Table Of ContentSchubert Varieties
P. Littelmann
May 21, 2012
Contents
Preface 1
1 SMT for Graßmann varieties 3
1.1 The Plu¨cker embedding . . . . . . . . . . . . . . . . . . . . 4
1.2 Monomials and tableaux . . . . . . . . . . . . . . . . . . . . 12
1.3 Straightening relation . . . . . . . . . . . . . . . . . . . . . 14
1.4 Schubert Varieties in Gr . . . . . . . . . . . . . . . . . . 16
d,n
1.5 SMT for Schubert varieties in the Graßmannian . . . . . . . 19
1.6 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 SMT for flag varieties 29
2.1 SMT for Schubert varieties in the flag variety . . . . . . . . 30
2.2 A basis for H0(X,L) and vanishing of higher cohomology
groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Ideal theory of Schubert varieties . . . . . . . . . . . . . . . 45
2.4 Lexicographic shellability and Cohen-Macaulayness . . . . . 49
References 55
2 Contents
1
Standard Monomial Theory for
Graßmann varieties
Standard Monomial theory (abbreviated as SMT) is the central theme of
this book. We think that the classical case of Schubert varieties in the
Graßmannian deserves special attention because the philosophy and the
strategyarethesameasinthegeneralcase,butthetechnicalrequirements
used in the constructions and in the proofs are elementary.
WeshallseehowonecandevelopSMTinthisclassicalcasebyusingjust
the Plu¨cker embedding and the associated Plu¨cker coordinates. It should
besupportivetohavethiselementarycasereadyasaguidelineforthelater
chapters.
SMT consists in constructing explicit bases for the homogeneous co-
ordinate rings of the Graßmann variety and its Schubert varieties. After
introducing the Graßmannian and its Schubert varieties, we present the
Standard Monomial Theory for Schubert varieties in the Graßmannian in
Section 1.5. As an application we present a proof of the “vanishing theo-
rems” for these Schubert varieties, where we also deduce their normality.
Let us fix some notation that we shall use throughout this chapter. Let
k be an algebraically closed field of arbitrary characteristic and set k∗ =
k \{0}. Let V = kn and let e ,...,e be the standard basis of V. The
1 n
group SL (k) acts naturally on V. Let T be the subgroup of of SL (k)
n n
consisting of diagonal matrices, and B the subgroup consisting of upper
triangularmatrices.WehaveB =TU =UT whereU isthesubgroupofB
consisting of upper triangular unipotent matrices. Given a positive integer
d, we denote by S the symmetric group on d letters.
d
4 1. SMT for Graßmann varieties
1.1 The Plu¨cker embedding
1.1.1. Graßmann variety. Let us start with the most simple example of
a Graßmann variety, the projective space Pn−1. Recall that the projective
space Pn−1 isdefined asthe setofall linesin V.Another way toformulate
thedefinitionistosaythattheprojectivespaceisthequotient(V\{0})/∼,
wheretheequivalencerelationisdefinedby:v ∼v(cid:48)ifthereexistsanelement
t∈k∗ such that tv =v(cid:48).
ThedefinitionofaGraßmannvarietyisastraightforwardgeneralization
oftheabove,onlyonehastoreplacelines,i.e.1-dimensionalsubspaces,by
d-dimensional subspaces.
Definition 1.1.2. Let 1≤d<n. The Graßmann variety Gr is defined
d,n
as the set of all d-dimensional subspaces in V.
In particular, Gr =Pn−1. To get a description of Gr as a quotient
1,n d,n
similar to the description of the projective space above, let U ∈ Gr be
d,n
a d-dimensional subspace of kn. Fix a basis {v ,...,v } of U, then we can
1 d
associate to U an n×d matrix A = (a ) of rank d such that the j-th
i,j
column consists of the coefficients of v with respect to the standard basis
j
{e ,...,e } of V, i.e. v =(cid:80)n a e .
1 n j i=1 ij i
Vice versa, to an n×d matrix A ∈ M (k) of rank d one associates
n,d
naturally the d-dimensional subspace U of V obtained as the span of the
column vectors. In this language we can give a description of Gr similar
d,n
to that of the projective space above: let Z be the set of n×d matrices
of rank strictly less than d, then Gr = (M (k)\Z)/ ∼, where the
d,n n,d
equivalence relation is defined by: A ∼ A(cid:48) if the column vectors span the
same subspace of V.
Above we defined the relation “∼” on V \ {0} in terms of the group
action of k∗ on V. Here we can do the same by using the fact that GL (k)
d
acts transitively on the set of bases of a d-dimensional subspace:
A(cid:48) =AC for some
Gr =(M (k)\Z)/∼, where A∼A(cid:48) ⇔
d,n n,d C ∈GL (k)
d
For d = 1, this is exactly the description of the projective space Pn−1 =
Gr given above.
1,n
1.1.3. Gr as homogeneous space. Another very useful description of
d,n
the Graßmann variety is that of Gr as a homogeneous space. If U ⊂ V
d,n
is a d-dimensional subspace and g ∈ SL (k), then gU = {gu | u ∈ U} is
n
again a d-dimensional subspace. In fact, given U,U(cid:48) ∈ Gr , there exists
d,n
always a g ∈SL (k) such that gU =U(cid:48).
n
Denote by F ⊂ V the j-dimensional subspace F = (cid:104)e ,e ...,e (cid:105)
j j 1 2 j
spannedbythefirstj elementsofthestandardbasis.Thenwecanidentify
Gr with the coset space SL (k)/P , where P is the isotropy group of
d,n n d d
1.1 The Plu¨cker embedding 5
the d-dimensional subspace F . Now g ∈SL (k) is an element of P if and
d n d
only if ge ∈F for 1≤j ≤d, and hence:
j d
(cid:26) (cid:12) (cid:18) (cid:19)(cid:27)
(cid:12) ∗ ∗
Grd,n =SLn(k)/Pd, where Pd = A∈SLn(k)(cid:12)(cid:12)A= 0(n−d)×d ∗ .
Note that the isotropy group P contains B.
d
1.1.4. Plu¨cker coordinates. To endow the Graßmann variety with the
structureofanalgebraicvariety,wewillidentifyGr withasubsetofthe
d,n
projective space P(ΛdV). A first step in this direction is the introduction
of Plu¨cker coordinates, which can be viewed as linear functions on ΛdV as
well as multilinear alternating functions on M (k).
n,d
Thed-foldwedgeproductisalternating,theorderedproductsofelements
in the canonical basis of V: e ∧···∧e , 1 ≤ i < ··· < i ≤ n, form a
i1 id 1 d
basis of ΛdV, called the canonical basis of ΛdV.
Definition 1.1.5. Let I := {i = (i ,...,i )|1 ≤ i < ··· < i ≤ n} be
d,n 1 d 1 d
the set of all strictly increasing sequences of length d between 1 and n. For
i=(i ,...,i )∈I wewritee =e ∧···∧e .Wedefineapartialorder
1 d d,n i i1 id
“≥” on I as follows: i≥j ⇔i ≥j for all t=1,...,d.
d,n t t
So the canonical basis of ΛdV can be written as {e | i ∈ I }. Denote
i d,n
by {p |i∈I } the dual basis of (ΛdV)∗, i.e., p (e )=δ .
i d,n i j i,j
Definition 1.1.6. The linear functions p , i ∈ I , on ΛdV are called
i d,n
Plu¨cker coordinates.
Bythedefinitionofthed-foldwedgeproductthespaceoflinearfunctions
onΛdV canbenaturallyidentifiedwiththespaceofmultilinearalternating
functions on d-copies of V, i.e., on M (k)=V ×...×V.
d,n
(cid:124) (cid:123)(cid:122) (cid:125)
d times
Remark 1.1.7. We use the same name Plu¨cker coordinates and the same
symbol p for the linear functions on ΛdV as well as the corresponding
i
multilinear alternating function on the space M (k).
n,d
To make this relationship more explicit, recall that we have a natural
map, the exterior product map:
π :M (k) → ΛdV
d n,d . (1.1)
A=(v ,...,v ) (cid:55)→ v ∧···∧v
1 d 1 d
Here v ,...,v are the column vectors of the matrix A. If we express the
1 d
productv ∧···∧v asalinearcombinationoftheelementsofthecanonical
1 d
basis, then, by the definition of the dual basis, we have
(cid:88)
v ∧···∧v = p (π (A))e .
1 d i d i
i∈Id,n
6 1. SMT for Graßmann varieties
ThealternatingmultilinearfunctiononM (k)associatedtop isjustthe
n,d i
i-th coordinate of the linear combination above, i.e., it is the composition
p ◦π . So by abuse of notation we write just p (A) instead of p (π (A)).
i d i i d
Denote by A the d×d submatrix of A consisting of the i -th, i -th, ...
i 1 2
and the i -th row of A. It follows that:
d
Lemma 1.1.8. p (A) is the determinant detA of the submatrix A of A.
i i i
1.1.9. Plu¨cker embedding. Our next step is to identify the Graßmann
variety with a subset of the projective space P(ΛdV).
For A∈M (k) of rang d let v ,...,v ∈kn be the column vectors, let
n,d 1 d
U ⊂V be the span of these column vectors and let u ,...,u ∈U. Denote
1 d
by C = (c ) the d×d-matrix expressing the u as linear combinations
i,j j
of the v . i.e., u = (cid:80)d c v . The exterior product is alternating, so we
i j i=1 i,j i
get v ∧...∧v = (detC)u ∧...∧u . As a consequence we see that the
1 d 1 d
exterior product map induces a well defined map:
π :Gr =((M (k)\Z)/∼)−→P(ΛdV)
d,n n,d
called the Plu¨cker embedding. We have a left action of SL (k) on M (k)
n n,d
defined by g(v ,...,v ) = (gv ,...,gv ), and we have a natural action of
1 d 1 d
SL (k)onΛdV givenbyg(v ∧···∧v )=(gv )∧···∧(gv ).Itfollowsthat
n 1 d 1 d
the exterior product map π : M (k) → ΛdV is equivariant with respect
d n,d
to these SL (k)-actions, and hence so is the Plu¨cker embedding. The term
n
embedding is justified because:
Proposition 1.1.10. The Plu¨cker map π :Gr →P(ΛdV) is injective.
d,n
Proof. Let F be the d-dimensional subspace of V spanned by e ,...,e .
d 1 d
By the homogeneity of the SL (k)-action on Gr , it is sufficient to show
n d,n
if π(U)=π(F ), then U =F .
d d
So suppose π(U) = π(F ) and let {v ,...,v } be a basis of U. Denote
d 1 d
by A ∈ M (k) the corresponding matrix. Since [π (A)] = [e ∧...∧e ],
n,d d 1 d
we can choose the basis such that π (A) = e ∧...∧e . It follows that
d 1 d
the submatrix A consisting of the first d-rows of A has determinant
1,...,d
one, so by replacing A by A·A−1 if necessary we can (and will) assume
1,...,d
that the submatrix of A consisting of the first d rows is the d×d identity
matrix.
Now all d×d minors except p (A) vanish. In particular, for i > d
1,2,...,d
we have ±a =p (A)=0 and hence U =F .
i,j 1,...,j−1,j+1,...,d,i d
1.1.11. Again Plu¨cker coordinates. Insection1.1.4weintroducedthe
namePlu¨ckercoordinateforthedualbasisp ofthecanonicalbasisofΛdV.
i
To simplify the notation we use p in the following for arbitrary d-tuples
i
and not only for elements i∈I .
d,n
1.1 The Plu¨cker embedding 7
Wegiveadescriptionofthefunctionsasalternatingmultilinearfunctions
on the columns of M (k) (instead of describing them as linear functions
n,d
on ΛdV).
For 1 ≤ i ,...,i ≤ n (not necessarily distinct nor in increasing order)
1 d
seti=(i ,...,i ).Forann×dmatrixAletA bethed×dmatrixhaving
1 d i
as first row the i -th row of A, as second row the i -th row of A and so on.
1 2
We set p (A)=detA .
i i
Clearly, p = 0 if the i ’s are not distinct, and if they are all distinct,
i j
then
p =sgn(σ)p (1.2)
i1,...,id σ(i1),...,σ(id)
where σ ∈S is such that (σ(i ),...,σ(i ))∈I .
d 1 d d,n
1.1.12. Alternating functions. In view of Proposition 1.1.10, we can
identify Gr with Imπ. In general the image will not be all of P(ΛdV),
d,n
so the Plu¨cker coordinates restricted to Gr must satisfy some relations.
d,n
By definition, the Plu¨cker coordinates are i) linear functions on ΛdV as
wellasii)multilinearalternatingfunctionsonthecolumnsofM (k)(the
n,d
latter being identified with d-copies of V).
These functions are defined as determinants of maximal submatrices, so
they have a third property: iii) the Plu¨cker coordinate p is a multilinear
i
and alternating function in the i -th, i -th etc. row of M (k).
1 2 n,d
Suppose now i∩j =∅, then the product p p is a quadratic function on
i j
ΛdV whichisdefinitelynotanymoremultilinearinthecolumnsofM (k).
n,d
But this function is still multilinear in the i -th, i -th, ..., j -th, j -th
1 2 1 2
etc. row of M (k), and alternating separately in the i and j . So if
n,d k (cid:96)
we alternate this function so that it becomes alternating in the rows say
i ,...,i ,j , then we have an alternating function on d + 1-copies of a
1 d 1
d-dimensional vector space (the space of row vectors of M (k)). Hence
n,d
this function is zero on M (k). Or, in other words, viewed as a quadratic
n,d
function on ΛdV, we have a function such that the restriction to Imπ
d
vanishes.
Example 1.1.13. Before starting with the formal approach consider the
example Gr and the product of Plu¨cker coordinates p p ∈ k[Λ2k4].
2,4 1,2 3,4
Thecompositionwithπ :M →Λ2k4 givesafunctionwhichisofcourse
2 4,2
not anymore multilinear in the columns of M (k), but which is still mul-
4,2
tilinear in the rows of this space of matrices. We will “formally alternate”
this function. For example (we will see below why this is the alternated
function)
p p +p p −p p (1.3)
1,2 3,4 2,3 1,4 2,4 1,3
isaquadraticpolynomialonΛ2k4.TherestrictiontoImπ isamultilinear
2
function on M (k) which is alternating in the first, the third and the
4,2
fourth row of M (k). The only function with this property (i.e. being
4,2
alternating on 3 copies of a 2-dimensional space) is the zero function, so
the function above vanishes identically on Imπ . But this means that the
2
8 1. SMT for Graßmann varieties
restriction of the quadratic polynomial in (1.3) to Gr is identically zero,
2,4
and hence the Plu¨cker coordinates satisfy on Gr a quadratic relation.
2,4
To formalize this idea, let us start with some generalities. We work in-
side the ring k[x ] of polynomial functions on M (k) and we write just
i,j n,d
x ,...,x for the vector variables corresponding to the rows of M (k).
1 n n,d
Let f(x ,...,x ) be a multilinear function, then we can alternate it by
1 n
setting:
(cid:88)
Alt(f):= sgn(σ)(σf)(x ,...,x ),
1 n
σ∈Sn
where σf(x ,...,x )=f(x ,...,x ).
1 n σ−1(1) σ−1(n)
Supposen≥d+1.Insteadofassumingthatthefunctionismultilinearin
allvectorvariables,fixasubsetM ={k ,...,k },1≤k ≤...≤k ≤
1 d+1 1 d+1
n, of pairwise different indices, and assume the function is multilinear in
the rows corresponding to the indices k ,...,k . The function
1 d+1
(cid:88)
Alt (f):= sgn(σ)f(...,x ,...,x ,...)
M kσ−1(1) kσ−1(d+1)
σ∈Sd+1
(i.e. all vector variables different from x ,...,x are not changed) is
k1 kd+1
alternating and multilinear in x ,...,x .
k1 kd+1
For 1 ≤ t < d+1 let M = M ∪M be a disjoint decomposition such
1 2
that (cid:93)M =t. If f is alternating separately in the variables {x |k ∈M }
1 k 1
and {x |(cid:96)∈M }, then
(cid:96) 2
sgn(σ)f(...,x , ... ,x ,...)
kσ−1(1) kσ−1(d+1)
= sgn(σ(cid:48))f(...,x ,...,x ,...)
kσ(cid:48)−1(1) kσ(cid:48)−1(d+1)
whenever σ and σ(cid:48) are in the same coset in S /S ×S . Here we
d+1 t d+1−t
identify the subgroup S ×S with the subgroup of permutations in
t d+1−t
S which separately permute only the elements in M and M among
d+1 1 2
themselves.
So to get an alternating function one has to take the sum
(cid:88)
Alt (f):= sgn(σ)f(...,x ,...,x ,...)
M1,M2 kσ−1(1) kσ−1(d+1)
σ∈Sd+1/St×Sd+1−t
only over a system of representatives of the cosets.
Example 1.1.14. Suppose n = 4 and d = 2. Let f(x ,x ,x ,x ) =
1 2 3 4
p p be the product of these two Plu¨cker coordinates, then f is a mul-
1,2 3,4
tilinear function on M (k), alternating separately in the 1st and 2nd and
4,2
the 3rd and 4th row. Set M = {1}, M = {3,4} and M = M ∪M , and
1 2 1 2
denote by S respectively S the permutation groups of the sets. Then
M Mi
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
134 134 134
id= ,σ = ,σ = ,
134 1 314 2 341
1.1 The Plu¨cker embedding 9
is a set of representatives of S /S × S (see section 1.1.20 for a
M M1 M2
procedure to get the representatives) and
AltM1,M2(f) = f +sgn(σ1)(σ1f)+sgn(σ2)(σ2f)
= f(x ,x ,x ,x )−f(x ,x ,x ,x )+f(x ,x ,x ,x )
1 2 3 4 3 2 1 4 4 2 1 3
= p p +p p −p p
1,2 3,4 2,3 1,4 2,4 1,3
is the function on M (k) in equation 1.3, which is alternating in the 1st,
4,2
3rd and 4th row.
1.1.15. Quadratic relations. Aproductf =p p ofPlu¨ckercoordinates
i j
isaquadraticpolynomialonΛdV.Supposenowallindicesi ,j arediffer-
k (cid:96)
ent.TheproductisafunctiononM (k)whichismultilinearwithrespect
n,d
to the rows of this space of matrices. Fix 1≤t<d, then f is, by construc-
tion, alternating separately in the (row) vector variables x ,...,x and
i1 it
x ,...,x .
jt jd
Givenσ ∈S ,notethatσ shufflestheindiceesi ,...,i andj ,...,j .
d+1 1 t t d
Denote by iσ and jσ the d-tuples (σ−1(i ),...,σ−1(i ),i ,...,i ) and
1 t t+1 d
(j ,...,j ,σ−1(j ),...,σ−1(j )).Recallthatthefunctionsgn(σ)(σf),σ ∈
1 t−1 t d
S /S ×S , is independent of the choice of a representative for σ.
d+1 t d+1−t
The function we get by alternating f =p p is:
i j
(cid:88)
Alt (p p )= sgn(σ)p p .
{i1,...,it},{jt,...,jd} i j iσ jσ
σ∈Sd+1/St×Sd+1−t
Lemma 1.1.16. Suppose n ≥ 2d. Let i,j be two d-tuples, 1 ≤ i ,j ≤ n,
k l
such that the entries are all distinct. Fix 1≤t<d, the homogeneous poly-
nomial Alt (p p )∈k[ΛdV] vanishes on Gr ⊂P[ΛdV].
{i1,...,it},{jt,...,jd} i j d,n
Proof. By composing the function with the exterior product map, we see
that the quadratic polynomial vanishes on Gr if and only if, viewed as
d,n
a sum of products of minors, the function vanishes on M (k). But this
n,d
function is multilinear and alternating in the d+1 row vector variables
x ,...,x , x ,...,x . The space of the row vectors is of dimension d, so
i1 it jt jd
this function vanishes on M (k).
n,d
To weaken the condition that all indices have to be different, consider
two arbitrary d-tuples i and j, 1 ≤ i ,j ≤ n. We will now define a new
k l
pair i(cid:48),j(cid:48) such that all entries are different. Set
i(cid:48) =i +mn where m=(cid:93){(cid:96)|(cid:96)<k,i =i }
k k k (cid:96)
j(cid:48) =j +mn where m=(cid:93){(cid:96)|j =i }+(cid:93){(cid:96)|(cid:96)<k,j =j }
k k k (cid:96) k (cid:96)
Forexample,supposei ,i ,j ,j arepairwisedifferent,thenthisprocedure
1 2 1 2
applied to the pair
i=(i ,i ,i ,i ,i ) j =(j ,i ,i ,j ,j )
1 2 1 1 2 1 2 1 1 2
↓
i(cid:48) =(i ,i ,i +n,i +2n,i +n) j(cid:48) =(j ,i +2n,i +3n,j +n,j )
1 2 1 1 2 1 2 1 1 2
