Carlsson’s rank conjecture and a conjecture on square-zero upper triangular matrices

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Contents lists available atScienceDirect

Journal of Pure and Applied Algebra

www.elsevier.com/locate/jpaa

Carlsson’s rank conjecture and a conjecture on square-zero upper triangular matrices

Berrin Şentürk^∗, Özgün Ünlü¹

DepartmentofMathematics,BilkentUniversity,Ankara,06800,Turkey

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received26May2017

Receivedinrevisedform20July 2018

Availableonlinexxxx CommunicatedbyS.Iyengar

MSC:

55M35;13D22;13D02

Keywords:

Rankconjecture Square-zeromatrices Projectivevariety Borelorbit

Letk bean algebraicallyclosedfieldandA thepolynomialalgebrainr variables withcoefficientsink.Incasethecharacteristicof k is2,Carlsson[9] conjectured thatforanyDG-A-moduleM ofdimensionN asafreeA-module,ifthehomology ofM isnontrivialandfinitedimensionalasak-vectorspace,then2^r≤ N.Herewe stateastrongerconjectureaboutvarietiesofsquare-zerouppertriangularN× N matriceswithentriesinA.UsingstratificationsofthesevarietiesviaBorelorbits, weshow that thestronger conjecture holds when N < 8 orr < 3 without any restrictiononthecharacteristicofk.Asaconsequence,weobtainanewprooffor manyoftheknowncasesofCarlsson’sconjectureandgivenewresultswhenN > 4 andr = 2.

1. Introduction

Awell-knownconjectureinalgebraictopologystatesthatif(Z/pZ)^ractsfreelyandcellularlyonafinite CW-complex homotopyequivalent to Sⁿ¹ × . . . × Sⁿ^m, thenr is lessthan orequal to m.This conjecture is known to be true in several cases: In the equidimensional case n1 = . . . = nm =: n, Carlsson [7], Browder [6], and Benson–Carlson[5] proved theconjectureunderthe assumptionthattheinduced action on homologyis trivial.Without the homologyassumption,the equidimensionalconjecture wasproved by Conner[11] for m= 2,Adem–Browder [1] for p= 2 or n= 1,3,7,andYalçın[25] for p= 2,n= 1. Inthe non-equidimensionalcase,theconjectureisprovedbySmith[23] form= 1, Heller[13] form= 2,Carlsson [10] forp= 2 andm= 3,Refai[20] forp= 2 andm= 4,andOkutan–Yalçın[19] forproductsinwhichthe averageofthe dimensionsissufficientlylargecompared tothe differencesbetweenthem.Thegeneralcase m≥ 5 is stillopen.

* Correspondingauthor.

E-mailaddresses:berrin@fen.bilkent.edu.tr(B. Şentürk),unluo@fen.bilkent.edu.tr(Ö. Ünlü).

1 ThesecondauthorispartiallysupportedbyTÜBA-GEBİP/2013-22.

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LetG= (Z/pZ)^randk beanalgebraicallyclosedfieldofcharacteristicp.AssumethatG actsfreelyand cellularlyonafiniteCW-complexX homotopyequivalenttoaproductofm spheres.Onecanconsiderthe cellularchaincomplexC_∗(X;k) asafinitechaincomplexoffreekG-moduleswithhomologyH_∗(X;k) that has dimension2^mas ak-vector space. Hence,astrongerandpurely algebraic conjecturecanbe stated as follows: IfC_∗ isafinite chaincomplexoffreekG-moduleswithnonzerohomologythendimkH_∗(C_∗)≥ 2^r. However,Iyengar–Walker in[15] disprovedthisalgebraic conjecturewhenp= 2,butthealgebraic version forp= 2 remainsopen.

LetR beagradedring.Apair(M,∂) isadiﬀerential graded R-moduleifM isagradedR-module and

∂ is anR-linearendomorphismofM thathasdegree−1 andsatisﬁes∂²= 0.Moreover,aDG-R-moduleis free if theunderlyingR-module isfree.

LetA= k[y₁,. . . ,y_r] bethepolynomialalgebrainr variables,wherek isaﬁeldandeachy_ihasdegree−1.

Using afunctorfrom thecategoryof chaincomplexes ofkG-modulestothecategory ofdiﬀerentialgraded A-modules, Carlsson showed in [8], [9] that the above algebraic conjecture is equivalent to the following conjecture whenthecharacteristicofk is2:

Conjecture1.Letk bean algebraicallyclosedfield, A thepolynomial algebrainr variables withcoefficients in k,andN apositiveinteger.If (M,∂) isafreeDG-A-module ofrankN whose homologyisnonzeroand finite dimensional asak-vectorspace, thenN ≥ 2^r.

When the characteristic of k is 2, Conjecture 1was proved by Carlsson [10] for r ≤ 3 and Refai [20]

for N ≤ 8. Avramov,Buchweitz, andIyengarin[4] dealtwith regularrings andinparticular theyproved Conjecture 1 for r ≤ 3 without any restriction on the characteristic of k. See also Proposition 1.1 and Corollary 1.2 in[2],Theorem 5.3 in[24] forresultsincharacteristicnotequalto2.

In thispaper weconsider the conjecturefrom theviewpoint ofalgebraic geometry. Weshow thatCon- jecture1isimpliedbythefollowinginSection2:

Conjecture 2. Let k be an algebraically closed ﬁeld, r a positive integer, and N = 2n an even positive integer.Assumethatthereexistsanonconstantmorphismψ fromtheprojectivevarietyP^rk⁻¹ totheweighted quasi-projectivevarietyofrankn square-zerouppertriangularN×N matrices(xij) withdeg(xij)= di−dj+1 forsome N -tupleofnonincreasing integers(d1,d2,. . . ,dN). ThenN ≥ 2^r.

Wewill giveamoreprecise statementofConjecture2inSection3after discussingnecessarydeﬁnitions and notation.Weproposethefollowing,whichisstrongerthanConjecture2:

Conjecture 3.Let k,r, N , n and ψ be asin Conjecture 2.Assume 1≤ R,C ≤ N andthe value of xij at every pointin theimageof ψ is 0 wheneveri≥ N − R+ 1 orj≤ C. ThenN ≥ 2^r−1(R+C).

Note thatinConjecture 3we have2^r−1(R+C)≥ 2^r because1≤ R and 1≤ C. The main result of the paper isaproof ofConjecture3whenN < 8 orr < 3,see Theorem2and Theorem6.As Conjecture3is thestrongestconjecturementionedabove,weobtainproofsofalltheconjecturesinthisintroductionunder the sameconditions, includingthe main result ofCarlsson in[9]. Alsonote thatfor r = 2, takingN > 4 gives novel results not coveredin the literature. However, when the characteristic of the ﬁeldk is not2, Iyengar–Walker [15] gaveacounterexampletoConjecture1foreachr≥ 8.HencebySection2,wecansay thatConjectures 2and 3are alsofalse whenr≥ 8 andthe characteristicof theﬁeldk isnot2.Allthese conjectures arestill open incasethe characteristicof k is2.InSection4, weconcludewith examplesand problems.

Wethanktherefereeforgivingus extensivefeedback, ashorterproofofTheorem 1andencouraging us toextendourresultstoﬁeldswithcharacteristicsotherthan2.WearealsogratefultoMatthewGelvinfor commentsandsuggestions.

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2. Some notesonConjectures1,2,and3

To show thatConjecture 3is thestrongest conjecture inSection1, itis enoughto prove thefollowing theorem.

Theorem1.Conjecture 2impliesConjecture1.

Proof. Letk,r, andA beasinConjecture1.Let(M,∂) beafreeDG-A-moduleofrankN whichsatisﬁes thehypothesisinConjecture1.Withoutlossofgenerality,wecanassumethatN isthesmallestrankofall suchDG-A-modules.

Suppose the image of the diﬀerential ∂ is not contained in (y1,. . . ,yr)M . Then, there exists a basis b₁,. . . ,b_N of M and there are some i and j so that ∂(b_i) = cb_j +

l=jg_lb_l for some non-zero c ∈ k and some gl ∈ A. Replacing bj with ∂(bi) gives a new basis b₁,...,b_N such that ∂(b_i) = b_j. Now form the acyclic sub-DG-A-module (E,∂) of (M,∂) spanned by b_i, b_j. The map (M,∂) → (M,∂)/(E,∂) is a surjective quasi-isomorphismand M/E isfreeof rankN− 2.This isacontradiction.Hence,theimageof thediﬀerential∂ iscontainedin(y₁,. . . ,y_r)M .

Now pick any basis c1,. . . ,cN of M such that deg(c1) ≤ · · · ≤ deg(cN). Let m be such that deg(cN−m+1) = · · · = deg(cN) and deg(ci) < deg(cN−m+1) for all i < N− m + 1. For each i, we have

∂(ci) =

jgijcj, for some homogeneous polynomials gij ∈ A. Since the image of ∂ is contained in (y1,. . . ,yr)M , no gij is anon-zero constant.Thus, whenever gij is non-zero, we havedeg(gij) ≤ −1 and hence deg(c_j) = deg(c_i)− 1 − deg(gij)≥ deg(ci). It follows thatthe diﬀerential on M restricts to one on thesubmodule

L = AcN−m+1⊕ · · · ⊕ AcN. Moreprecisely,foralli∈ {N − m+ 1,. . . ,N} wehave∂(ci) =N

j=N−m+1gijcj whereeachnonzerogij isa linearpolynomial.Hence,relativetothebasisc_N−m+1,. . . ,c_N,thedifferential∂ on L isgivenbyamatrix intheform y1X1+· · · + yrXr whereeachXi isanm× m matrixwith entriesink.Since ∂² = 0 wehave X_i²= 0 andX_iX_j+ X_jX_i = 0 foralli,j.Incasethecharacteristicof thefieldk is2,byaclassicalresult aboutcommutingsetofmatrices,forexampleseeTheorem7 onpage207 in[14],thereexistsaninvertible m×m matrixT withcoefficientsink suchthatT⁻¹X_iT isuppertriangularforalli∈ {1,. . . ,r}.Incasethe characteristicofthefieldk isnot2,foreverypolynomialQ innoncommutativer variables,thesquareofthe matrixQ(X₁,. . . X_r)(X_iX_j−XjX_i) iszero.Therefore,byatheoremofMcCoyasstatedin[22] (seealso[12], [17]), againthere existsamatrixT as abovewhichsimultaneouslyconjugates allXi’sto uppertriangular matrices.Inotherwords,thereisak-linearchangeofbasisinwhicheachX_iisuppertriangular.Itfollows that,relativetothisnewbasisc_N−m+1,. . . ,c_NofL onehas∂(c_i)∈ ⊕^Nj=i+1Ac_jforalli∈ {N −m+1,. . . N}.

Note thatM/L is afree DG-A-module whose diﬀerential has an image in (y1,. . . ,yr)(M/L) and so, by inductivearguments onrank,wemayassumethatM/L admits abasiswhich makesitsdiﬀerential upper triangular. Theunion ofany lift ofthis basis to M withthe basisc_N_−m+1,. . . ,c_N givesabasis B forM where ∂ is representedbyanupper triangularmatrix Ψ. Moreover,Propositions8 and 9 in [10] workfor any characteristic. HenceN is divisibleby 2 andfor any γ ink^r− {0} the evaluation of Ψ at γ gives a matrixofrankN/2.

LetS = k[x1,. . . ,xr] bethepolynomialalgebrawithdeg(xi)= 1.For1≤ i≤ r,replaceyi withxiinΨ toobtainΨ.NotethatΨ canbeconsideredasa nonconstantmorphismfromtheprojectivevarietyP^r−1_k to theweightedquasi-projectivevarietyofrankN/2 square-zero uppertriangularN× N matrices(xij) with deg(x_ij)= d_i− dj+ 1 whered_i=−(degree of theith element in B). 2

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3. Varietiesofsquare-zeromatrices

Weassumethatk isanalgebraically closedﬁeld,n apositiveinteger,N = 2n,andd= (d₁,d₂,. . . ,d_N) anN -tupleofnonincreasing integers.

3.1. Statementsofconjectures

Wegive herethenotation fortheaffineandprojective varietiesusedto provetheconjecturesdiscussed above.FirstwefixanaffinevarietyUN,aringR(UN),andasubvarietyVN as follows:

• U_N istheaﬃnevarietyofstrictly uppertriangularN× N matricesover k.

• R(U_N)= k[ x_ij | 1≤ i< j≤ N ] isthecoordinateringofU_N.

• VN isthesubvarietyofsquarezeromatricesinUN.

Deﬁne an actionof theunitgroupk^∗ on UN byλ· (xij)= (λ^dⁱ^−d^j⁺¹xij) for λ∈ k^∗.Using thisaction we set twomorenotation.

• U (d) istheweightedprojectivespacegivenbythequotientofU_N− {0} bytheactionofk^∗.

• R(U (d)) isthehomogeneouscoordinateringofU (d).Inother words,R(U (d)) isR(U_N) consideredas agraded ringwithdeg(xij)= di− dj+ 1.

Note thatthepolynomialp_ij =

j−1

m=i+1

x_imx_mj inR(U (d)) ishomogeneous ofdegreed_i− dj+ 2 whenever 1≤ i< j ≤ N.Similarly, then× n-minorsof (xij) are homogeneouspolynomials inR(U (d)). Hence,we deﬁne twosubvarietiesofU (d) asfollows:

• V (d) istheprojective varietyofsquare zeromatricesinU (d).

• L(d) isthesubvarietyofmatrices ofranklessthann inV (d).

Wecanusethisterminologytorestate Conjecture2:

Conjecture4.Letk beanalgebraicallyclosed ﬁeld, r apositiveinteger,andd an N -tupleof nonincreasing integers. If there existsa nonconstantmorphismψ from theprojective varietyP^rk⁻¹ tothequasi-projective variety V (d)− L(d), thenN ≥ 2^r.

Let U beanopen subsetof V (d).We sayψ :P^rk⁻¹ → U isanonconstantmorphismifψ isrepresentedby (ψij) sothatthefollowing conditionsaresatisﬁed:

(I) there exists a positive integer m so that each ψ_ij is a homogeneous polynomial in the variables x₁,x₂,. . . ,x_r inS ofdegreem(d_i− dj+ 1) for1≤ i< j≤ N,

(II) foreveryγ∈ P^rk⁻¹ thereexistsi,j suchthatψ_ij(γ)= 0.

Inparticular, ifψ :P^rk⁻¹→ U isanonconstantmorphism,ψ canbeconsideredas afunctionfrom P^rk⁻¹ to U representedbyanonconstantpolynomialmapψ from A^rk totheconeoverU suchthatψ(A^rk− {0}) does not contain thezero matrixin VN.Each indeterminatexij canbe viewedas homogeneouspolynomialin R(U (d)).Hencefor1≤ R,C ≤ N wedeﬁne animportantsubvarietyofV (d):

• V (d)_RC isthesubvarietyofV (d) givenbytheequationsx_ij= 0 for i≥ N − R+ 1 orj≤ C.

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Nowwerestate theConjecture 3as follows:

Conjecture5.Letk bean algebraicallyclosedﬁeld, r a positiveinteger,andd anN -tupleof nonincreasing integers. If thereexists anonconstant morphismψ fromthe projectivevariety P^r−1k to thequasi-projective varietyV (d)_RC− L(d), thenN ≥ 2^r⁻¹(R+C).

Hence,thesevarietiesarethemaininterestinthispaper.

3.2. Action ofaBorelsubgroup onVN

HereweintroduceanactionofaBorelsubgroupinthegroupofinvertibleN×N matricesonthevarieties discussedintheprevioussubsection.FirstwesetanotationfortheBorelsubgroup.

• BN isthegroupofinvertibleuppertriangularN× N matriceswithcoeﬃcientsink.

ThegroupBN actsonVN byconjugation.

• VN/BN denotesthesetoforbitsoftheactionofBN onVN.

• BX denotes theBN-orbitthatcontainsX ∈ VN.

A partial permutation matrix is amatrix having at mostone nonzero entry, which is 1, ineach row and column. A result of Rothbach (Theorem 1 in [21]) implies that each BN-orbit of VN contains a unique partialpermutationmatrix.Henceweintroducethefollowingnotation:

• PM(N ) denotes the set of nonzero N× N strictly upper triangular square-zero partial permutation matrices.

Thereisaone-to-onecorrespondencebetweenPM(N ) andV_N/B_NsendingP toB_P.Wecanidentifythese partialpermutationmatriceswith asubsetofthesymmetricgroupSym(N ):

• P(N ) isthesetofinvolutionsinSym(N );i.e., thesetofnon-identitypermutationswhosesquareisthe identity ().

ForP ∈ PM(N) andσ∈ P(N),

• σP denotesthepermutationinP(N ) thatsendsi to j ifPij = 1;

• Pσ denotesthe partialpermutation matrixinPM(N ) that satisﬁes(Pσ)ij = 1 if andonly ifσ(i)= j and i< j.

ClearlytheassignmentsP → σP andσ→ Pσaremutualinversesandsodeﬁneaone-to-onecorrespondence betweenP(N ) andPM(N ).

3.3. A partialorder onthesetof orbits

There are important partial orders on VN/BN, P(N ), PM(N ), all of which are equivalent under the one-to-one correspondence mentioned above (cf. [21]). We begin with V_N/B_N. For Borel orbits B,B ∈ V_N/B_N,

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• B≤ B meanstheclosureofB, consideredas asubspaceofV_N,containsB.

Second, we deﬁne apartial order on PM(N ). To do this, we consider ranksof certain minors of partial permutationmatrices. Ingeneral,foranN× N matrixX,

• rij(X) denotestherankofthelowerleft ((N− i+ 1)× j) submatrix ofX,where1≤ i< j≤ N.

Forpartial permutationmatricesP,P ∈ PM(N),

• P≤ P meansrij(P)≤ rij(P ) foralli,j.

Third, wedeﬁne apartial order onP(N ).For positiveintegersi< j, letσ(i,j) denotetheproduct ofthe permutationsσ and (i,j) andσ^(i,j) theconjugateofσ by(i,j).Forσ,σ ∈ P(N),

• σ≤ σ ifσ canbeobtainedfrom σ byasequenceofmovesofthefollowingform:

– TypeIreplacesσ with σ(i,j) ifσ(i)= j andi= j.

– TypeIIreplacesσ withσ^(i,i⁾ ifσ(i)= i< i< σ(i).

– TypeIIIreplacesσ withσ^(j,j⁾ ifσ(j)< σ(j)< j < j.

– TypeIVreplacesσ withσ^(j,j⁾ifσ(j)< j< j = σ(j).

– TypeVreplacesσ withσ^(i,j) ifi< σ(i)< σ(j)< j.

The ideaof describingorderviathese movescomesfrom [16]. Althoughweuse diﬀerentnamesformoves, the set of possible moves are same. We represent apermutation (i₁,j₁)(i₂,j₂). . . (i_s,j_s) in P(N ) by the matrix

i1 i2 . . . is

j1 j2 . . . js

.

Forexample,wedrawtheHassediagramofP(4) inwhicheachedgeislabelled bythetypeofthemove itrepresents(seeFig.1).

WhenN ≥ 6,theHasse diagramforP(N ) is toolargeto drawhere.Weareactuallyonlyinterestedin asmallpartofthisdiagram,whichwediscussinSection3.6.

One canconsiderFig.1as astratiﬁcation of V4. Inthenext section,we usethe stratiﬁcationof VN to stratifyV (d).

3.4. Stratiﬁcationof V (d)

Ford= (d₁,d₂,. . . ,d_N) anN -tupleofnonincreasingintegers,λ∈ k^∗,and X = (x_ij)∈ VN, wehave λ· X = λ · (xij) = (λ^dⁱ^−d^j⁺¹x_ij) = D_λI_λXD⁻¹_λ

where D_λ denotes the diagonal matrix with entries λ^d¹,λ^d²,. . . ,λ^d^N and I_λ is the scalarmatrix with all diagonal entries λ. Let P_X ∈ PM(N) be the unique partial permutation matrixin theBorel orbitof X.

Considerb∈ BN suchthat

P_X= b⁻¹Xb.

Let I_λ,X be thediagonal matrixwhose jth entry is λ if (P_X)_ij = 1 for somei and 1 otherwise. Then we have

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Fig. 1. Hasse diagram of P(4).

IλPX = I_λ,X⁻¹ PXIλ,X. Hence,wehave

λ· X = Dλb I_λ,X⁻¹ b⁻¹X b Iλ,Xb⁻¹D_λ⁻¹= Z⁻¹XZ

where Z = bI_λ,Xb⁻¹D_λ⁻¹ is in B_N. Thus, for any X ∈ V (d) there exists a well-deﬁned Borel orbit in VN/BN thatcontainsarepresentative ofX inVN.Hencewecanset thefollowingnotation.ForX ∈ V (d),

• BX denotes theBorelorbitinVN/BN thatcontainsarepresentativeofX inVN.

Letψ :P^rk⁻¹ → V (d)− L(d) beanonconstantmorphism.Thereisaliftofthismorphismtoamorphism from A^rk− {0} to the cone over V (d)− L(d) that can be extended to a morphism ψ : A^rk → VN. Since A^rk is anirreducible aﬃne variety,there exists auniquemaximal Borel orbitamong theBorel orbitsthat intersects the image of ψ nontrivially. Note that this maximal Borel orbit is independent of the lift and extension we selectedbecause it is also maximal in the set {BX|X ∈ V (d)}. Hencewe may associate a permutationtothenonconstantmorphismψ:

• σ_ψ isthepermutationthatcorrespondstotheuniquemaximalBorelorbitB_X whereX isintheimage of ψ.

Note that every point in the image of a morphism ψ as above must have rank n. Hence σψ must be a productofn distinct transpositions.InSection3.6,wewillrestrictourattentiontosuchpermutations.

3.5. Operations onpolynomial mapsfromA^rk toV_N

Another way to see that BX is well-deﬁned for X ∈ V (d) is to consider the fact that a minor of a representativeofX iszeroifandonlyifthecorrespondingminor ofanotherrepresentativeiszero.Weuse thisfactseveraltimesto proveourmainresult.Henceweintroducethefollowingnotation.ForX ∈ VN,

• mⁱ_j¹₁ⁱ_j²₂^{... i}_{... j}^k_k(X) denotesthedeterminantofthek×k submatrixobtainedbytakingthei₁th,i₂th,. . . , i_kth rowsandj₁th, j₂th,. . . , j_kth columnsofX.

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Firstnotethatmⁱ_j¹₁ⁱ_j²₂^{... i}_{... j}^k_k canbeconsideredasamorphismfromV_N tok,andhencecanbecomposedwith the morphism ψ mentioned in theprevious subsection. Here we discuss several other morphisms thatwe cancompose withsuchmorphisms.Foru∈ k,

• Ri,j(u) is thefunctionthattakesasquare matrixM and multipliestheith rowofM byu andaddsit tothejth rowofM whilemultiplyingthejth columnofM byu andaddingittotheith columnofM . Note thatRi,j(u)(M ) is aconjugate of M . In fact,they are in the sameBorel orbit when M ∈ VN and i > j. Hence, for i > j, we canconsider Ri,j(u) as an operation that takes a morphism from A^sk to VN

and transformsit toamorphism from A^s+1k to V_N byconsideringu as anewindeterminateand applying Ri,j(u) tothemorphism. Forv∈ k^∗,

• D_i(v) denotesthefunctionthattakesasquarematrixM andmultipliestheith rowofM byv andthe ith columnofM by1/v.

Letq beapolynomialins indeterminates.WedeﬁneD_i(q) asanoperationthattakes arationalmapfrom thequasi-aﬃnevarietyA^sk− Z toVN andtransformsitinto arationalmapfrom A^sk− Z ∪ V (q) toVN by applying Di(q),usingthefollowing notation:

• V (q₁,q₂,. . . ,q_k) isthevarietydetermined bytheequationsq₁= q₂= . . . q_k= 0.

Weusetheabovenotationalsoforvarietiesinprojectivespacesdeterminedbythehomogeneouspolynomials q₁,q₂,. . . ,q_k.

3.6. Rankof orbitsandproof ofﬁrst mainresult

Each σ∈ P(N) is aproductof disjoint transpositions.Henceforσ∈ P(N), we deﬁnethe rank ofσ to be thenumberof transpositionsinσ. Note thatunder theone-to-one correspondence between P(N ) and PM(N ),therankofapermutationisequaltotherankofthecorrespondingpartialpermutationmatrix.

• RP(N ) denotesthepermutationsinP(N ) of rankn.

NotethatallmovesotherthantypeIpreservetherankofσ.Indeed,theonlywayofobtainingσ ofsmaller rankbyapplying ourmovesisbydeletingatransposition,whichisthe eﬀectof moveoftypeI.Alsonote thatitisimpossibleto haveamoveoftypeII oramoveof typeIVbetweentwopermutationsinRP(N ).

Forexample,wedrawtheHasse diagramforRP(6) whereeachdottedlinedenotes amoveoftypeIIIand solidlinedenotesamoveoftypeV(seeFig.2).SuchHassediagrams,withparticularattentionpaidtothe maximal elements,will leadto theproof ofourﬁrstmain result.

Theorem 2.Conjecture5holdsforN < 8.

Proof. TakeN < 8,d= (d1,d2,. . . ,dN) anN -tupleofnonincreasingintegers,andψ :P^rk⁻¹→ V (d)− L(d) anonconstantmorphism.Thenσ = σψisinRP(N ).ByconsideringFigs.1and2,wenotethatthereexists a uniquemaximal σ ∈ RP(N) such thatσ can be obtainedfrom σ by asequence of movesof type III.

Since moves of type III do not change the number of leading zero rows and ending zero columns of the corresponding partial permutationmatrices, the Borelorbit corresponding to σ iscontained inV (d)_RC if and only ifthe Borelorbitcorresponding toσ is containedin V (d)_RC forall R,C. Henceit isenough to

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Fig. 2. Hasse diagram of RP(6).

consider thecaseswhere σ islessthanor equaltoamaximalelement inRP(N ) forN = 2,4,6.Wecover thesecasesbyprovinginthefollowingeightstatements:

(i) Ifσ = (1,2) thenr < 2.

Assumetothecontrarythatσ = (1,2) andr≥ 2.Ifwealsowriteψ foritsrestrictiontoP¹k ⊆ P^r−1_k ,weget amapoftheform

ψ(x : y) =

0 p₁₂

0 0

,

where p12 is ahomogeneouspolynomialink[x,y]. Since k isalgebraicallyclosed, there existsγ ∈ P¹k such thatp12(γ)= 0.Thismeansψ(γ) isinL(d),whichisacontradiction.

(ii) Ifσ≤ (1,2)(3,4) thenr < 3.

Supposetothecontrarythatσ≤ (1,2)(3,4) andr≥ 3.Whenwerestrictψ toP²k,wegetamapoftheform

ψ(x : y : z) =

⎡

⎢⎢

⎢⎣

0 p₁₂ p₁₃ p₁₄

0 0 0 p₂₄

0 0 0 p34

0 0 0 0

⎤

⎥⎥

⎥⎦.

Notethatthereexists γ inP²k suchthat

p₁₂(γ) = 0 and p₁₃(γ) = 0.

Againthismeansψ(γ)∈ L(d).Hencethis caseisprovedbycontradictionas well.

(iii) Ifσ≤ (1,4)(2,3) thenr < 2.

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Supposewehave

ψ(x : y) =

⎡

⎢⎢

⎢⎣

0 0 p13 p14

0 0 p₂₃ p₂₄

0 0 0 0

⎤

⎥⎥

⎥⎦.

Letmⁱ_j¹₁ⁱ_j²₂^{... i}_{... j}^k_k beasinSection3.5andusethesamenotationtodenoteitscompositionwithψ.Thenthere exists γ inP¹k suchthat

m¹²₃₄(γ) = (p₁₃p₂₄− p23p₁₄)(γ) = 0.

This againgivesacontradiction.

(iv) Ifσ≤ (1,2)(3,4)(5,6) then r < 3.

Supposeotherwise.Wehave

ψ(x : y : z) =

⎡

⎢⎢

⎢⎣

0 p12 p13 p14 p15 p16

0 0 0 p₂₄ p₂₅ p₂₆ 0 0 0 p₃₄ p₃₅ p₃₆

0 0 0 0 0 p₄₆

0 0 0 0 0 p56

0 0 0 0 0 0

⎤

⎥⎥

⎥⎦ .

If p₁₂ andp₁₃ arenotrelativelyprimehomogeneouspolynomialsthenthere existsγ∈ P²k suchthat

p12(γ) = 0, p13(γ) = 0, and m¹²³₄₅₆(γ) = 0.

Moreover, ifp46(γ)= 0 andp56(γ)= 0,thentherankofψ(γ) isat most2,whichleadstoacontradiction.

Hencewehavep46(γ)= 0 orp56(γ)= 0.Let

c₄(γ) :=

⎡

⎢⎣ p14(γ) p24(γ) p34(γ)

⎤

⎥⎦ and c⁵(γ) :=

⎡

⎢⎣ p15(γ) p25(γ) p35(γ)

⎤

⎥⎦ .

Since ψ²= 0,c4(γ)p46(γ)+ c5(γ)p56(γ)= 0.Bythefactthatp46(γ)= 0 orp56(γ)= 0,c4(γ) and c5(γ) are linearlydependent.Thustherankof ψ(γ) isatmost2,whichisacontradiction.

Therefore we mayassumep₁₂ and p₁₃ are relatively prime.Since ψ² = 0,we havep₁₂p₂₄+ p₁₃p₃₄= 0 and p12p25+ p13p35= 0. This implies thatp12 dividesp34 andp35,and similarly p13 dividesp24 andp25. Then thereexists γ in P²k suchthat

p₁₂(γ) = 0 and p₁₃(γ) = 0.

This meansp₁₂,p₁₃,p₂₄,p₂₅,p₃₄,andp₃₅ allvanishat γ.Henceψ(γ)∈ L(d),whichisacontradiction.

(v) Ifσ≤ (1,2)(3,6)(4,5) thenr < 3,and (vi) Ifσ≤ (1,4)(2,3)(5,6) thenr < 3.

(11)

Thesecasesaresymmetric,so itisenoughtoprove (v).Consider

ψ(x : y : z) =

⎡

⎢⎢

⎢⎣

0 p12 p13 p14 p15 p16

0 0 0 0 p25 p26

0 0 0 0 p₃₅ p₃₆ 0 0 0 0 p₄₅ p₄₆

0 0 0 0 0 0

⎤

⎥⎥

⎥⎦ .

Wemodifyψ bytheoperationsinSection3.5.FirstapplyR_6,5(u) toψ foranewvariableu.Ifp₄₆+up₄₅= 0, applyD5(1/(p46+ up45)) andthen R5,6(−p45) toobtainamatrixoftheform

⎡

⎢⎢

⎢⎣

0 p12 p13 p14 ∗ ∗ 0 0 0 0 m²⁴₅₆ p26+ up25

0 0 0 0 m³⁴₅₆ p₃₆+ up₃₅ 0 0 0 0 0 p₄₆+ up₄₅

0 0 0 0 0 0

⎤

⎥⎥

⎥⎦ .

Hencebyselectingacorrectvalueforu wewouldbe doneifV (m²⁴₅₆,m³⁴₅₆) V (p45).Wemayassume V (m²⁴₅₆, m³⁴₅₆)⊆ V (p45).

Similarly,wemayalsoassume

V (m²³₅₆, m³⁴₅₆)⊆ V (p35) and V (m²³₅₆, m²⁴₅₆)⊆ V (p25).

Therefore,

V (m²³₅₆, m³⁴₅₆, m²⁴₅₆)⊆ V (p25, p35, p45) =∅.

Thus,{m²³56,m³⁴₅₆,m²⁴₅₆} isaregularsequenceink[x,y,z].Ifp45andp46arenotrelativelyprime,thereexists γ suchthatm²³₅₆(γ)= 0,andp45(γ)= p46(γ)= 0.Hence,wemayassumep45and p46 arerelatively prime, whichleadsacontradictionas wehave

p45m²³₅₆+ p25m³⁴₅₆− p35m²⁴₅₆= 0.

(vii) Ifσ≤ (1,6)(2,3)(4,5) thenr < 2.

Toprovethiscase,consider

ψ(x : y) =

⎡

⎢⎢

⎢⎣

0 0 p13 p14 p15 p16

0 0 p23 p24 p25 p26

0 0 0 0 p₃₅ p₃₆ 0 0 0 0 p₄₅ p₄₆

0 0 0 0 0 0

⎤

⎥⎥

⎥⎦ .

AgainbyapplyingR_i,j(u) and D_i(v) for someu,v wemayassumethat

(12)

V (m¹²⁴₃₅₆)⊆ V (p13, p₂₃) and V (m¹²³₄₅₆)⊆ V (p14, p₂₄).

Hence{m¹²⁴356,m¹²³₄₅₆} mustbearegularsequenceink[x,y].Howeverthisisimpossiblebecausethedeterminant of

⎡

⎢⎣

gcd(p₁₃, p₁₄) p₁₅ p₁₆ gcd(p23, p24) p25 p26

0 gcd(p35, p45) gcd(p36, p46)

⎤

⎥⎦

dividesbothm¹²⁴₃₅₆andm¹²³₄₅₆.

(viii) If σ≤ (1,6)(2,5)(3,4) then r < 2.

It isenoughtoconsiderarootofm¹²³₄₅₆ toprovethiscase. 2

Note thatintheaboveproof thelast two casesproveConjecture5when N ≤ 6 andr≤ 2. Inthe rest of thepaper we will generalize these ideas to prove the conjecture for r≤ 2. To do this we examine the dimensionsofthese varieties.

3.7. Orbitdimensions andproofof second mainresult

Wenowintroducenotationfordimensionsofthesevarieties.Inthissubsection,forσ∈ P(N),iftherank of σ iss,thenweobtaintwosequencesofnumbersi₁,. . . ,i_s andj₁,. . . ,j_s satisfyingthefollowing:

σ = (i1, j1)(i2, j2) . . . (is, js)

with i1< i2<· · · < is andia< ja forall1≤ a≤ s.In[18],Melnikovgivesaformulaforthedimensionof aBorelorbitB_σ forσ inP(N ) asfollows:

• ft(σ):= #{jp| p< t,jp< jt} + #{jp| p< t,jp< it} for2≤ t≤ s,

• dim(B_σ)= N s+

s t=1

(i_t− jt)−

s t=2

f_t(σ).

Wedeﬁne anewsubsetofRP(N ):

• DP(N ) isthesetofallσ inRP(N ) such thatdim(Bσ)= dim(Bσ)− 1 wheneverσ isapermutation obtainedbyapplying asinglemoveoftypeItoσ.

For instance, the following is the Hasse diagram of DP(8) (Fig. 3). Note that in the Hasse diagram of DP(8) all movesareoftypeV. Thisisgenerallythe case,whichwewillprovebelow.Beforewedoso,we will proveaneasierresultthatwill introducethenotationandargument stylethatwillbe necessary.

Fixσ∈ DP(N).Weusetheourconventionforσ atthebeginningofthissubsectionwhichimpliesi₁= 1.

For q∈ {1,. . . ,n},letσ be theresultof applyingthemoveoftypeI thatdeletestheqth transposition of σ,sothat

σ= (1, j1) . . . (iq−1, jq−1)(iq+1, jq+1) . . . (in, jn) =

⎛

⎜⎝

1 i2 . . . iq−1 iq iq+1 . . . in

j₁ j₂ . . . j_q−1 j_q j_q+1 . . . j_n

⎞

⎟⎠ .

(13)

Fig. 3. Hasse diagram of DP(8).

ThenbyMelnikov’sformulawehave

dim(Bσ) = N n +

n t=1

(it− jt)−

n t=2

ft(σ) , and

dim(Bσ) = N (n− 1) +

n−1

t=1

(it− jt)−

n−1

t=2

ft(σ).

Tosimplifyourcalculation,wewritef_t(σ)= f_t¹(σ)+ f_t²(σ) for2≤ t≤ n,where f_t¹(σ) = #{jp| p < t, jp< j_t} and ft²(σ) = #{jp| p < t, jp< i_t}, andweusethenotation:

f_t,q^l (σ) =

⎧⎪

⎨

⎪⎩

f_t^l(σ) if t≤ q − 1

0 if t = q

f_t−1^l (σ) if t≥ q + 1 forl = 1,2.

Lemma3.If N = 2 and thetransposition(1,N ) appears in σ,thenσ /∈ DP(N).

Proof. If (1,N ) appears in σ and σ ∈ DP(N), let q = 2 and σ be the result of deleting the second transpositionfromσ.Since thei’s areincreasingandi_a< j_aforall1≤ a≤ n,wehavei₂= 2,so

(14)

σ =

⎛

⎜⎝

1 2 i₃ . . . i_n N j2j3 . . . jn

⎞

⎟⎠ .

Thus 3≤ j2≤ N − 1.Letj2= N− b for some1≤ b≤ N − 3.Then

σ =

⎛

⎜⎝

1 2 . . . i_n N N− b . . . jn

⎞

⎟⎠ ,

and anynumberbetweenN− b andN has toappearasaj or ani thatisbigger thanj2. Therefore,

_n

t=2

f_t¹(σ)− ft,2¹ (σ)

+

_n

t=2

f_t²(σ)− ft,2² (σ)

= b− 1.

Hence,dim(Bσ)− dim(Bσ)= N + 2− N + b− (b− 1)= 3,so σ /∈ DP(N). 2 Now weprovethemain propositionof thissubsection.

Proposition 4.If σ∈ DP(N), thenj_p< j_t forallp< t, andtherefore wecannot apply amoveof type III to σ.Conversely,ifσ∈ RP(N) and we cannotapply amoveof typeof IIItoσ,then σ∈ DP(N).

Proof. Assumethatσ∈ DP(N).Wewillprovethefollowing statementbyinductiononk:

jn−k< . . . < jn−1< jn and∀ (p < n − k), jp< jn−k. (∗) Suppose k = 0.To prove(∗), we needto show that∀p< n,jp < jn.Let σ be obtainedby deletingnth transpositionofσ.

σ =

⎛

⎜⎝

1 i₂ . . . i_n−1 i_n j1 j2. . . jn−1 jn

⎞

⎟⎠ .

Since σ∈ DP(N),wehave

1 = dim(B_σ)− dim(Bσ) = N + (i_n− jn)−

f_n¹(σ) + f_n²(σ)

. (1)

Since thetotalnumberof possiblej except j_n isn− 1, andany numberbetweeni_n and j_n hastoappear as j, wehavef_n²(σ) = n− 1 − (jn− in− 1). Bytheequation (1),f_n¹(σ)= n− 1,so that∀p< n,jp < jn

is true.Thereforejn= N .

Now assumethestatement(∗) istruefork.Thenwe canvisualiseσ asfollows:

σ =

⎛

⎜⎝

1 < i₂ < . . . < i_n−k−1 < i_n−k < i_n−k+1 < . . . < i_n j₁ j₂ . . . j_n−k−1 j_n−k < j_n−k+1 < . . . < j_n

⎞

⎟⎠

Weneedto prove(∗) fork + 1, thatis,

j_n−k−1 < . . . < j_n−1< j_n and∀ (p < n − k − 1), jp < j_n−k−1.

(15)

By the second part of the inductive hypothesis for k, we have j_n−k−1< j_n−k so the ﬁrst part of (∗) is already true,and we only need to show thatthe second partholds. Inother words, it is enoughto show thatf_n¹_−k−1(σ)= n− k − 2.Letσ beobtainedbydeleting(n− k − 1)thtranspositionofσ.Thenwehave

n t=n−k

f_t¹(σ)− ft,n¹ −k−1(σ) = k + 1.

Letw := #{ip| in−k−1 < ip< jn−k−1}.Then

f_n−k−1² (σ) = n− (k + 2) − (jn−k−1− in−k−1− 1 − w), and

n t=n−k

f_t²(σ)− ft,n² −k−1(σ) = k + 1− w.

Bythefactthatdim(Bσ)− dim(Bσ)= 1,wehavef_n−k−1¹ (σ)= n− k − 2.Thustheﬁrstclaimisproved.

Conversely,given σ∈ RP(N), supposethatσ istheresultof applyingthemoveof typeI thatdeletes theq-thtranspositionfromσ. Notethatf_q¹(σ)= q− 1 andn

t=q+1f_t¹(σ)− ft,q¹ (σ)= n− q. Hence,

n t=2

f_t¹(σ)− ft,q¹ (σ) = n− 1.

Then,

dim(B_σ)− dim(Bσ) = N + (i_q− jq)− (n − 1) −

n t=2

f_t²(σ)− ft,q² (σ)

.

Wealsohavethediﬀerence f_t²(σ)− ft²(σ)= 0 whent∈ {1,. . . ,q− 1}.Therefore,

n t=2

f_t²(σ)− ft,q² (σ) =

n t=q

#{jp| p < t, jp< it}

−

n t=q+1

#{jp| p < t, p = q, jp < it}

= #{jp| jp< iq} + #{it| jq< it}.

Let F = #{jp | jp < iq} and G = #{it | jq < it}. Note that numbers between iq and jq must appear as j_l for l < q or as i_s where s > q. Let a = #{jp | iq < j_p < j_q} and b = #{it | iq < i_t < j_q}.Then a+ b= j_q− iq− 1. LetA = #{jp | iq < j_p} and B = #{it | it< j_q}.We haveA− a+ B− b− 1= n.

Therefore,A+ B = n+ jq− iq.

σ =

⎛

⎜⎜

⎝

B

i1, . . . , iq. . . b

G . . . , in

j₁, . . .

F

. . . ja _q, . . . , j_n

A

⎞

⎟⎟

⎠