Poset pinball, highest forms, and (n-2,2) Springer varieties

0.37 MB·English

by  Barry Dewitt

POSET PINBALL, HIGHEST FORMS, AND (n−2,2) SPRINGER VARIETIES
BARRYDEWITTANDMEGUMIHARADA
1 ABSTRACT. In thismanuscript we study type A nilpotent Hessenberg varieties equipped with a natural S1-
1 action using techniquesintroduced byTymoczko, Harada-Tymoczko, and Bayegan-Harada, with aparticular
0 emphasison a special class of nilpotent Springervarieties corresponding tothe partition λ = (n−2,2) for
2 n ≥ 4. Firstwedefinetheadjacent-pairmatrixcorrespondingtoanyfillingofaYoungdiagramwithnboxes
with thealphabet{1,2,...,n}. Usingtheadjacent-pairmatrixwe makemore explicit andalso extendsome
n statementsconcerninghighestformsoflinearoperatorsinpreviousworkofTymoczko.Second,foranilpotent
a operator N and Hessenberg function h, we construct an explicit bijection between theS1-fixed pointsof the
J
nilpotentHessenbergvarietyHess(N,h)andthesetof(h,λN)-permissiblefillingsoftheYoungdiagramλN.
7 Third, weuse posetpinball, thecombinatorial gameintroducedby HaradaandTymoczko, tostudytheS1-
equivariantcohomologyoftypeASpringervarietiesS(n−2,2)associatedtoYoungdiagramsofshape(n−2,2)
] forn ≥ 4. Specifically,weusethedimensionpairalgorithmforBetti-acceptablepinballdescribedbyBayegan
G
andHaradatospecifyasubsetoftheequivariantSchubertclassesintheT-equivariantcohomologyoftheflag
A varietyFℓags(Cn) ∼= GL(n,C)/B whichmapstoamodulebasisofHS∗1(S(n−2,2))undertheprojectionmap
. HT∗(Fℓags(Cn))→HS∗1(S(n−2,2)).Ourposetpinballmodulebasisisnotposet-upper-triangular;thisisthefirst
h concretesuchexampleintheliterature. Astraightforwardconsequenceofourproofisthatthereexistsasimple
t andexplicitchangeofbasiswhichtransformsourposetpinballbasistoaposet-upper-triangularmodulebasis
ma forHS∗1(S(n−2,2)).Weclosewithopenquestionsforfuturework.
[

2
v
5 CONTENTS
6
2 1. Introduction 1
5 2. NilpotentHessenbergvarietiesandS1-actions 3
. 3. Adjacent-pairmatricesandhighestformsofnilpotentoperators 4
2
1 4. S1-fixedpointsinHessenbergvarietiesandpermissiblefillings 12
0 5. Betti-acceptablepinballandlinearindependence 14
1 6. Small-ncases: n=4andn=5 17
:
v 7. Aposetpinballmodulebasisfor(n−2,2)Springervarieties 18
i 8. Openquestions 24
X
References 24
r
a
1. INTRODUCTION
ThestudyofHessenbergvarietiesisanactivefieldofmodernmathematicalresearch. Indeed,Hessen-
berg varieties arise in many areas of mathematics, including geometric representation theory [8,15,16],
numericalanalysis[6],mathematicalphysics[12,14],combinatorics[7],andalgebraicgeometry[4,5],soit
isofinteresttoexplicitlyanalyzetheirtopology,e.g. thestructureoftheir(equivariant)cohomologyrings.
In this paper we further develop the approach, initiated and developed in [1,2,9,10], which studies the
topologyofHessenbergvarietiesthroughposetpinballandSchubertcalculustechniques.
In this manuscript we focus on the case of nilpotent Hessenberg varieties, and more particularly on
nilpotent Springer varieties. We begin by briefly recalling the setting of our results; for more details we
referthereadertoSection2. LetN : Cn → Cn beanilpotentoperator. Leth : {1,2,...,n} → {1,2,...,n}
Date:January10,2011.
2000MathematicsSubjectClassification. Primary:14M17;Secondary:55N91.
ThesecondauthorispartiallysupportedbyanNSERCDiscoveryGrant,anNSERCUniversityFacultyAward,andanOntario
MinistryofResearchandInnovationEarlyResearcherAward.
1
2 BARRYDEWITTANDMEGUMIHARADA
beafunctionsatisfyingh(i)≥ iforall1 ≤ i≤ nandh(i+1)≥ h(i)forall1 ≤ i < n. IntypeA,nilpotent
HessenbergvarietiescanbedefinedasthefollowingsubvarietyofFℓags(Cn):
Hess(N,h):={V• =(0⊆V1 ⊆V2 ⊆···⊆Vn−1 ⊆Vn =Cn) | NVi ⊆Vh(i) foralli=1,...,n}.
WeequipHess(N,h)withanaturalS1-action(describedpreciselyinSection2)inducedfromthediagonal
torus subgroup T of U(n,C)acting in the usual fashion on GL(n,C)/B ∼= Fℓags(Cn). In the specialcase
whentheHessenbergfunctionh : {1,2,...,n} → {1,2,...,n}istheidentityh(i) = iforall1 ≤ i ≤ n,we
callHess(N,h)anilpotentSpringervarietyanddenoteitbyS .
N
OurfirsttworesultsapplytogeneraltypeAnilpotentHessenbergvarieties. LetN beanilpotentn×n
matrixinJordancanonicalformwithweaklydecreasingblocksizesandletλdenotethe Young diagram1
(equivalentlythepartition)withrowlengthstheJordanblocksizesofN listedinweaklydecreasingorder.
In [17, Theorem 6.1] Tymoczko builds a paving-by-affines of a nilpotent Hessenberg variety Hess(N,h),
wherethenilpotentoperatorN isrequiredtobeinhighestform(see[17,Definition4.1]). Muchtopolog-
icalinformationaboutavarietyisencodedinapaving-by-affines,soitisusefultobuildtools fordealing
withthe technicalcondition thattheoperatorN beinhighest form. Ourfirstcontribution istointroduce
what we call the adjacency-pairmatrix, which is an n×n matrix constructed from a filling of a Young
diagramλ with n boxesbythe alphabet{1,2,...,n}. This thenallows usto makemore explicitand also
generalizesaprocedureforproducinghighestformsoflinearoperatorssketchedin[17,Section4]. Inpar-
ticularourmethodsallowsustostraightforwardlyderivetheexplicitchange-of-basispermutationmatrix
whichputsN intoanychoiceofhighestform(includingthatusedbyTymoczkoin[17],cf.Corollary3.23).
Theadjacent-pairmatricesalsoallowsustoseepreciselythesetofpermutationmatriceswhichconjugate
N tohighestform(Theorem3.21). Theexplicitnatureofourresultsallowsforothercomputationsrelated
to these nilpotent Hessenbergvarieties. Asanexample, we derivein Lemma 3.28anexplicitformula for
the Lie algebra projection induced by the inclusion of the S1 subgroup acting on a special case of nilpo-
tent Springer variety into the diagonal subgroup T of U(n,C) acting on Fℓags(Cn). Thus we expect our
proceduretobeusefulforfutureposetpinballanalysisoftypeAnilpotentHessenbergvarieties.
The affine cells in Tymoczko’s paving-by-affines of Hess(N,h) are in one-to-one correspondence with
permissiblefillingsofYoungdiagrams(definedpreciselyinSection4);thisisausefulcombinatorialenu-
merationoftheaffinecells. ThecorrespondencearisessincetheaffinecellsareintersectionsofHess(N,h)
with certainSchubertcellsBwB ⊆ GL(n,C)/B ∼= Fℓags(Cn). EachsuchSchubertcellcontains a unique
(cosetofa)permutationmatrixwB,andeachpermutationwcanbeassociatedtoapermissiblefillingofλ.
Oursecondcontributionistoextendthisrelationshipbetweenthepermutations(whichinthismanuscript
wethinkofasS1-fixedpointsofHess(N,h))andthepermissiblefillingsasfollows. Forthepurposeofthe
discussionbelowassumethatN isinJordancanonicalformwithweaklydecreasingblocksizes. Wedefine
foreachpermutationσ ∈ S abijectionφ betweenthesetFiℓℓ(λ)offillingsofλwiththesetofpermu-
n λ,σ
tationsS (Definition4.3). Eachφ theninducesabijectionbetweenthepermissiblefillingsPFiℓℓ(λ)of
n λ,σ
λandtheS1-fixedpointsofthetranslatedHessenbergvarietyHess(σNσ−1,h). Inparticularthisyieldsan
explicitformulaforthisbijectionforallthepossiblehighestformsofN inTheorem3.21. Ourresultsalso
provideproofsofstatementsquotedin[2].
Ourthirdcontributionisanexplicitconstruction ofacomputationallyconvenientmodulebasisforthe
S1-equivariant cohomology2 of a special class of type A nilpotent Springer varieties, namely, the 2-block
(alsoknownas2-row)nilpotentSpringervarietiesassociatedtoYoungdiagramsoftheform(n−2,2),e.g.
.
Hereandbelowwewillalwaysassumen≥4,sothesmallestSpringervarietyweconsidercorrespondsto
the2×2block
Morespecifically, we use the posetpinballmethodsintroduced in[9] andthe dimensionpairalgorithm
for determining pinball rolldowns described in [2] to construct our combinatorially naturalmodule basis
forH∗ (S ). Ourargumentsuseourresultsaboveonhighestformsandtheexplicitcorrespondence
S1 (n−2,2)
1WeuseEnglishnotationforYoungdiagrams.
2WeworkwithcohomologywithcoefficientsinCthroughout,andhenceomititfromournotation.
POSETPINBALL,HIGHESTFORMS,AND(n−2,2)SPRINGERVARIETIES 3
betweenpermissible fillings andS1-fixedpoints of the Springervariety. Themodule basisisobtained by
takingimagesunderthenaturalprojectionmapH∗(Fℓags(Cn)) →H∗ (S ),tobedescribedindetail
T S1 (n−2,2)
below,ofasubsetoftheT-equivariantSchubertclassesinH∗(Fℓags(Cn)). AsimilaranalysisbyBayegan
T
and the second author in a special case of regular nilpotent Hessenberg varieties [2] yields a poset-upper-
triangularbasis in the sense of [9]. In contrastto the resultsin [2], in the presentmanuscript we find that
themodulebasisisnotposet-upper-triangular;thisisthefirstsuchexampleintheliterature.Inaddition,a
straightforwardconsequenceofourproofisthatasimplechangeofvariablesyieldsamodulebasiswhichis
notaposetpinballbasisbutisposet-upper-triangular.Theseresultsprovidesfurtherevidenceforthepoint
of view, explained in [9], that geometrically natural GKM-type module bases in equivariant cohomology
maynotalwaysbeposet-upper-triangular,butstillcomputationallyconvenient.
Wenowoutlinethecontentsofthepaper.InSection2weprovidethenecessarydefinitionsandsetsome
notation. In Section 3 we define the adjacent-pair matrix and prove results concerning highest forms of
linearoperators. Asasimpleapplicationwederivethechange-of-variablematrixrequiredtodescribethe
circlesubgroupofT ⊆U(n,C)actingonatranslatedSpringervariety.Section4containsourresultsonthe
bijectionbetweenpermissiblefillingsofaYoungdiagramandtheS1-fixedpointsofHessenbergvarieties.
Section5isamainlyexpositorysectionwhichrecallstheterminologyanddefinitionsofposetpinballand
thedimensionpairalgorithmin[2,9]. InSections6and7,posetpinballforthecaseof(n−2,2)Springer
varieties is studied in detail. The small-n cases n = 4 and n = 5 are explicitly computed and recorded
in Section 6. The main pinball result is in Section 7, where we prove that the dimension pair algorithm
yieldsalinearlyindependentsetofclassesinH∗ (S )andhenceamodulebasis. Weclosewithsome
S1 (n−2,2)
directionsforfutureinvestigationinSection8.
Acknowledgements. We thank Darius Bayegan, Erik Insko, and Aba Mbirika for helpful conversations
andinterestinthisproject. WeareparticularlygratefultoJuliannaTymoczkoforherongoingsupport,as
wellasforfinding (andsuggesting waystofix!) errorsinanearlierdraftofthispaperandmakingmany
excellentsuggestionsforimprovingexposition.
2. NILPOTENT HESSENBERG VARIETIES ANDS1-ACTIONS
WebeginwiththedefinitionofthetypeAnilpotentHessenbergvarieties,ofwhichthenilpotentSpringer
varieties are a special case. We also recall the definition of a circle subgroup of the maximal torus T of
U(n,C)whichactsonanynilpotentHessenbergvariety.Sincesomeofthediscussionbelowappliestoany
nilpotentHessenbergvariety,wepresentthe generaldefinitionhere. WeworkexclusivelywithtypeAin
thismanuscriptandhenceomititfromourterminologybelow.
Given a nilpotent operator N : Cn → Cn, consider its Jordan canonical form with weakly decreasing
sizes of Jordan blocks. Let λ denote the partition of n with entries the sizes of the Jordan blocks of N.
N
ThroughoutthismanuscriptweidentifypartitionsofnwiththecorrespondingYoungdiagram.Forexam-
ple,ifN :C5 →C5istheoperatorwithcorrespondingmatrix
0 1 0 0 0
0 0 1 0 0
 
0 0 0 0 0
0 0 0 0 1
 
0 0 0 0 0
 
 
withrespecttothestandardbasisofC5,thensincethematrixhas2Jordanblocksofsizes3and2respec-
tively,ithasassociatedYoungdiagram
whichinturncorrespondstothepartitionλ =(3,2).
N
AHessenbergfunctionisafunctionh : {1,2,...,n} →{1,2,...,n}satisfyingh(i)≥ iforall1 ≤i ≤ n
andh(i+1) ≥ h(i)forall1 ≤ i < n. We frequentlydenoteaHessenbergfunctionbylisting itsvaluesin
sequence,h=(h(1),h(2),...,h(n)=n).
The(nilpotent)HessenbergvarietyHess(N,h)associatedtoN andaHessenbergfunctionhisasubva-
rietyoftheflagvarietyFℓags(Cn) ∼= GL(n,C)/B. RecallthatFℓags(Cn)istheprojectivevarietyofnested
4 BARRYDEWITTANDMEGUMIHARADA
subspacesinCn,i.e.
Fℓags(Cn)={V• =(Vi):0⊆V1 ⊆V2 ⊆···⊆Vn−1 ⊆Vn =Cnsuchthat dimC(Vi)=i}.
ThenHess(N,h)isdefinedtobethefollowingsubvarietyofFℓags(Cn):
(2.1) Hess(N,h):={V• ∈Fℓags(Cn)|NVi ⊆Vh(i) forall1≤i≤n}.
The(nilpotent)Springervarieties3areHessenbergvarietiesforthespecialcasewheretheHessenbergfunc-
tionistheidentityfunctionh(i)=iforall1≤i≤n:
Definition 2.1. Let N : Cn → Cn be a nilpotent operator. The Springer variety S associated to N is
N
definedas
SN :={V• ∈Fℓags(Cn)|NVi ⊆Viforall1≤i≤n}.
For any g ∈ GL(n,C), it is straightforward to see that the Hessenberg variety Hess(gNg−1,h) for the
conjugate gNg−1 of N is homeomorphic (in fact, isomorphic as algebraic varieties) to Hess(N,h), with
explicithomeomorphismgivenbytranslationbyg,i.e.,
(2.2) Hess(N,h) // Hess(gNg−1,h)
h (cid:31) // gh
whereh∈GL(n,C)denotesaflag[h]∈GL(n,C)/B ∼=Fℓags(Cn).
ThereexistsacircleactiononanynilpotentHessenbergvariety. RecallfirstthatthemaximaltorusT of
U(n,C),identifiedwiththediagonalsubgroupofU(n,C),actsontheflagvarietyFℓags(Cn). Considerthe
followingcirclesubgroupofT:
tn 0 ··· 0
0 tn−1 0
(2.3) S1 :=0 0 ... 0(cid:12)(cid:12)(cid:12)(cid:12) t∈C, ktk=1⊆T ⊆U(n,C).
0 0 t(cid:12)
 (cid:12)
FItℓiasgssh(oCwn)nwinhe[9n,tLheemnmilpao5te.1n]ttohpaetrtahteorSN1 ohfa(s2m.3)atprri(cid:12)(cid:12)(cid:12)exsienrvJoersdtahnecnainlpoontiecnatlfHoermssewnibtehrrgevspaericetttyoHtheesss(tNan,hd)ar⊆d
basis of Cn. Moreover, the S1-fixed points Hess(N,h)S1 are isolated and are a subset of Fℓags(Cn)T, the
T-fixed points of Fℓags(Cn). Using the identification Fℓags(Cn)T ∼= Sn we henceforth think of S1-fixed
pointsofHess(N,h)aspermutationsinS .
n
3. ADJACENT-PAIR MATRICES ANDHIGHEST FORMS OF NILPOTENT OPERATORS
SupposegivenanilpotentmatrixN instandardJordancanonicalformwithweaklydecreasingJordan
0
block sizes. We think of N as a linear operator on Cn written with respect to the standard basis of Cn.
0
As mentioned in Section 2, in addition to the Hessenberg variety Hess(N ,h) we may also consider the
0
translated HessenbergvarietiesHess(gN g−1,h) = g·Hess(N ,h) for variousg ∈ GL(n,C). For the pur-
0 0
posesofposetpinball(discussedinmoredetailinSection5)itturnsouttobenecessarytouseconjugates
σNσ−1 whereσisapermutationmatrixandσNσ−1isinso-calledhighestform[17,Definition4.2];thisis
because Tymoczko’s construction of a paving-by-affinesof a HessenbergvarietyHess(N,h) [17, Theorem
6.1]assumesthatN isinhighestform. Motivatedbythis,inthissectionwedevelopatheorywhichrelates
highestformsofN withfillingsofthecorrespondingYoungdiagramλ = λ . Firstweintroduceabijec-
0 N0
tion φ : Fiℓℓ(λ) → S from the set of fillings Fiℓℓ(λ) of λ and the permutation group S . Secondly we
λ n n
associatetoeachfilling T ofλamatrixN whichwecallthe adjacent-pairmatrixofT. Themainresults
T
ofthissectionareTheorems3.16and3.21. Theorem3.16observesthattheadjacent-pairmatrixN ispre-
T
ciselytheconjugateσN σ−1whereσ =φ (T)isthepermutationcorrespondingtoT underthebijectionφ .
0 λ λ
ThisgivesacomputationallyeasyandexplicitmethodforspecifyingtheconjugatesofN bypermutation
0
matrices. In Theorem 3.21 we then prove that N = σN σ−1 is in highest form precisely when T arises
T 0
fromacertainsimplealgorithmwhichwedescribebelow. Thisyieldsastraightforwardenumerationofall
3IntheliteraturetheyarealsocalledSpringerfibresbecausetheyariseasfibresofthesymplecticresolutionT∗Fℓags(Cn) → N
whereNdenotesthesubspaceofnilpotentmatricesingl(n,C),butwedonotneedorusethisperspectivehere.
POSETPINBALL,HIGHESTFORMS,AND(n−2,2)SPRINGERVARIETIES 5
permutationmatricesσ forwhichσN σ−1 isinhighestform,andinparticularinCorollary3.25wegivea
0
countofthenumberofconjugatesσN σ−1 forσ ∈S whichareinhighestform.
0 n
Thediscussioninthissectionhasseveralmotivationsandconsequences. Firstly,ourresults(e.g. Corol-
lary3.23)bothmakeexplicitandalsogeneralizeaprocedureforproducinghighestformsoflinearoperators
whichissketchedin[17,Section4,textnearFigure4]. Secondly,ourexplicitcorrespondencebetweencer-
tainfillingsofλandhighestformsofN allowsustoeasilydeterminethepermutationσ =φ (T)(seee.g.
0 λ
Example3.26)andthusmakefurtherexplicitcomputationswithσ. Asasamplesuchcomputationandfor
use inSection7, atthe endof thissection wegive aconcretedescriptionincoordinatesof the conjugated
circle σS1σ−1 which acts on the Springer varietySσN0σ−1 for N0 corresponding to λ = (n−2,2), as well
asacomputationoftheassociatedLiealgebraprojectionLie(T) → Lie(σS1σ−1). Thussomeoftheresults
in this section are preliminary to the arguments in the sections below. Third, we believe that the theory
initiatedhereofhighestformsinrelationtoSpringervarietiesisofindependentinterest;wedescribesome
openquestionsmotivatedbythistheoryinSection8.
Werecallsomedefinitions.
Definition3.1. ([17,Definition4.1])LetX beanym×nmatrix. WecalltheentryX apivotofX ifX
ik ik
isnonzeroandifallentriesbelowandtoitsleftvanish,i.e.,X =0ifj <kandX =0ifj >i.
ij jk
Moreover,giveni,definer tobetherowofX iftheentryisapivot,and0otherwise.
i ri,i
Example3.2. Let
0 1 1 0
0 0 5 0
X = .
0 1 0 0
 0 0 0 3 
 
 
Thenr =0,r =3,r =2,andr =4.
1 2 3 4
Definition3.3. ( [17, Definition4.2])Anupper-triangularnilpotent n×n matrix isin highestform if its
pivotsformanondecreasingsequence,namelyr ≤r ≤···≤r .
1 2 n
Example3.4. Thenilpotentmatrix
0 0 0 0 0 0
0 0 0 1 0 0
 
0 0 0 0 1 0
N =
0 0 0 0 0 0
 
0 0 0 0 0 1
 
0 0 0 0 0 0
 
 
isinhighestformsincer =r =r =0,r =2,r =3,r =5.
1 2 3 4 5 6
Recall that a filling of λ by the alphabet [n] := {1,2,...,n} is an injective placing of the integers
{1,2,...,n}intotheboxesofλ. FollowingtableauxnotationwedenotebyT afillingofλby[n]. Wedenote
byFiℓℓ(λ)thesetofallfillingsofλby[n]. ForλaYoungdiagramwithnboxes,wehave|Fiℓℓ(λ)|=n!. In
the theorybelowwe useaparticularbijectivecorrespondencebetweenFiℓℓ(λ)and S . We introducethe
n
followingterminology.
Definition3.5. LetλbeaYoungdiagram. LetT beafillingofλwithalphabet[n]forsomen ∈ N. Bythe
EnglishreadingofT wemeanthereadingoftheentriesofT fromlefttorightalongrows,startingatthe
toprowandproceedinginsequencetothebottomrow. ThewordofT obtainedviatheEnglishreadingof
T iscalledtheEnglishwordofT.IfλisaYoungdiagramwithnboxesthenwedefine
(3.1) φ :Fiℓℓ(λ)→S
λ n
whereφ (T)isthepermutationwhoseone-linenotationisgivenbytheEnglishwordofT.Finally,ifλhas
λ
nboxesthentheEnglishfillingofλisthefillingT suchthatφ (T)istheidentitypermutationinS .
λ n
ForλaYoungdiagramwithnboxes,itisimmediatefromthedefinitionthatφ isabijectionfromFiℓℓ(λ)
λ
toS .
n
6 BARRYDEWITTANDMEGUMIHARADA
Example3.6. For
1 2 3 4 3 5 6 7
T = 5 6 and T′ = 2 4
7 1
we have that φ (T) and φ (T′) are respectively the permutations (in one-line notation) 1234567 and 3567241.
λ λ
MoreoverT istheEnglishfillingofλ=(4,2,1).
NextweintroduceadifferentreadingoffillingswhichappearsinthetheoryofhighestformsandHes-
senbergvarietiesdeveloped by Tymoczko in [17] (butthe terminology we use is new). In particular, this
readingplaysasignificantroleinourposetpinballmethodsinSections5-7(cf. inparticularTheorem5.3).
Definition3.7. LetλbeaYoungdiagram. LetT beafillingofaYoungdiagramwithalphabet[n]forsome
n∈N. BytherotatedEnglishreadingofT wemeanthereadingoftheentriesofT fromthebottomtothe
topalongcolumns,startingattheleftmostcolumnandproceedingtotherightmostcolumn. ThewordofT
obtainedviatherotatedEnglishreadingistherotatedEnglishwordofT. LetλbeaYoungdiagramwith
nboxes. TherotatedEnglishfillingofλisthefillingT ofλwith[n]suchthatitsrotatedEnglishreadingis
theidentitypermutationinS .
n
3 5 6 7
Example3.8. Supposethatλ= . ThentherotatedEnglishfillingofλisthefilling 2 4 .
1
Remark3.9. NotethattherotatedEnglishfillingisnotthesamethingastheconjugateoftheEnglishfillingofthe
conjugateYoungdiagram. ForinstancefortheλinExample3.8theconjugateoftheEnglishfillingoftheconjugate
1 4 6 7 3 5 6 7
Youngdiagramλ˜is 2 5 whereastherotatedEnglishfillingofλis 2 4
3 1 .
Remark 3.10. In the next section we develop a more general framework in which both Definition 3.5 and Defini-
tion3.7arespecialcases,butwedonotneedthisperspectivehere.
Given a Young diagramwith n boxes and any filling T of λ by [n], we now construct a matrix we call
theadjacent-pairmatrix.OurconstructionisageneralizationofaproceduresketchedbyTymoczkoin[17,
Section4](seeinparticular[17,Figure4]). WebeginbydefiningadjacencyinλandinafillingT.
Definition3.11. LetλbeaYoungdiagram. Wesaythattwoboxesofλareadjacentifthetwoboxesarein
thesamerow, andone boxisdirectlytotheleftof theother. Thatis, the twoboxesareoftheform
withintheYoungdiagramλ. Similarly,givenafillingT ofλ,wesaythattwoentriesofMareadjacent,or
thattheyformanadjacentpair,iftheyoccurinadjacentboxes.
Example3.12. For
1 2 3
4 5
T = 6
thepairs{1,2},{2,3},and{4,5}aretheadjacentpairsofentriesofT.
Definition3.13. LetλbeaYoungdiagramwithnboxesandT afillingofλwithentriesfrom[n]. Thenwe
definetheadjacent-pairmatrixcorrespondingtoT,denotedNT,tobethematrixNT = (aij)1≤i,j≤n such
thatits(i,j)-thentryisgivenby
1 ifiandj areadjacentinT andiisleftofj,
a :=
ij
(0 otherwise.
POSETPINBALL,HIGHESTFORMS,AND(n−2,2)SPRINGERVARIETIES 7
3 2 4
1 5
Example3.14. Supposethatλ= andthatT= 6 .Then
0 0 0 0 1 0
0 0 0 1 0 0
 
0 1 0 0 0 0
N = .
T  0 0 0 0 0 0 
 
 0 0 0 0 0 0 
 
 0 0 0 0 0 0 
 
 
Remark3.15. Theadjacent-pairmatrixN correspondingtotheEnglishfillingofλisthenilpotentmatrixinJordan
T
1 2 3
4 5
canonicalformcorrespondingtoλ. ForexampleifT= 6 then
0 1 0 0 0 0
0 0 1 0 0 0
 
0 0 0 0 0 0
N = .
T 0 0 0 0 1 0
 
0 0 0 0 0 0
 
0 0 0 0 0 0
 
 
The following is a basic computation which relates adjacent-pair matrices to highest forms. Given a
permutation σ ∈ S by slight abuse of notation we denote also by σ its n×n permutation matrix with
n
respecttothestandardbasisofCn,i.e.,thematrixwithi-thcolumnequaltothestandardbasisvectore .
σ(i)
Theorem3.16. LetN beann×nnilpotentmatrixinJordancanonicalformwithweaklydecreasingsizesofJordan
blocks. Let λ be the corresponding Young diagram and let L : Cn → Cn be the linear operator with matrix
N N
N with respect to the standard basis of Cn. Let T be a filling on λ with alphabet [n] and σ := φ (T) ∈ S the
N λ n
permutationgivenby theEnglish word ofT. Thentheadjacent-pairmatrixN corresponding to T is equal to the
T
conjugateσNσ−1,i.e.,NT isthematrixofLN withrespecttothebasis{eσ−1(1),...,eσ−1(n)}.
Proof. By the definition of the adjacent-pairmatrix, given a Young diagram λ with n boxes and ℓ rows,
N
N contains a 1 in n−ℓ entries, and all other entries are 0. Similarly, an n×n nilpotent matrix N, with
T
correspondingYoungdiagramλ ,containsa1inn−ℓentriesand0’selsewhere,andsodoesanyconjugate
N
σNσ−1forσapermutation(matrix).
Nowleta denotethe(i,j)-thentryofM. Theprecedingdiscussionimpliesthatinordertoprovethe
ij
proposition it suffices to check that if a = 1 for some i and j, then the matrix of L with respect to the
ij
basis{eσ−1(1),...,eσ−1(n)}alsocontainsa1atthe(i,j)-thentry. Supposethataij =1. Byconstructionthis
means that i and j are adjacent in the filling T with i to the left of j. Hence by definition of the English
readingtheone-linenotationofσ isoftheformσ = ···ij···. Supposethattheioccursattheℓ-thspotof
theone-linenotation,soσ(ℓ)= i,σ(ℓ+1)= j. Thenσ−1(i)= ℓ,σ−1(j) =ℓ+1. Sinceiandj areadjacent,
wealsoknowthatL(eℓ+1)=eℓ (cf. Remark3.15)orequivalentlyL(eσ−1(j))=eσ−1(i). Thisimpliesthatthe
matrixofLwrittenwithrespecttothebasis{eσ−1(1),...,eσ−1(n)}hasa1inthe(i,j)-thentry,asdesired.
(cid:3)
WenowwishtodeterminethesetoffillingsT suchthattheadjacent-pairmatrixN isinhighestform.
T
Throughoutthisdiscussionweusethefollowingassumptionsandnotation. LetλbeaYoungdiagramwith
nboxes,ℓrows,krowsofdistinctlength,andrcolumns. Ifλ >λ >···>λ arethedistinctrowlengths
1 2 k
ofλweletd for1≤i≤kdenotethenumberofrowsofλwithlengthλ . Thustherowlengthsofλare
i i
(λ ,λ ,...,λ ,λ ,λ ,...,λ ,··· ,λ ,λ ,...,λ )
1 1 1 2 2 2 k k k
d1 d2 dk
with k d =ℓ. Wealsolet(|µ ≥µ{z ≥··}·|≥µ ){dzenot}etheco|lumn{zlength}sofλ. Noteµ =ℓ.
i=1 i 1 2 r 1
Webeginwithsomeobservationsaboutthepivotsofanadjacent-pairmatrixN .
T
P
8 BARRYDEWITTANDMEGUMIHARADA
Lemma3.17. LetλbeaYoungdiagramwithnboxesandT afillingofλby[n]. LetN betheadjacent-pairmatrix
T
ofT.TheneachmatrixentryinN whichisequalto1isapivotofN .
T T
Proof. Bydefinitionoftheadjacent-pairmatrix,itsnon-zeroentriesareinone-to-onecorrespondencewith
the distinctadjacentpairs i j whichappearin T. Eachbox of λwhichis notinthe leftmost(i.e. first)
column of λ is the right hand box of precisely one such adjacent pair of boxes. Hence each such
accountsforpreciselyoneentryofN equalto1.
T
Bydefinitionoffillings,eachentryinT appearsonlyonce. Inparticularthismeansthatforanyi ∈ [n],
theindexiappearsatmostonceaseithertherighthandbox j i orthelefthandbox i j inanadjacent
pairinT.Thusbydefinitionoftheadjacent-pairmatrixthereexistsatmostoneentryequalto1ineachrow
andeachcolumnofN . Sinceallotherentriesareequalto0,thisinturnimpliesthateach1thatappears
T
inN isinfactapivot. (cid:3)
T
Lemma3.18. LetλbeaYoungdiagramwithnboxesandT afillingofλby[n]. LetN betheadjacent-pairmatrix
T
ofT.ThenN isinhighestformifandonlyifT satisfiesthefollowingconditions:
T
(a) theleftmostcolumnofλisfilledwiththeintegers{1,2,...,µ =ℓ},and
1
(b) if i j and i j bothappearasadjacentpairsinT then
1 1 2 2
i <i ifandonlyifj <j .
1 2 1 2
Proof. FirstsupposeN isinhighestform. ByLemma3.17if i j appearsasanadjacentpairinT then
T
r = i > 0. Forj ∈ [n],theindexj doesnotappearintherighthandboxofanyadjacentpairinT (sothe
j
j-thcolumnofN isidentically0)preciselywhenj appearsintheleftmost(i.e. first)columnofλ. Inthis
T
case,bydefinitionofpivots, r = 0. SinceN isinhighestformwemusthaver ≤ r ≤ ··· ≤ r andin
j T 1 2 n
particularanyr with r = 0 mustoccur beforeanyr withr > 0. We conclude thatj is inthe leftmost
j j j j
column of λ precisely when 1 ≤ j ≤ µ = ℓ. This proves (a). Now suppose i j and i j both
1 1 1 2 2
appearasadjacentpairsinT. ThenagainfromLemma3.17weknowr = i ,r = i . IfN isinhighest
j1 1 j2 2 T
form then the pivots must be increasing so j < j if and only if i < i . This proves (b). If T satisfies
1 2 1 2
conditions(a)and(b)thenreversingthisreasoningshowsthatN mustbeinhighestform. (cid:3)
T
WenowdescribeanalgorithmwhichproducesafillingT ofλwhichsatisfiescertainconditions,starting
from the data of a filling of the leftmost column of λ. As we show in Theorem 3.21below, the algorithm
gives an explicit method for producing precisely those fillings T for which the corresponding N are in
T
highestform. Wefollownotationestablishedabove.
(1) Fix an arbitrary filling of the leftmost (i.e. first) column of λ with
thealphabet[µ ]. Thisfillingspecifiesalinearorderingoftherows
1
ofλ.
(3.2)
(2) For the s-th column of λ for 2 ≤ s ≤ r, place the µ integers
s
{( s−1µ )+1,..., s µ }intheµ boxesofthes-thcolumnin
t=1 t t=1 t s
thelinearorderspecifiedbystep(1).
P P
Notethat,bydefinitionofthisalgorithm,thefillingoftheleftmostcolumncompletelyspecifiestherestof
thefilling.
Example3.19. IftheYoungdiagramλandtheinitialfillingofitsleftmostcolumnare
5
1
4
3
and 2
thenthealgorithm(3.2)determinestherestofthefillingtobe
5 9 12 15 16
1 6 10 13
4 8 11 14
3 7
2 .
POSETPINBALL,HIGHESTFORMS,AND(n−2,2)SPRINGERVARIETIES 9
Remark3.20. Supposethefillingoftheleftmostcolumnofλisgivenbyplacingtheintegeri,for1≤i≤µ ,inthe
1
i-thboxfromthebottom.Thenthefillingofλobtainedbyapplyingthealgorithm(3.2)ispreciselytherotatedEnglish
fillingofDefinition3.7.
WenowprovethatthefillingsT forwhichN isinhighestformarepreciselythoseproducedfromthe
T
algorithm(3.2).
Theorem 3.21. Let λbe a Young diagramwith n boxesand T a filling ofλby [n]. Thentheadjacent-pairmatrix
N isinhighestformifandonlyifthealgorithm(3.2)appliedtothefillingoftheleftmostcolumnofT producesthe
T
fillingT.
Toprovethepropositionweusethefollowinglemma. Wefollownotationestablishedabove.
Lemma3.22. LetλbeaYoungdiagramwithnboxesandT afillingofλby[n]. SupposeT satisfiestheconditions
(a)and(b)ofLemma3.18. Thenthes-thcolumnofλfor1 ≤ s ≤ r containspreciselytheintegers{( s−1µ )+
t=1 t
s
1,..., µ }.
t=1 t P
Proof. PWearguebyinduction. Condition(a)alreadyimpliestheleftmostcolumnisfilledwith[µ1],which
provesthebasecases=1.Nowsupposethefirstscolumnscontainpreciselytheintegers{1,2,..., s µ }.
t=1 t
Supposeforacontradictionthatsomeelementuin{( s µ )+1,..., s+1µ }appearsinthev-thcolumn
t=1 t t=1 t P
forsome v > s+1. Sincetherearepreciselyµ boxesinthe (s+1)thcolumn, this inturnimpliesthat
s+1
theremust existsome u′ > s+1µ thatappearsinPthe (s+1)th coluPmn. Thus thereexistadjacentpairs
t=1 t
i u′ and i u withthepropertiesthat
1 2
P
s
• i ≤ µ and
1 t=1 t
s
• i > µ
2 Pt=1 t
sinceu′ appearsinthe(s+1)thcolumnandallentriesinthesthcolumnarelessthanorequalto s µ
P t=1 t
byassumption,andsinceuappearsinacolumnstrictlytotherightofthe(s+1)thcolumn. Thusi < i
1 2
butu′ >u,whichcontradictscondition(b).Theresultfollows. P (cid:3)
ProofofTheorem3.21. By Lemma 3.18 it suffices to prove that a filling T satisfies conditions (a) and (b)
of Lemma 3.18 if and only if it arises from (3.2). So suppose T satisfies Lemma 3.18(a) and (b). From
Lemma3.22wealreadyknowthatthesetofentriesineachcolumnagreeswiththatspecifiedby(3.2),so
it remains to show that the ordering of the entries also agrees, i.e. that the entries of the s-th column for
2 ≤ s ≤ r respectsthe linear orderimposed on the rows by the filling of the leftmost column. We argue
by induction. Suppose s = 2. Then the entries of the 2nd column respectthe orderingin the 1st column
preciselywhen the following holds: if i j and i j are two adjacentpairs with j ,j in the 2nd
1 1 2 2 1 2
column of λ then i < i if and only if j < j . But this follows from condition (b). Moreover if this
1 2 1 2
condition holds it follows that the linear ordering of the boxes in the 2nd column given by its filling by
{µ +1,...,µ +µ }agreeswiththatinducedbythelinearorderingoftherowsofλcorrespondingtothe
1 1 2
fillingofthe1stcolumn. Assumingthefirstscolumnsareobtainedby(3.2),thesameargumentasabove
showsthatthe(s+1)stcolumnmustalsobefilledaccordingto(3.2),asdesired.
Conversely, suppose T is obtained from (3.2). By construction T satisfies condition (a). Now suppose
i j and i j aretwoadjacentpairsappearinginT. Weconsidercases. Supposei andi appear
1 1 2 2 1 2
in the sth and s′th columns of T. Without loss of generality we may assume s < s′. Then i ≤ s µ
1 t=1 t
while i ≥ ( s′−1µ )+1 ≥ i . Thus we wish toshow j < j . This follows because the adjacencywith
2 t=1 t 1 1 2 P
i and i respectively implies that j is in the (s+1)th column and j is in the (s′ +1)th column. Since
1 2 1 2
s+1 < s′ +P1 an argumentsimilar to that above implies j < j asdesired. On the other hand suppose
1 2
i and i appearin the same column, saythe sth. Then j and j appearin the s+1thcolumn. Suppose
1 2 1 2
furtherthati appearsinther throwandi appearsinther throw. Ifi < i thenbydefinitionofthe
1 i1 2 i2 1 2
algorithm(3.2)theentryinther throwofthefirstcolumnislessthanthatinther throw,whichinturn
i1 i2
impliesj <j . Similarlyj <j impliesi <i . Thisconcludestheproof.
1 2 1 2 1 2
(cid:3)
Thefollowing,assertedin[17,Section4,seee.g. Figure4],isnowastraightforwardconsequence.
Corollary3.23. LetλbeaYoungdiagramwithnboxesandT betherotatedEnglishfillingofλ.Letσ :=φ (T )
RE λ RE
bethepermutationgivenbytheEnglishreadingofT . ThenN =σNσ−1 isinhighestform.
RE TRE
10 BARRYDEWITTANDMEGUMIHARADA
Proof. ImmediatefromTheorems3.16and3.21andRemark3.20. (cid:3)
WehavejustseenthateachfillingT obtainedfrom(3.2)yieldsaconjugateN =σNσ−1ofN inhighest
T
form.Sinceafillinggivenin(3.2)isspecifiedbythefillingofitsleftmostcolumn,thereareµ !=|S |many
1 µ1
suchfillings. However,differentsuchfillingsT andT′mayyieldthesameadjacent-pairmatrixNT =NT′.
The next lemma makes this precise, for the purpose of which we use the following terminology. We say
a filling T′ is obtained from T by a row swap if the entries of 2 equal-length rows of T have been inter-
changed; moreprecisely,ifboththe athrowandthebthrowofλhavedboxesandentries a1 ··· ad
and b1 ··· bd respectively,thenT′isobtainedfromT byswappingtheathrowandbthrowifT′con-
tainsthesameentriesasinT inallotherrows,andtheathrowofT′ hasentries b1 ··· bd andthebth
rowhasentries a1 ··· ad .
Lemma 3.24. Let λ be a Young diagram with n boxes and let T and T′ be fillings of λ obtained from (3.2). Then
NT =NT′ ifandonlyifT′isobtainedfromT byasequenceofrowswaps.
Proof. If T and T′ differ only by a sequence of row swaps, then T and T′ have precisely the same sets of
adjacentpairs. Thusfromthedefinitionoftheadjacent-pairmatrixitfollowsthatNT =NT′. Nowsuppose
T and T′ differbymore than a sequence of rowswaps. Since both T and T′ areobtained from(3.2), this
means that there exists an element s ∈ [µ ] which appears in T in a row of length d and appears in T′
1
in a row of length d′, with d 6= d′. Without loss of generality we assume d′ > d. We wish to show that
NT 6= NT′. For this it suffices to show that there exists some adjacent pair i j which occurs in T but
notinT′,orviceversa. ConsidertheentriesintherowofT andT′ whichcontains. Byassumptionthese
areoftheform a1 =s a2 ··· ad and a′1 =s a′2 ··· a′d ··· a′d′ respectivelywhered′ >d. We
take cases. Suppose there exists an index 1 < i ≤ d for which a 6= a′. Then in particular there exists a
i i
minimalsuch,denoteiti. Thenthereisanadjacentpair ai−1 ai inT andapair a′i−1 a′i inT′where
ai−1 = a′i−1 butai 6= a′i,soNT 6= NT′. Nowsupposeai = a′i forall1 ≤ i ≤ d. Inparticularad = a′d. Then
a′d =ad a′d+1 isanadjacentpairinT′whichdoesnotoccurinT.HenceNT 6=NT′ alsointhiscase.The
resultfollows. (cid:3)
Thefollowingisnowstraightforward. Recallµ = ℓisthetotalnumberofrowsofλandd ,...,d are
1 1 k
thenumbersofrowsofλoflengthλ ,...,λ respectively.
1 k
Corollary3.25. Thereexistprecisely
ℓ!
d !d !···d !
1 2 k
highestformsofN obtainedasσNσ−1 forapermutationmatrixσ ∈S .
n
Proof. There are µ ! = ℓ! fillings T arising from the algorithm (3.2). From Lemma 3.24we know that the
1
matricesN donot change preciselywhen the entries in the first column contained in equal-length rows
T
arepermuted. Thed countthenumbersofequal-lengthrowssotheresultfollows. (cid:3)
i
Our constructions allow us to do explicit computations. For instance, given the discussion above it is
straightforwardtolistthepermutationmatricesσ forwhichtheassociatedconjugate σNσ−1 isinhighest
form. For instance, let T be the rotated English filling of λ. It follows from the results above that the
RE
permutation σ for which σNσ−1 is the choice of highest form of N used in [17, Section 4] is precisely
σ :=φ (T).
λ
Example3.26. SupposetheYoungdiagramis
correspondingtothenilpotentmatrixN inExample3.4. ThentherotatedEnglishfillingis
3 5 6
2 4
1
andthepermutationσsuchthatσNσ−1 isinthehighestformusedin[17,Section4]isσ =356241.