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

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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


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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

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
).
(3)Weusecohomologicaltechniquestoprovethattherestrictionmap
00H
(
Z
i
,L
i
,
m
)

H

i
,L
i
,
m
)
issurjective.Moreexplicitly,wedefineafamily(
Y
i
u
)ofsubvarietiesof
Z
i
windexedby
S
n
,withthepropertythat
Y
i
e
=
Z
i
and
Y
i
0

i
.Weconstruct
uasequencein
S
n
,
e
=
u
0
<u
1
<
∙∙∙
<u
N
=
w
0
,suchthatforevery
t
,
Y
i
t
+1
isdefinedin
Y
i
u
t
bythevanishingofasinglePlu¨ckercoordinate
p
κ
,insuch
awaythateachrestrictionmap
H
0
(
Y
i
u
t
,L
i
,
m
)

H
0
(
Y
i
u
t
+1
,L
i
,
m
)canbe
showntobesurjectiveusingvanishingtheorems(Corollary5.7andTheo-
rem5.23).Thisshowsthatthe
w
0
-standardmonomialsspan
H
0

i
,L
i
,
m
).
NotethatalternatebasesforcertainBott-Samelsonvarietieshavebeencon-
structedin[39],andthefibredproducts
Z
×
F`
(
n
)
Z
0
havebeenstudiedfromthis
pointofviewin[11].
Sectionsareorganizedasfollows:inSection1,wefirstfixnotationandre-
callinformationonBott-Samelsonvarieties
Z
i
,andthenshowthatthefibreΓ
i
of
Z
i

F`
(
n
)over
F
opcan
isadesingularizationoftheRichardsonvariety
X
wv
;this
factismostcertainlyknowntoexperts,buthasnot,toourknowledge,appeared
intheliterature.InSection2,werecallthemainresultsaboutSMTforBott-
Samelsonvarietiesfrom[28],inparticularthedefinitionofstandardtableaux.In
Section3,wedefine
w
0
-standardmonomialsandweprovethattheyarelinearly
independentinΓ
i
.InSection4,weprovethatwhen
m
isregular,astandard
monomialdoesnotvanishidenticallyonΓ
i
ifandonlyifitis
w
0
-standard.Finally,
weproveinSection5that
w
0
-standardmonomialsgeneratethespaceofsections
H
0

i
,L
i
,
m
).
Acknowledgements.
IwouldliketothankChristianOhnforhelpfuldiscus-
sions.IamalsogratefultoMichelBrionforhisvaluableremarks,andespecially
forpointingoutgapsintheproofsofProposition5.16andTheorem5.23.
1.
DesingularizedRichardsonvarieties
ThenotationsareasintheIntroduction.Inaddition,if
k,l

Z
,thenwedenote
by[
k,l
]theset
{
k,k
+1
,...,l
}
,andby[
l
]theset[1
,l
].
WefirstrecallanumberofresultsonBott-Samelsonvarieties(see
e.g.
[35]).
Definition1.1.
Twoflags
F,G
in
F`
(
n
)arecalled
i
-adjacent
iftheycoincide
iexcept(possibly)attheircomponentsofdimension
i
,asituationdenotedby
F

G
.

4MICHAE¨LBALAN
Notations1.2.
For
i

[
n
],wedenoteby
F`
(
i
ˆ)thevarietyofpartialflags
V
1

V
2
⊂∙∙∙⊂
V
i

1

V
i
+1
⊂∙∙∙⊂
V
n
,
(dim
V
j
=
j
)
,
andby
ψ
i
ˆ
:
F`
(
n
)

F`
(
i
ˆ)thenaturalprojection.
Then
F
and
G
are
i
-adjacentifandonlyiftheyhavethesameimageby
ψ
i
ˆ
.
Consideraword
i
=
i
1
...i
r
in[
n

1],with
w
(
i
)=
s
i
1
...s
i
r

S
n
notnecessarily
reduced.A
galleryoftype
i
isasequenceoftheform
iii(1)
F
0

1
F
1

2
...

r
F
r
.
Foragivenflag
F
0
,the
Bott-Samelsonvariety
oftype
i
startingat
F
0
isthesetof
allgalleries(1),
i.e.
thefibredproduct
Z
i
(
F
0
)=
{
F
0

F`
(
i
ˆ
1
)
F`
(
n
)
×
F`
(
i
ˆ
2
)
∙∙∙×
F`
(
i
ˆ
r
)
F`
(
n
)
(asubvarietyof
F`
(
n
)
r
).Inparticular,
Z
i
1
...i
r
(
F
0
)isa
P
1
-fibrationover
Z
i
1
...i
r

1
(
F
0
),
whichshowsbyinductionover
r
thatBott-Samelsonvarietiesaresmooth.
Eachsubset
J
=
{
j
1
<
∙∙∙
<j
k
}⊂
[
r
]definesasubword
i
(
J
)=(
i
j
1
,...,i
j
k
)of
i
.Wethenwrite
Z
J
(
F
0
)insteadof
Z
i
(
J
)
(
F
0
),andweviewitasthesubvarietyof
Z
i
(
F
0
)consistingofallgalleries(1)suchthat
F
j

1
=
F
j
whenever
j
6∈
J
.
Wedenoteby
F
can
:
h
e
1
i⊂h
e
1
,e
2
i⊂∙∙∙⊂h
e
1
,e
2
,...,e
n
i
theflagassociatedto
thecanonicalbasis,andby
F
opcan
:
h
e
n
i⊂h
e
n
,e
n

1
i⊂∙∙∙⊂h
e
n
,e
n

1
,...,e
1
i
the
oppositecanonicalflag.Notethat
F
opcan
=
e
w
0
.
Inthesequel,weshallonlyneedgalleriesstartingat
F
can
orat
F
opcan
;in
particular,wewrite
Z
i
=
Z
i
(
F
can
).
The(diagonal)
B
-actionon
F`
(
n
)
r
leaves
Z
i
invariant.Inparticular,the
T
-fixed
pointsof
Z
i
arethegalleriesoftheform
i
1
i
2
i
3
i
r
F
can

e
u
1

e
u
1
u
2

...

e
u
1
...u
r
,
whereeach
u
j

S
n
iseither
e
or
s
i
j
.Thisgallerywillbedenoted
e
J

Z
i
,where
J
=
{
j
|
u
j
=
s
i
j
}
=
{
j
1
<
∙∙∙
<j
k
}
.
For
j

[
r
],wedenotebypr
j
:
Z
i

F`
(
n
)theprojectionsendingthegallery(1)
to
F
j
.Notethat
w
(
i
(
J
))=
s
i
j
1
...s
i
jk
=
u
1
...u
r
,sopr
r
(
e
J
)=
e
u
1
...u
r
=
e
w
(
i
(
J
))
.
When
i
isreduced,
i.e.
w
=
s
i
1
...s
i
r
isareducedexpressionin
S
n
,aflag
F
lies
intheSchubertvariety
X
w
ifandonlyifthereisagalleryoftype
i
=
i
1
...i
r
from
F
can
to
F
,hencethelastprojectionpr
r
takes
Z
i
surjectivelyto
X
w
.Moreover,
thissurjectionisbirational:itrestrictstoanisomorphismovertheSchubertcell
C
w
:thus,pr
r
:
Z
i

X
w
isadesingularizationof
X
w
,andlikewiseforthelast
projection
Z
i
(
F
opcan
)

X
w
0
w
.
When
i
isnotnecessarilyreduced,pr
r
(
Z
i
)maybedescribedasfollows.Re-
call[28,Definition-Lemma1]thattheposet
{
w
(
i
(
J
))
|
J

[
r
]
}
admitsaunique
maximalelement,denotedby
w
max
(
i
)(so
w
max
(
i
)=
w
(
i
)ifandonlyif
i
isreduced):
Proposition1.3.
Let
i
beanarbitraryword.Then
pr
r
(
Z
i
)
istheSchubertvariety
X
w
,where
w
=
w
max
(
i
)
.
Proof.
Sincepr
r
(
Z
i
)is
B
-stable,itisaunionofSchubertcells.But
Z
i
isapro-
jectivevariety,sothemorphismpr
r
isclosed,hencepr
r
(
Z
i
)isaunionofSchubert
varieties,andthereforeasingleSchubertvariety
X
w
since
Z
i
isirreducible.More-
over,the
T
-fixedpoints
e
J
in
Z
i
projecttothe
T
-fixedpoints
e
w
(
i
(
J
))
in
X
w
,and

SMTFORDESINGULARIZEDRICHARDSONVARIETIES5
all
T
-fixedpointsof
X
w
areobtainedinthisway(indeed,if
e
v
issuchapoint,then
thefibrepr
r

1
(
e
v
)is
T
-stable,soitmustcontainsome
e
J
byBorel’sfixedpoint
theorem).Inparticular,
e
w
correspondstoachoiceof
J
⊂{
1
,...,r
}
suchthat
w
(
i
(
J
))ismaximal,hencetheresult.

WenowturntothedescriptionofadesingularizationofaRichardsonvariety
X
wv
=
X
w

X
v
,
v

w

S
n
.Let
Z
=
Z
i
1
...i
d
forsomereduceddecomposition
w
=
s
i
1
...s
i
d
and
Z
0
=
Z
i
r
...i
d
+1
(
F
opcan
)forsomereduceddecomposition
w
0
v
=
s
i
r
s
i
r

1
...s
i
d
+1
.Since
Z
desingularizes
X
w
and
Z
0
desingularizes
X
v
,anatural
candidateforadesingularizationof
X
wv
isthefibredproduct
Z
×
F`
(
n
)
Z
0
.However,
wewishtoseethisvarietyinaslightlydifferentway:anelementof
Z
×
Z
0
isapair
ofgalleries
i
1
i
2
i
d
F
can

F
1

...

F
d
,
i
r
i
r

1
i
d
+1
F
opcan

G
r

1

...

G
d
,
anditbelongsto
Z
×
F`
(
n
)
Z
0
whentheendpoints
F
d
and
G
d
coincide:inthiscase,
byreversingthesecondgallery,theyconcatenatetoformalongergallery
i
1
i
2
i
d
i
d
+1
i
r
F
can

F
1

...

F
d

...

F
opcan
.
Thus,
Z
×
F`
(
n
)
Z
0
identifieswiththesetofallgalleriesin
Z
i
=
Z
i
1
...i
r
thatendin
F
opcan
,
i.e.
withthefibre
Γ
i
=pr
r

1
(
F
opcan
)
ofthelastprojectionpr
r
:
Z
i

F`
(
n
).Byconstruction,the
d
thprojectionpr
d
thenmapsΓ
i
ontotheRichardsonvariety
X
wv
.
Proposition1.4.
Intheabovenotation,the
d
thprojection
pr
d

i

X
wv
isa
desingularization,
i.e.pr
d
isbirational,andthevariety
Γ
i
issmoothandirreducible.
Proof.
WefirstcomputethedimensionofΓ
i
:sincepr
r
issurjective,thereexists
anon-emptyopenset
O
in
F`
(
n
)suchthateverypoint
F

O
hasafibreofpure
dimensiondim(
Z
i
)

dim(
F`
(
n
)).Sincetheflagvariety
F`
(
n
)isirreducible,
O
meetstheopenSchubertcell
C
w
0
.Let
F

O

C
w
0
.Sincepr
r
is
B
-equivariant,
thefibresof
F
and
F
opcan
=
e
w
0
areisomorphic.Inparticular,theyhavethesame
dimension,sodim(Γ
i
)=dim(
Z
i
)

dim(
F`
(
n
)).
NextweshowthatΓ
i
issmooth.Let
γ

Γ
i
.Wewanttoprovethatthetangent
space
T
γ

i
)ofΓ
i
at
γ
andΓ
i
havethesamedimension.LetΩ=pr
r

1
(
C
w
0
).Let
U
bethemaximalunipotentsubgroupof
B
.Thissubgroupactssimplytransitively
ontheSchubertcell
C
w
0
.Considerthemorphism
s
:
C
w
0
=
U.e
w
0

Ω
u.e
w
0
7→
u.γ
Sincepr
r
is
U
-equivariant,wehavepr
r

s
=id
C
w
0
.Differentiatingthisequalityin
e
w
0
givesdpr
r
(
γ
)

d
s
(
e
w
0
)=id
T
ew
F`
(
n
)
.Inparticular,thelinearmapdpr
r
(
γ
):
0T
γ
(
Z
i
)

T
e
w
0
(
F`
(
n
))issurjective.Moreover,
T
γ

i
)

ker(dpr
r
(
γ
)).Fromthis,
wededuce
dim(Γ
i
)

dim
T
γ

i
)

dim
T
γ
(
Z
i
)

dim
T
e
w
0
(
F`
(
n
))

dim
Z
i

dim
F`
(
n
)(since
Z
i
and
F`
(
n
)arebothsmooth)

dimΓ
i
,

6MICHA¨ELBALAN
henceΓ
i
issmooth.
NowweshowthatΓ
i
isirreducible.Let
C
1
,...,C
e
betheirreduciblecomponents
ofΓ
i
.SinceΓ
i
issmooth,the
C
j
arealsotheconnectedcomponentsofΓ
i
.The
varietyΩisopenin
Z
i
.Inparticular,Ωisirreducible.Sincepr
r
is
B
-equivariant,
e[[Ω=
bC
i
.
i
=1
b

B
SLetΩ
i
=
b

B
bC
i
.Themorphism
f
:
U
×
Γ
i

Ω
,
(
b,γ
)
7→
b.γ
isanisom
S
orphism.
Inparticular,Ω
i
=
f
(
U
×
C
i
)isanirreducibleclosedsetinΩ.SoΩ=
ie
=1
Ω
i
is
adisjointdecompositionofΩintoirreducibles.Hence
e
=1,andΓ
i
isirreducible.
Finally,toshowthatΓ
i

X
wv
isbirational,weconsidertheprojectionspr
d
:
Z

X
w
andpr
r

d
:
Z
0

X
v
.Sincetheyarebirational,thereexistopensubsets
U
w

X
w
and
O

Z
isomorphicunderpr
d
,andopensubsets
U
v

X
v
and
O
0

Z
0
isomorphicunderpr
r

d
.Thentheopenset(
O
×
O
0
)

(
Z
×
F`
(
n
)
Z
0
)of
Z
×
F`
(
n
)
Z
0
isisomorphictotheopenset
U
w

U
v
of
X
wv
underpr
d
:
Z
×
F`
(
n
)
Z
0

X
wv
.Since
X
wv
and
Z
×
F`
(
n
)
Z
0
=

Γ
i
areirreducible,theseopensubsetsmustbedense.The
birationalityofpr
d

i

X
wv
follows.

Remark
1.5
.
Incharacteristic0,itcanbeprovedmoredirectlythatthefibred
0product
Z
×
F`
(
n
)
Z
issmoothusingKleiman’stransversalitytheorem(
cf.
[16],
Theorem10.8).Moreover,thistheoremalsostatesthateveryirreduciblecomponent
of
Z
×
F`
(
n
)
Z
0
isofdimensiondim(
Z
)+dim(
Z
0
)

dim(
F`
(
n
)).Toprovethe
irreducibilityof
Z
×
F`
(
n
)
Z
0
,consider
∂Z
(resp.
∂Z
0
)theunionofallBott-Samelson
varieties
X
with
X
(
Z
(resp.
X
(
Z
0
).ByKleiman’stransversalitytheorem,the
000dimensionof(
∂Z
×
F`
(
n
)
Z
)

(
Z
×
F`
(
n
)
∂Z
)islessthandim(
Z
×
F`
(
n
)
Z
).So,on
onehand,thefibred-product
O
=(
Z
\
∂Z
)
×
F`
(
n
)
(
Z
0
\
∂Z
0
)meetseachirreducible
0componentof
Z
×
F`
(
n
)
Z
,hence
O
isdense.Ontheotherhand,
O
isisomorphic
totheopenRichardsonvariety
C
wv
=
C
w

C
v
,hence
O
isirreducible.Therefore,
0Z
×
F`
(
n
)
Z
isirreducible.

For
i
anarbitraryword,wemaystillconsiderthevarietyΓ
i
ofgalleriesoftype
i
,beginningat
F
can
andendingat
F
opcan
.Ingeneralthisvarietyisnolongerbira-
tionaltoaRichardsonvariety.Butwestillhave
Proposition1.6.
Let
i
=
i
1
...i
r
beanarbitraryword,andconsiderthepro-
jection
pr
j

i

F`
(
n
)
.Then
pr
j

i
)
istheRichardsonvariety
X
yx
where
y
=
w
max
(
i
1
...i
j
)
and
x
=
w
0
w
max
(
i
j
+1
...i
r
)

1
.Moreover,
Γ
i
issmoothand
irreducible.
Proof.
ThevarietyΓ
i
isisomorphictothefibredproduct
Z
i
1
...i
j
×
F`
(
n
)
Z
i
r
...i
j
+1
(
F
opcan
)
,
ecnehpr
j

i
)=pr
j
(
Z
i
1
...i
j
)

pr
r

j
(
Z
i
r
...i
j
+1
(
F
opcan
))
=
X
w
max
(
i
1
...i
j
)

w
0
X
w
max
(
i
r
...i
j
+1
)
x.X=y

SMTFORDESINGULARIZEDRICHARDSONVARIETIES7
Eventually,wemayprovethatΓ
i
issmoothandirreducibleexactlyasintheproof
ofProposition1.4.

Example1.7.
WeconsidertheRichardsonvariety
X
wv

F`
(4)with
w
=[4231]
and
v
=[2143].Aflag
F
=(
F
1

F
2

F
3

F
4
=
k
4
)belongstotheSchubert
variety
X
w
ifandonlyif
F
2
meets
h
e
1
,e
2
i
.
Since
w
=
s
1
s
2
s
3
s
2
s
1
isareduceddecomposition,theBott-Samelsonvariety
Z
12321
desingularizes
X
w
.Anelementof
Z
12321
isagallery
12321F
can

F
1

F
2

F
3

F
4

F
5
.
Aflag
G
belongstotheoppositeSchubertvariety
X
v
ifandonlyif
G
1
⊂h
e
2
,e
3
,e
4
i
and
G
3
⊃h
e
4
i
.
Similarly,
w
0
v
=
s
2
s
1
s
3
s
2
isareduceddecomposition,sotheBott-Samelson
variety
Z
2312
(
F
opcan
)desingularizestheoppositeSchubertvariety
X
v
.Anelement
of
Z
2132
(
F
opcan
)isagallery
2312F
opcan

G
8

G
7

G
6

G
5
.
Therefore,anelementofthevarietyΓ
123212312
hastheform

123212312

γ
=
F
can

F
1

F
2

F
3

F
4

F
5

G
6

G
7

G
8

F
opcan
.
Theprojection
pr
5
:
γ
7→
F
5
=
G
5
mapsΓ
123212312
birationallyto
X
wv
.
Thereareonlytwosingularpointson
X
wv
,namely
e
w
and
e
v
.Theirfibres
pr
5

1
(
e
w
)andpr
5

1
(
e
v
)are1-dimensional.Indeed,givenagallery
γ

Γ
i
,let
V
j
be
ijthe
i
j
-thcomponentofpr
j
(
γ
).Sincepr
j

1
(
γ
)—pr
j
(
γ
),weknowpr
j
(
γ
)assoonas
weknowpr
j

1
(
γ
)and
V
j
.Thus,agallerycanbegivenbythesequence
V
1
,...,V
9
.
Withthisdescription,agalleryinthefibreof
e
w
isthengivenby
h
e
2
i
,
h
e
2
,e
3
i
,
h
e
2
,e
3
,e
4
i
,
h
e
2
,e
4
i
,
h
e
4
i
,
h
e
4
,xe
2
+
ye
3
i
,
h
e
2
,e
3
,e
4
i
,
h
e
4
i
,
h
e
3
,e
4
i
,
with[
x
:
y
]

P
1
.
Similarly,thefibreof
e
v
isgivenby
h
xe
1
+
ye
2
i
,
h
e
1
,e
2
i
,
h
e
1
,e
2
,e
3
i
,
h
e
1
,e
2
i
,
h
e
2
i
,
h
e
2
,e
4
i
,
h
e
2
,e
3
,e
4
i
,
h
e
4
i
,
h
e
3
,e
4
i
,
with[
x
:
y
]

P
1
.
2.
BackgroundonSMTforBott-Samelsonvarieties
Inthissection,werecallfrom[28]themaindefinitionsandresultsaboutStan-
dardMonomialTheoryforBott-Samelsonvarieties.
Definitions2.1.
A
tableau
isasequence
T
=
t
1
...t
p
with
t
j

[
n
].If
T
=
t
1
...t
p
and
T
0
=
t
0
1
...t
0
p
0
aretwotableaux,thenthe
concatenation
T

T
0
isthetableau
t
1
...t
p
t
0
1
...t
0
p
0
.Wedenoteby

theemptytableau,sothat
T
∗∅
=
∅∗
T
=
T
.
A
column
κ
ofsize
i
isatableau
κ
=
t
1
...t
i
with1

t
1
<
∙∙∙
<t
i

n
.The
setofallcolumnsofsize
i
isdenotedby
I
i,n
.The
Bruhatorder
on
I
i,n
isdefined
ybκ
=
t
1
...t
i

κ
0
=
t
0
1
...t
i
0
⇐⇒
t
1

t
0
1
,...,t
i

t
i
0
.

8MICHAE¨LBALAN
Thesymmetricgroup
S
n
actson
I
i,n
:if
w

S
n
and
κ
=
t
1
...t
i

I
i,n
,then

isthecolumnobtainedbyrearrangingthetableau
w
(
t
1
)
...w
(
t
i
)inanincreasing
sequence.
For
i

[
n
],the
fundamentalweightcolumn
$
i
isthesequence12
...i
.
Weshallbeinterestedinaparticulartypeoftableau,calledstandard.
rDefinitions2.2.
Let
i
=
i
1
...i
r
,and
m
=
m
1
...m
r

Z

0
.A
tableauofshape
(
i
,
m
)isatableauoftheform
κ
11
∗∙∙∙∗
κ
1
m
1

κ
21
∗∙∙∙∗
κ
2
m
2
∗∙∙∙∗
κ
r
1
∗∙∙∙∗
κ
rm
r
where
κ
km
isacolumnofsize
i
k
forevery
k,m
.(If
m
k
=0,thereisnocolumnin
thecorrespondingpositionof
T
.)
A
lifting
of
T
isasequenceofsubwordsof
i
J
11
⊃∙∙∙⊃
J
1
m
1

J
21
⊃∙∙∙⊃
J
2
m
2
⊃∙∙∙⊃
J
r
1
⊃∙∙∙⊃
J
rm
r
suchthat
J
km

[
k
]isareducedsubwordof
i
and
w
(
i
(
J
km

[
k
]))
$
i
k
=
κ
km
.If
suchaliftingexists,thenthetableau
T
issaidtobe
standard
.
Remark
2.3
.
Thelastequalityinthedefinitionofaliftingmaybeviewedge-
ometricallyasfollows.If
J

[
r
]and
j

[
r
],thenpr
j
:
Z
i

F`
(
n
)maps
Z
J

Z
i
ontoaSchubertvariety
X
w

F`
(
n
)(
cf.
ProofofProposition1.3).In
thenotationsofSection1,theimagesof
T
-fixedpointsof
Z
J
underpr
j
areofthe
formpr
j
(
e
K
)=
e
u
1
...u
j
=
e
w
(
i
(
K

[
j
]))
with
K
runningoverallsubsetsof
J
,hence
w
=
w
max
(
i
(
J

[
j
])).Inturn,theimageofpr
j
(
Z
J
)bytheprojection
F`
(
n
)

G
i
j
,n
isequaltotheSchubertvariety
X
w$
ij
:for
J
=
J
km
intheabovelifting,thispro-
jectionisthereforeequalto
X
κ
km
.Weshallfollowuponthispointofviewin
Remark4.7.
Notation2.4.
Eachcolumn
κ

I
i,n
identifieswithaweightof
GL
(
n
),insucha
waythatthefundamentalweightcolumn
$
i
correspondstothe
i
thfundamental
weightof
GL
(
n
).Therefore,wealsodenoteby
$
i
thisfundamentalweight.
WerecallthePlu¨ckerembedding:givenan
i
-subspace
V
of
k
n
,chooseabasis
v
1
,...,v
i
of
V
,andlet
M
bethematrixofthevectors
v
1
,...,v
i
writteninthebasis
(
e
1
,...,e
n
).Weassociatetoeachcolumn
κ
=
t
1
...t
i
theminor
p
κ
(
V
)of
M
on
rows
t
1
,...,t
i
.Thenthemap
p
:
V
7→
[
p
κ
(
V
)
|
κ

I
i,n
]isthePlu¨ckerembedding.
Let
π
i
:
F`
(
n
)

G
i,n
bethenaturalprojection.Wedenoteby
L
$
i
theline
bundle(
p

π
i
)

O
(1).
Nowconsiderthetensorproduct
L
$

m
1
⊗∙∙∙⊗
L
$

m
r
on
F`
(
n
)
r
,anddenoteby
iir1L
i
,
m
itsrestrictionto
Z
i

F`
(
n
)
r
.
Definition2.5.
Toatableau
T
=
κ
11
∗∙∙∙∗
κ
1
m
1
∗∙∙∙∗
κ
r
1
∗∙∙∙∗
k
rm
r
,one
associatesthesection
p
T
=
p
κ
11
⊗∙∙∙⊗
p
κ
1
m
1
⊗∙∙∙⊗
p
κ
r
1
⊗∙∙∙⊗
p
κ
rmr
of
L
i
,
m
.If
T
isstandardofshape(
i
,
m
),then
p
T
iscalleda
standardmonomialofshape
(
i
,
m
).
Theorem2.6
([28])
.
(1)Thestandardmonomialsofshape
(
i
,
m
)
formabasisofthespaceofsections
H
0
(
Z
i
,L
i
,
m
)
.
(2)For
i>
0
,
H
i
(
Z
i
,L
i
,
m
)=0
.
(3)Thevariety
Z
i
isprojectivelynormalforanyembeddinginducedbyavery
amplelinebundle
L
i
,
m
.

SMTFORDESINGULARIZEDRICHARDSONVARIETIES9
3.
LinearIndependence

Example3.1.
WewanttoseeonExample1.7howonemayconstructanSMT
forthevarietiesΓ
i
.
Considerthelinebundle
L
i
,
m
on
Z
i
where
i
=123212312and
m
=200010111.
Weconsidertherest
0
rictionmap
H
0
(
Z
i
,L
i
,
m
)

H
0

i
,L
i
,
m
),andanaturalidea
togetabasisof
H

i
,L
i
,
m
)istotakeallthestandardmonomialsthatdonot
belongtoitskernel.
Solet
T
=
κ
11

κ
12

κ
51

κ
71

κ
81

κ
91
beatableauofshape(
i
,
m
).The
monomial
p
T
doesnotvanishidenticallyonΓ
i
ifandonlyif
κ
11

12
∈{
1
,
2
}
,
κ
51
6
=1,
κ
71
=234,
κ
81
=4,
κ
91
=34.
Onemaycheck(bycomputer)thatthereare708standardtableaux.Among
thesetableaux,9donotvanishidentically:
T
1
=2

2
∗∅∗∅∗∅∗
4
∗∅∗
234

4

34
T
4
=2

1
∗∅∗∅∗∅∗
4
∗∅∗
234

4

34
T
2
=2

2
∗∅∗∅∗∅∗
3
∗∅∗
234

4

34
T
5
=2

1
∗∅∗∅∗∅∗
3
∗∅∗
234

4

34
T
3
=2

2
∗∅∗∅∗∅∗
2
∗∅∗
234

4

34
T
6
=2

1
∗∅∗∅∗∅∗
2
∗∅∗
234

4

34
T
7
=1

1
∗∅∗∅∗∅∗
4
∗∅∗
234

4

34
T
8
=1

1
∗∅∗∅∗∅∗
3
∗∅∗
234

4

34
T
9
=1

1
∗∅∗∅∗∅∗
2
∗∅∗
234

4

34
Moreover,thetableaux
T
i
admitthefollowingliftings(
J
kim
)
J
111
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
141
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
171
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
211
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
241
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
271
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
511
=
{
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
541
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
571
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
711
=
{
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
741
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
771
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
811
=
{
3
,
4
,
5
,
6
,
7
,
9
}
J
841
=
{
2
,
3
,
4
,
5
,
6
,
7
,
9
}
J
871
=
{
2
,
3
,
4
,
5
,
6
,
7
,
9
}
J
911
=
{
3
,
4
,
5
,
6
,
7
,
9
}
J
941
=
{
2
,
3
,
4
,
5
,
6
,
7
}
J
971
=
{
2
,
3
,
4
,
5
,
6
,
7
}
J
121
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
151
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
181
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
221
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
251
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
281
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
521
=
{
1
,
2
,
3
,
5
,
6
,
7
,
8
,
9
}
J
551
=
{
2
,
3
,
5
,
6
,
7
,
8
,
9
}
J
581
=
{
2
,
3
,
5
,
6
,
7
,
8
,
9
}
J
721
=
{
1
,
2
,
3
,
5
,
6
,
8
,
9
}
J
751
=
{
2
,
3
,
5
,
6
,
7
,
8
,
9
}
J
781
=
{
2
,
3
,
5
,
6
,
7
,
8
,
9
}
J
821
=
{
1
,
2
,
3
,
5
,
6
,
8
,
9
}
J
851
=
{
2
,
3
,
5
,
6
,
7
,
8
,
9
}
J
881
=
{
2
,
3
,
5
,
6
,
7
,
8
,
9
}
J
921
=
{
1
,
2
,
3
,
5
,
6
,
8
}
J
951
=
{
2
,
3
,
5
,
6
,
7
,
8
}
J
981
=
{
2
,
3
,
5
,
6
,
7
,
8
}
J
131
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
161
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
191
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
231
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
261
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
291
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
J
531
=
{
1
,
2
,
3
,
4
,
6
,
7
,
8
,
9
}
J
561
=
{
3
,
5
,
6
,
7
,
8
,
9
}
J
591
=
{
3
,
5
,
6
,
7
,
8
,
9
}
J
731
=
{
1
,
2
,
3
,
4
,
8
,
9
}
J
761
=
{
3
,
5
,
6
,
7
,
8
,
9
}
J
791
=
{
3
,
5
,
6
,
7
,
8
,
9
}
J
831
=
{
1
,
2
,
3
,
4
,
8
,
9
}
J
861
=
{
3
,
5
,
6
,
7
,
8
,
9
}
J
891
=
{
3
,
5
,
6
,
7
,
8
,
9
}
J
931
=
{
1
,
2
,
3
,
4
,
8
,
9
}
J
961
=
{
3
,
5
,
6
,
7
,
8
,
9
}
J
991
=
{
3
,
5
,
6
,
7
,
8
,
9
}

10MICHAE¨LBALAN
iTheseliftingshavethefollowingproperty:
w
max
(
i
(
J
km
))=
w
0
foreach
k,m
.We
thensaythat
T
i
is
w
0
-standard.Itcanbecheckedthatthestandardtableauxthat
arenot
w
0
-standardvanishidenticallyonΓ
i
.
Toseethattheremainingmonomials
p
T
i
arelinearlyindependent,wemaywork
onanopenaffineset.ThereexistsanopensetΩof
Z
i
suchthatΓ
i

Ωisisomorphic
totheaffinespace
k
3
(seeDefinition5.10andProposition5.19).Here,wehave
ϕ
:(
x,y,z
)
7→
(
V
1
,...,V
9
)
,
rofV
1
=
h
xe
1
+
e
2
i
V
2
=
h
xe
1
+
e
2
,

xye
1
+
e
3
i
V
3
=
h
xe
1
+
e
2
,

xye
1
+
e
3
,e
4
i
V
4
=
h
xe
1
+
e
2
,

xyze
1
+
ze
3
+
e
4
i
V
5
=
h
yze
2
+
ze
3
+
e
4
i
V
6
=
h
yze
2
+
ze
3
+
e
4
,ye
2
+
e
3
i
V
7
=
h
e
2
,e
3
,e
4
i
V
8
=
h
e
4
i
V
9
=
h
e
3
,e
4
i
Wedenoteagainby
p
T
thepolynomial
ϕ

(
p
T
)
|
Ω
.Wethenhave
p
T
1
=1
,p
T
4
=
x,p
T
7
=
x
2
,
p
T
2
=
z,p
T
5
=
xz,p
T
8
=
x
2
z,
p
T
3
=
yz,p
T
6
=
xyz,p
T
9
=
x
2
yz.
Itisclearthatthesemonomialsarelinearlyindependentin
k
[
x,y,z
].
Definitions3.2belowwillgeneralizethebehaviouroftheliftings(
J
kim
)observed
inthisexample.
Definitions3.2.
Let
T
beastandardtableauofshape(
i
,
m
).Wesaythat
T
(or
themonomial
p
T
)is
w
0
-standard
ifthereexistsalifting(
J
km
)of
T
suchthateach
subword
J
km
containsareducedexpressionof
w
0
.
Moregenerally,if
J

[
r
]containsareducedexpressionfor
w
0
,thenΓ
J
=
Z
J

Γ
i
6
=

,andwesaythat
T
(or
p
T
)is
w
0
-standardon
Γ
J
ifthereexistsa
lifting(
J
km
)of
T
suchthatforevery
k,m
,
J

J
km
and
J
km
containsareduced
expressionof
w
0
.
Similarly,
T
(or
p
T
)issaidtobe
w
0
-standardonaunion
Γ=Γ
J
1
∪∙∙∙∪
Γ
J
k
if
T
is
w
0
-standardonatleastoneofthecomponentsΓ
J
1
,...,
Γ
J
k
.Wethendenote
by
S
(Γ)thesetofall
w
0
-standardtableauxonΓ.
Weneedsomeresultsabout
positroidvarieties
.Referencesforthesevarieties
canbefoundin[21].
Let
π
i
bethecanonicalprojection
F`
(
n
)

G
i,n
.Ingeneral,theprojectionof
aRichardsonvariety
X
wv

F`
(
n
)isnolongeraRichardsonvariety.But
π
i
(
X
wv
)
isstilldefinedinsidetheGrassmannian
G
i,n
bythevanishingofsomePlu¨cker
coordinates.Moreprecisely,considertheset
M
=
{
κ

I
i,n
|
e
κ

π
i
(
X
wv
)
}
.Then
vΠ=
π
i
(
X
w
)=
{
V

G
i,n
|
κ

/
M
=

p
κ
(
V
)=0
}
.
Theposet
M
isa
positroid
(seetheparagraphfollowingLemma3.20in[21]),and
thevarietyΠiscalleda
positroidvariety
.
Lemma3.3.
Withthenotationabove,
M
=
{
κ

I
i,n
|∃
u

[
v,w
]
,u$
i
=
κ
}
.