 12,12Immp213Immq312, (2)where at most four of the exponents a, g, A , G, p, q are positive

It is worth noting that each extended cluster provides a criterion for testing totalpositivity of a matrix y in GL(3, C) or SL(3, C) Specifically, y is totally positive (allminors of y are positive) if and only if each element of an (arbitrary) extended clusterevaluates positively on y (See [2, Sec 2.4], [11, Fig 8].) Of course, the inequality det(y) >

0 may be omitted for y ∈ SL(3, C)

Observe that any permutation of x1,1, , x3,3 induces an automorphism of the ring

Z[x1,1, , x3,3] In particular, we will consider three natural permutations and the sponding involutive automorphisms defined by the usual matrix transposition x 7→ x>, bymatrix antitransposition x 7→ x⊥ (transposition across the antidiagonal)

on modified triangulations and therefore on clusters Specifically, twenty-five pairs {C, D}

of clusters satisfy D = C⊥ 6= C, and each such pair occupies a single row of the tableabove No cluster C satisfies C = C⊥

The transposition map may be interpreted geometrically as a swapping of colors ondiameters which fixes all pairs of non-diameter diagonals Again, this clearly induces aninvolution on modified triangulations and therefore on clusters Specifically, fifteen pairs{C, D} of clusters satisfy D = C>6= C Eleven of these pairs occupy the consecutive rows

of the table containing clusters cdeA, , gBDF and four more such pairs are

{bACp, cABp}, {acBp, abCp}, {eFGq, fEGq}, {egFq, fgEq}

The twenty clusters C not included in these fifteen pairs satisfy C = C>

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Using the diagrams (1) and the definition of modified triangulations of the octogon,

we can identify certain pairs of cluster variables which never appear together in a singlecluster, and therefore never appear together in a cluster monomial In particular, we shalluse the following facts

Observation 2.2 A product xi 1 ,j 1xi 2 ,j 2 of cluster variables is a cluster monomial if andonly if we have (i1− i2)(j1− j2) ≤ 0

By (2), the lowercase letters a, , g in a cluster name correspond to cluster variableswhich are single matrix entries Observation 2.2 therefore says that the pairs of letters inthis range which appear together in a cluster name are precisely

ab, ac, bc, bd, bf, cd, ce, de, df, ef, eg, fg

Observation 2.3 Each product ∆{i 1 ,i 2 },{j 1 ,j 2 }xi 1 ,j 1 and ∆{i 1 ,i 2 },{j 1 ,j 2 }xi 2 ,j 2 of cluster ables is a cluster monomial

vari-By (2), the capital letters A, , G in a cluster name correspond to cluster variableswhich are 2 × 2 minors Observation 2.3 therefore says that the pairs of cluster variablessatisfying the claimed conditions are precisely those corersponding to the pairs of letters

aD, aE, aF, aG, bC, cB, dA, dG, eF, fE, gA, gB, gC, gD

3 A correspondence between cluster monomials and matrices

Let M be the set of cluster monomials of Z[x1,1, , x3,3] Let Mat3(N) be the set of 3 × 3matrices with entries in N, and let Ei,j ∈ Mat3(N) be the matrix whose (i, j) entry is

1 and whose other entries are 0 Let φ : M → Mat3(N) be the map defined on clustervariables by

φ(∆{i 1 , ,i k },{j 1 , ,j k }(x)) = Ei 1 ,j 1 + · · · + Ei k ,j k,

φ(Imm213(x)) = E1,2+ E2,1+ E3,3,φ(Imm132(x)) = E1,1+ E2,3+ E3,2,and extended to cluster monomials in Z[x1,1, , x3,3] by

By definition we have φ(1) = 0, and it is clear that φ maps each cluster monomial

of degree r in x1,1, , x3,3 to a matrix whose entries sum to r It is also clear that φcommutes with the transposition and antitransposition maps,

φ(x>) = φ(x)>, φ(x⊥) = φ(x)⊥

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To begin to establish that φ is a bijection, we partition the fifty clusters into twelve blocksdefined in terms of φ Specifically, each block consists of the clusters C = {z1, z2, z3, z4}with the property that for every cluster monomial Z = z1

1 z2

2 z3

3 z4

4 (which contains nofrozen factors) the matrix φ(Z) has five specific entries which are equal to zero

Two blocks of clusters produce cluster monomials Z for which the matrices φ(Z) have

0 0 0

i

Observation 3.1 Applying φ to the cluster monomials

23,23 if p3,3 ≤ p2,2,

xe 2,3xf3,2xg3,3∆A

23,23 if p2,2 < p3,3,where

A = min{p2,2, p3,3}, d = p2,2− A, g = p3,3− A, e = p2,3, f = p3,2

Note that Observation 3.1 does not assert the existence of a unique cluster monomial

Z satifying φ(Z) = P for a matrix P of the stated form, except when Z is assumed toappear on the list (3) We in fact will make the stronger assertion in Theorem 3.17

xb 1,2xc 2,1xd 2,2∆G 12,12, xa

1,1xb 1,2xc 2,1∆G

we obtain matrices P satisfying p1,3 = p2,3 = p3,1 = p3,2 = p3,3 = 0 Conversely, if

P ∈ Mat3(N) has the stated form then there is a unique cluster monomial Z in (4)satisfying φ(Z) = P

Proof: Apply the antitransposition map to Observation 3.1, or use straightforwardcomputation

Four blocks of clusters produce cluster monomials Z for which the matrices φ(Z) have

0 ∗ 0

i, h0 ∗ 00 ∗ 0

0 ∗ ∗

i, h∗ 0 0∗ ∗ ∗

0 0 0

i

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xc 2,1xd 2,2xe 2,3∆A 23,23, xc

2,1xe 2,3∆A 23,23∆B 23,13, xe

xc 2,1xe 2,3∆A 23,23∆B 23,13 if p2,2 < p3,3 ≤ p2,1+ p2,2,

xe 2,3xg3,3∆A

23,23∆B 23,13 if p2,1+ p2,2 < p3,3,where

A = min{p2,2, p3,3}, B = min{p3,3− A, p2,1}, e = p2,3,

c = p2,1− B, d = p2,2− A, g = p3,3− A − B

xb 1,2xd 2,2xf3,2∆G

12,12, xb

1,2xf3,2∆E

13,12∆G 12,12, xa

1,1xb 1,2∆E 13,12∆G

Proof: Apply the antitransposition map to Observation 3.3

xb1,2xd2,2xf3,2∆A23,23, xb1,2xf3,2∆A23,23∆C13,23, xf3,2xg3,3∆A23,23∆C13,23, (7)

Proof: Apply the transposition map to Observation 3.3

xc2,1xd2,2xe2,3∆G12,12, xc2,1xe2,3∆F12,13∆G12,12, xa1,1xc2,1∆F12,13∆G12,12, (8)

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Proof: Apply the transposition and antitransposition maps to Observation 3.3 Two blocks of clusters produce cluster monomials Z for which the matrices φ(Z) have

0 ∗ ∗

i.Observation 3.7 Applying φ to the cluster monomials

xc2,1xe2,3∆B23,13∆F12,13, xa1,1xc2,1∆B23,13∆F12,13, xe2,3xg3,3∆B23,13∆F12,13,

xa 1,1∆B 23,13∆D 13,13∆F 12,13, xg3,3∆B

23,13∆D 13,13∆F

xa 1,1xc 2,1∆B 23,13∆F 12,13 if p3,3− p2,1 ≤ 0 < p1,1− p2,3,

xe 2,3xg3,3∆B

23,13∆F 12,13 if p1,1− p2,3 ≤ 0 < p3,3− p2,1,

xa 1,1∆B 23,13∆D 13,13∆F 12,13 if 0 < p3,3− p2,1 ≤ p1,1− p2,3,

xg3,3∆B 23,13∆D 13,13∆F 12,13 if 0 < p1,1− p2,3 < p3,3− p2,1,where

B = min{p2,1, p3,3}, F = min{p1,1, p2,3}, D = min{p1,1− F, p3,3− B},

e = p2,3− F, c = p2,1− B, a = p1,1− D − F, g = p3,3− B − D

xb 1,2xf3,2∆C

13,23∆E 13,12, xf3,2xg3,3∆C

13,23∆E 13,12, xa

1,1xb 1,2∆C 13,23∆E 13,12,

xg3,3∆C13,23∆D13,13∆E13,12, xa1,1∆C13,23∆D13,13∆E13,12, (10)

Proof: Apply the transposition or antitransposition map to Observation 3.7

0 ∗ 0

i

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Observation 3.9 Applying φ to the clusters

xb

1,2xc

2,1xd 2,2∆A 23,23, xb

1,2xc 2,1∆A 23,23Immp213, xb

1,2∆A 23,23∆C 13,23Immp213,

23,23∆B 23,13∆C

xb

1,2xc

2,1∆A 23,23Immp213 if p2,2 < p3,3 ≤ p1,2+ p2,2, p2,1+ p2,2

xb

1,2∆A

23,23∆C 13,23Immp213 if p2,1+ p2,2 < p3,3 ≤ p1,2+ p2,2

xc

2,1∆A

23,23∆B 23,13Immp213 if p1,2+ p2,2 < p3,3 ≤ p2,1+ p2,2

∆A

23,23∆B

23,13∆C 13,23Immp213 if p1,2+ p2,2, p2,1+ p2,2 < p3,3 ≤ p1,2+ p2,1+ p2,2

xg3,3∆A

23,23∆B 23,13∆C 13,23 if p1,2+ p2,1 + p2,2 < p3,3,where

A = min{p2,2, p3,3}, b = max{p1,2+ A − p3,3, 0}, c = max{p2,1+ A − p3,3, 0},

1,1∆E 13,12∆F 12,13∆G

0 ∗ ∗

i

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xa

1,1xb 1,2xc 2,1Immp213, xa

1,1xc 2,1∆B 23,13Immp213, xa

1,1xb 1,2∆C 13,23Immp213,

xa

1,1∆B

23,13∆C 13,23Immp213, xa

1,1∆B 23,13∆C 13,23∆D 13,13, xg3,3∆B

23,13∆C 13,23∆D

xa 1,1xc 2,1∆B 23,13Immp213 if p1,2 < p3,3 ≤ p2,1

xa 1,1xb 1,2∆C 13,23Immp213 if p2,1 < p3,3 ≤ p1,2

xa 1,1∆B 23,13∆C 13,23Immp213 if p1,2, p2,1 < p3,3 ≤ p1,2+ p2,1

xa 1,1∆B 23,13∆C 13,23∆D 13,13 if p1,2+ p2,1 < p3,3 ≤ p1,1+ p1,2+ p2,1

xg3,3∆B 23,13∆C 13,23∆D 13,13 if p1,1+ p1,2+ p2,1 < p3,3,where

b = max{p1,2− p3,3, 0}, c = max{p2,1− p3,3, 0}, g = max{p3,3− p1,1− p1,2− p2,1, 0},

D = max{p3,3− p1,2− p2,1− g, 0}, a = p1,1− D, B = p3,3− p1,2+ b − g − D

C = p3,3 − p2,1+ c − g − D p = p3,3− g − D − B − C

Observation 3.12 Applying the map φ to the cluster monomials

xe2,3xf3,2xg3,3Immq132, xf3,2xg3,3∆E13,12Immq132, xe2,3xg3,3∆F12,13Immq132,

xg3,3∆E13,12∆F12,13Immq132, xg3,3∆D13,13∆E13,12∆F12,13, xa1,1∆D13,13∆E13,12∆F12,13, (14)

Combining Observations 3.1 - 3.12, we now have the following

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