Notes on Schubert Polynomials

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 3 July 2013

Schubert varieties

Let V be a vector space of dimension n over a field K, and let (e1,,en) be a basis of V, fixed once and for all. A flag in V is a sequence U=(Ui)0in of subspaces of V such that

0=U0U1 Un=V

with strict inclusions at each stage, so that dimUi=i for each i. In particular, if Vi is the subspace of V spanned by e1,,ei, then V=(Vi)0in is a flag in V, called the standard flag.

The set F=F(V) of flags in V is called the flag manifold of V.

Let G be the group of all automorphisims of the vector space V. Since we have fixed a basis of V, we may identify G with the general linear group GLn(k): if gG and

gej=i=1n gijei (1jn)

then g is identified with the matrix (gij).

The group G acts on F: if U=(Ui) and gG, then gU is the flag (gUi). Let B be the subgroup of G that fixes the standard flag V. Then gB if and only if gej is a linear combination of e1,,ej, for 1jn, that is to say if and only if gij=0 whenever i>j, so that B is the group of upper triangular matrices in GLn(k).

A basis of a flag U=(Ui) is a sequence (u1,,un) in V such that uiUi-Ui-1 1in, or equivalently such that u1,,ui is a basis of Ui for each i. Given such a basis of U, there is a unique gG such that gei=ui for each i, and we have U=gV. Hence G acts transitively on the flag manifold F, and the mapping gVgB is a bijection of F onto the coset space G/B.

For a flag U=(Ui), let

Ei=Ei(U)= { j:1jnand UiVjUi Vj-1 }

for 0in. Then (E0,,En) is a 'flag of sets', i.e. we have

(i) Card(Ei)=i for 0in,
(ii) Ei-1Ei for 1in.


(i) Fix i and let dj=dim(UiVj). Since

ViVj UiVj-1 = UiVj (UiVj)Vj-1 (UiVj)+Vj-1 Vj-1 VjVj-1

it follows that dj-dj-1=0 or 1. Since d0=0 and dn=i, there are therefore i jumps in the sequence (d0,d1,,dn), which proves (i).

(ii) Suppose that jEi, so that UiVj=UiVj-1. Intersecting with Ui-1, we see that jEi-1. Hence Ei-1Ei.

From (A.1) it follows that that each UF determines a permutation wSn as follows: w(i) is the unique element of Ei-Ei-1, for i=1,2,,n. Let ϕ:FSn denote the mapping so defined.

The symmetric group acts on V by permuting the basis elements ei:

w(ei)= ew(i)

for wSn and 1in. Hence we may regard Sn as a subgroup of G.

(A.2) Let UF, wSn. Then ϕ(U)=w if and only if U=bwV for some bB.


Suppose ϕ(U)=w. Then for i-1,,n we have

(1) UiVw(i) UiVw(i)-1


(2) Ui-1 Vw(i)= Ui-1 Vw(i)-1

By virtue of (1) we can choose uiUi of the form

(3) ui=ew(i)+ lower terms

where by 'lower terms' is meant a linear combination of e1,,ew(i)-1; and uiUi-1 by virtue of (2).

By rewriting (3) in the form

uw-1(j)= ej+lower terms (1jn)

we see that there exists bB such that uw-1(j)=bej for all j, or equivalently

ui=bew(i)= bwei.

Hence U-bwV as required.

For the converse it is enough to show that (i) ϕ(wV)=w and (ii) ϕ(bU)=ϕ(U) for all bB and UF. As to (i), wViVj is spanned by the basis vectors ew(k) such that ki and w(k)j, and therefore wViVjwViVj-1 if and only if j=w(k) for some ki. Thus the set Ei(wV) consists of w(1),,w(i), which establishes (i). Finally as to (ii), we have bUiVj=b(UiVj) if bB, so that Ei(bU)=Ei(U) and hence ϕ(bV)=ϕ(U) as required.

From (A2) we have immediately

(A3) (Bruhat decomposition) G is the disjoint union of the double cosets BwB, wSn.

For each wSn, let

Cw=(BwB)/B G/B=F.

The subsets Cw are the Schubert cells in the flag manifold F. By (A.3), F is the disjoint union of the Cw.

Let UF. Then UCw if and only if U has a basis (u1,,un) such that uiVw(i)-Vw(i)-1 for each i. We may normalize the ui by taking

ui=ew(i)+ lower terms.

We can then subtract from ui suitable multiples of the uk for which k<i and w(k)<w(i), so as to make the coefficient of ew(k) in ui zero for each such k. Then ui is replaced by a vector of the form

ew(i)+j aijej

where the sum is over j<w(i) such that jw(k) for any k<i, i.e., such that j<w(i) and w-1(j)>i, or equivalently (i,j)D(w), the diagram of w.

(A.4) Let UF. Then UCw if and only if U has a basis (u1,,un) of the form

ui=ew(i)+ jaijej

where the sum is over all j in the ith row of the diagram of w, and the coefficients aij are arbitrary elements of the field K. Moreover, the aij are uniquely determined by the flag U, and the mapping CwKD(w) so defined is a bijection.


Clearly each "matrix" a=(aij) of shape D(w) determines a basis (u1,,un) of V as above, and hence a flag UCw. If a*=(aij*) determines (u1*,,un*) and the same flag U, then each ui* must be expressible as

ui*=ui+ j<icij uj,

and from the form of ui* and the uj it follows that ui*=ui for each i, and hence a*=a.

Since CardD(w)=(w) it follows from (A.4) that the Schubert cell Cw is isomorphic to affine space of dimension (w).

Let UF and let (u1,,un) be any basis of U. Since u1,,ui is a basis for Ui for each i=1,,n-1, the flag U determines each of the exterior products u1uiΛi(V) up to a nonzero scalar multiple, and hence U determines the vector

(1) u1 (u1u2) (u1un-1) E

up to a nonzero scalar multiple, where E=VΛ2VΛn-1V. If P(E) denotes the projective space of E (i.e. the space whose points are the lines in E), we have an injective mapping


(the Plücker embedding) for which π(U) is the line in E generated by the vector (1).

Assume from now on that the field K is the field of complex numbers. Then the embedding π realizes the flag manifold F as a complex projective algebraic variety, which is smooth because F has a transitive group of automorphisms (namely G). Each Schubert cell Cw is a locally closed subvariety of F, isomorphic to affine space of dimension (w).

For each wSn let


be the closure of Cw in F. The Xw are the Schubert varieties in F, and a flag U lies in Xw if and only if U has a basis (u1,,un) such that uiVw(i) for each i. Each Xw is in fact a union of Schubert cells Cv: if (a1,,ap) is a reduced word for w, then CvXw if and only if v is of the form sb1sbq where (b1,,bq) is a subsequence of (a1,,ap), that is to say if and only if vw in the Bruhat order. In particular, X1=C1 is the single point VF. At the other extreme, if w0 is the longest element of Sn, then Xw0 is the whole of F, and the dimension of F is (w0)=12n(n-1).

Let H*(F;) be the cohomology ring (with integral coefficients) of the projective variety F. Each closed subvariety X of F determines an element [X]H*(F;), and cup-product in H*(F;) corresponds, roughly speaking, to intersection of subvarieties. In particular, for each wSn, we have a cohomology class [Xw]H*(F;), and it is a consequence of the cell decomposition (A.3) of F that the [Xw] form a -basis of H*(F;). In particular, [Xw0] is the identity element.

The connection between the classes [Xw] and the Schubert polynomials 𝔖w(wSn) is given by

(A.5) There is a surjective ring homomorphism

α:[x1,,xn] H*(F;)

such that

α(𝔖w)= [Xw0w]

for each wSn.


Let us temporarily write

σw= [Xw0w]

for wSn. Monk [Mon1959] proved that for all wSn and r=1,,n-1

(1) σw· σsr= tσwt

where the sum on the right hand side is over all transpositions t=tij such that ir<jn and (wt)=(w)+1, as in (4.15''),

Define ξ1,,ξnH*(F,) by

ξ1 = σ1 ξi = σi-σi-1 (2in-1) ξn = -σn-1

From (1) we deduce the counterpart of (4.16): if r is the last descent of w (so that rn-1), then we have

(2) σw=σvξr+ wσw

where v,w are as in (4.16). Now iteration of (4.16) will ultimately express 𝔖w as a sum of monomiais, i.e. as a polynomial in x1,,xn-1; and iteration of (2) will express σw as the same polynomial in ξ1,,ξn-1. Hence if we define α:PnH*(F;) by α(xi)=ξi (1in), we have σw=α(𝔖w) for all wSn, and the proof of (A.5) is complete.

In fact the kernel of the homomorphism α is generated by the elementary symmetric functions e1,,en of the x's.

We shall draw one consequence of (A.5) that we have not succeeded in deriving directly from the definition (4.1) of the Schubert polynomials. Since the σw,wSn, form a -basis of H*(F;), any product σuσv(u,vSn) is uniquely a linear combination of the σw, and it follows from intersection theory on F that the coefficient of σw in σuσv is a non-negative integer. From this we deduce

(A.6) Let u,v be permutations, and write 𝔖u𝔖v as an integral linear combination of the 𝔖w, say

(1) 𝔖u𝔖v=w cuvw𝔖w.

Then the coefficients cuvw are non-negative.

We have only to choose n sufficiently large so that u,v and all the permutations w such that cuvw0 lie in Sn, and then apply the homomorphism α of (A,5).

Remark. The coefficients cuvw in (A.6) are zero unless

(a) (w)=(u)+ (v),
(b) uw and vw.

For 𝔖u𝔖v is homogeneous of degree (u)+(v), which gives condition (a). Also we have

cuvw = w(𝔖u𝔖v) = v1wv1 w/v1 (𝔖u)v1 (𝔖v)

by (2.17), and the only possible nonzero term in this sum is that corresponding to v1=v. Hence if cuvw0 we must have vw, and by symmetry also uw.

Notes and References

This is a typed excerpt of the book Notes on Schubert Polynomials by I. G. Macdonald.

page history