STANDARD MONOMIAL THEORY FOR DESINGULARIZED RICHARDSON VARIETIES IN THE FLAG VARIETY GL n B

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Niveau: Supérieur, Doctorat, Bac+8
STANDARD MONOMIAL THEORY FOR DESINGULARIZED RICHARDSON VARIETIES IN THE FLAG VARIETY GL(n)/B MICHAEL BALAN Abstract. We consider a desingularization ? of a Richardson variety Xvw = Xw ? Xv in the flag variety F(n) = GL(n)/B, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. We then construct a basis of the homogeneous coordinate ring of ? inside Z, indexed by combinatorial objects which we call w0-standard tableaux. Introduction Standard Monomial Theory (SMT) originated in the work of Hodge [19], who considered it in the case of the Grassmannian Gd,n of d-subspaces of a (complex) vector space of dimension n. The homogeneous coordinate ring C[Gd,n] is the quotient of the polynomial ring in the Plucker coordinates pi1...id by the Plucker re- lations, and Hodge provided a combinatorial rule to select, among all monomials in the pi1...id , a subset that forms a basis of C[Gd,n]: these (so-called standard) mono- mials are parametrized by semi-standard Young tableaux. Moreover, he showed that this basis is compatible with any Schubert variety X ? Gd,n, in the sense that those basis elements that remain non-zero when restricted to X can be char- acterized combinatorially, and still form a basis of C[X].

• called standard


• line bundle


• projection prd


• v2 ?


• schubert variety


• standard monomial


• has been developed

Sujets

• Richardson
• Line bundle
• Schubert variety

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 32
• Langue: English

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STANDARDMONOMIALTHEORYFORDESINGULARIZED
RICHARDSONVARIETIESINTHEFLAGVARIETY
GL
(
n
)
/B

MICHAE¨LBALAN
Abstract.
WeconsideradesingularizationΓofaRichardsonvariety
X
wv
=
X
w
∩
X
v
intheflagvariety
F`
(
n
)=
GL
(
n
)
/B
,obtainedasafibreofa
projectionfromacertainBott-Samelsonvariety
Z
.Wethenconstructabasis
ofthehomogeneouscoordinateringofΓinside
Z
,indexedbycombinatorial
objectswhichwecall
w
0
-standardtableaux
.

Introduction
StandardMonomialTheory(SMT)originatedintheworkofHodge[19],who
considereditinthecaseoftheGrassmannian
G
d,n
of
d
-subspacesofa(complex)
vectorspaceofdimension
n
.Thehomogeneouscoordinatering
C
[
G
d,n
]isthe
quotientofthepolynomialringinthePlu¨ckercoordinates
p
i
1
...i
d
bythePlu¨ckerre-
lations,andHodgeprovidedacombinatorialruletoselect,amongallmonomialsin
the
p
i
1
...i
d
,asubsetthatformsabasisof
C
[
G
d,n
]:these(so-calledstandard)mono-
mialsareparametrizedbysemi-standardYoungtableaux.Moreover,heshowed
thatthisbasisiscompatiblewithanySchubertvariety
X
⊂
G
d,n
,inthesense
thatthosebasiselementsthatremainnon-zerowhenrestrictedto
X
canbechar-
acterizedcombinatorially,andstillformabasisof
C
[
X
].TheaimofSMTisthen
togeneralizeHodge’sresulttoanyflagvariety
G/P
(
G
aconnectedsemi-simple
group,
P
aparabolicsubgroup):inamoremodernformulation,theproblemcon-
sists,givenalinebundle
L
on
G/P
,inproducinga“nice”basisofthespaceof
sections
H
0
(
X,L
)(
X
⊂
G/P
aSchubertvariety),parametrizedbysomecombina-
torialobjects.SMTwasdevelopedbyLakshmibaiandSeshadri(see[29,30])for
groupsofclassicaltype,andLittelmannextendedittogroupsofarbitrarytype
(includingintheKac-Moodysetting),usingtechniquessuchasthepathmodel
inrepresentationtheory[32,33]andLusztig’sFrobeniusmapforquantumgroups
atrootsofunity[34].StandardMonomialTheoryhasnumerousapplicationsin
thegeometryofSchubertvarieties:normality,vanishingtheorems,idealtheory,
singularities,andsoon[26].
Richardsonvarieties,namedafter[36],areintersectionsofaSchubertvarietyand
anoppositeSchubertvarietyinsideaflagvariety
G/P
.Theypreviouslyappeared
in[20,Ch.XIV,
§
4]and[38],aswellasthecorrespondingopensubvarietiesin
[10].Theyhavesinceplayedaroleindifferentcontexts,suchasequivariantK-
theory[25],positivityinGrothendieckgroups[5],standardmonomialtheory[7],
Poissongeometry[13],positroidvarieties[21],andtheirgeneralizations[22,2].In
particular,SMTon
G/P
isknowntobecompatiblewithRichardsonvarieties[25]
(atleastforaveryamplelinebundleon
G/P
).
Date
:December2,2011.
2010
MathematicsSubjectClassification.
Primary14M15,Secondary05E1014M1720G05.
1

2MICHAE¨LBALAN
LikeSchubertvarieties,Richardsonvarietiesmaybesingular[24,23,40,1].
DesingularizationsofSchubertvarietiesarewellknown:theyaretheBott-Samelson
varieties[4,9,14],whicharealsousedforexampletoestablishsomeproperties
ofSchubertpolynomials[35],ortogivecriteriaforthesmoothnessofSchubert
varieties[12,8].AnSMThasbeendevelopedforBott-Samelsonvarietiesoftype
A
n
in[28],andofarbitrarytypein[27]usingthepathmodel[32,33].
Inthepresentpaper,weshalldescribeaStandardMonomialTheoryforadesin-
gularizationofaRichardsonvariety.Tobemoreprecise,weintroducesomeno-
tations.Let
G
=
GL
(
n,k
)where
k
isanalgebraicallyclosedfieldofarbitrary
characteristic,
B
theBorelsubgroupofuppertriangularmatrices,and
T
⊂
B
themaximaltorusofdiagonalmatrices.Thequotient
G/B
identifieswiththe
variety
F`
(
n
)ofallcompleteflagsin
k
n
.Let(
e
1
,...,e
n
)bethecanonicalba-
sisof
k
n
.Toeachpermutation
w
∈
S
n
,wecanassociatea
T
-fixedpoint
e
w
in
F`
(
n
):its
i
thconstituentisthespacegeneratedby
e
w
(1)
,...,e
w
(
i
)
.Wede-
noteby
F
can
the
T
-fixedpointcorrespondingtotheidentity
e
of
S
n
,and
F
opcan
the
T
-fixedpoint
e
w
0
,where
w
0
isthelongestelementof
S
n
.Thesymmetric
group
S
n
isgeneratedbythesimpletranspositions
s
i
=(
i,i
+1),
i
=1
,...,n
.
Wedenoteapermutation
u
∈
S
n
withtheone-linenotation[
u
(1)
u
(2)
...u
(
n
)].
Denoteby
B
−
thesubgroupof
G
oflowertriangularmatricesandconsiderthe
Schubertcells
C
w
=
B.e
w
andtheoppositeSchubertcells
C
v
=
B
−
.e
v
.The
Richardsonvariety
X
wv
⊂
F`
(
n
)istheintersectionofthedirectSchubertvariety
X
w
=
C
w
withtheoppositeSchubertvariety
X
v
=
C
v
=
w
0
X
w
0
v
.Fixareduced
decomposition
w
=
s
i
1
...s
i
d
andconsidertheBott-Samelsondesingularization
Z
=
Z
i
1
...i
d
(
F
can
)
→
X
w
,andsimilarly
Z
0
=
Z
i
r
i
r
−
1
...i
d
+1
(
F
opcan
)
→
X
v
forare-
duceddecomposition
w
0
v
=
s
i
r
s
i
r
−
1
...s
i
d
+1
.Thenthefibredproduct
Z
×
F`
(
n
)
Z
0
hasbeenconsideredasadesingularizationof
X
wv
in[6],butforourpurposes,itwill
bemoreconvenienttorealizeitasthefibreΓ
i
(
i
=
i
1
...i
d
i
d
+1
...i
r
)oftheprojec-
tion
Z
i
=
Z
i
(
F
can
)
→
F`
(
n
)over
F
opcan
(seeSection1forthepreciseconnection
betweenthosetwoconstructions).
In[28,27],Lakshmibai,Littelmann,andMagyardefineafamilyoflinebundles
rL
i
,
m
(
m
=
m
1
...m
r
∈
Z
≥
0
)on
Z
i
(theyaretheonlygloballygeneratedline
bundleson
Z
i
,aspointedoutin[31]),andgiveabasisforthespaceofsections
H
0
(
Z
i
,L
i
,
m
).In[28],theelements
p
T
ofthisbasis,calledstandardmonomials,are
indexedbycombinatorialobjects
T
calledstandardtableaux:thelatter’sdefinition
involvescertainsequences
J
11
⊃∙∙∙⊃
J
1
m
1
⊃∙∙∙⊃
J
r
1
⊃∙∙∙⊃
J
rm
r
ofsubwords
of
i
,calledliftingsof
T
(seeSection2forprecisedefinitions—actually,twoequivalent
definitionsofstandardtableauxaregivenin[28],butwewillonlyusetheonein
termsofliftings).Notealsothat
L
i
,
m
isveryamplepreciselywhen
m
j
>
0forall
j
(see[31],Theorem3.1),inwhichcase
m
iscalledregular.
Themainresultofthispaperstatesthatinthiscase,SMTon
Z
i
iscompatible
withΓ
i
.
Theorem0.1.
Assumethat
m
isregular.Withtheabovenotation,thestandard
0monomials
p
T
suchthat
(
p
T
)
|
Γ
i
6
=0
stillformabasisof
H
(Γ
i
,L
i
,
m
)
.
Moreover,
(
p
T
)
|
Γ
i
6
=0
ifandonlyif
T
admitsalifting
J
11
⊃∙∙∙⊃
J
rm
r
suchthat
eachsubword
J
km
containsareducedexpressionof
w
0
.

SMTFORDESINGULARIZEDRICHARDSONVARIETIES3
Weprovethistheoreminthreesteps.
(1)Call
T
(or
p
T
)
w
0
-standar
P
diftheaboveconditionon(
J
km
)holds.Weprove
rbyinductionover
M
=
j
=1
m
j
thatthe
w
0
-standardmonomials
p
T
are
linearlyindependentonΓ
i
.(Heretheassumptionthat
m
isregularisnot
necessary.)
(2)Intheregularcase,weprovethatastandardmonomial
p
T
doesnotvanish
identicallyonΓ
i
ifandonlyifitis
w
0
-standard,usingthecombinatorics
oftheDemazureproduct(seeDefinition4.2).Itfollowsthat
w
0
-standard
monomialsformabasisofthehomogeneouscoordinateringofΓ
i
(whenΓ
i
isembeddedinaprojectivespaceviatheveryamplelinebundle
L
i
,
m
).