Notes on Schubert Polynomials
Chapter 7

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

Last update: 2 July 2013

Schubert Polynomials (2)

Recall the decomposition (4.17) of a Schubert polynomial 𝔖w:

𝔖w(x1,x2,) =u,vduvw 𝔖u (x1,,xm) 𝔖v (xm+1,xm+2,)

Our first aim in this Chapter will be to give a method for calculating the coefficients duvw. We shall then apply our results to the calculation of Card(R(w)), the number of reduced decompositions w=sa1sap (where p=(w)) of a permutation w.

For this purpose, we introduce the operators i*, i1, defined by

(7.1) i*𝔖w= { 𝔖siw if(siw) <(w), 0 otherwise.

Remarks. 1. If ω is the (linear) involution defined by ω(𝔖w)=𝔖w-1 for each permutation w, it follows from (4.2) that i*=ωiω. Hence we may define w*=ωwω for any permutation w, and we have w*=a1*ap*. whenever (a1,,ap) is a reduced word for w.

2. If wSn we have i*𝔖w=0 for all i>n, because i*𝔖w= ωi𝔖w-1, which is zero because w-1(i)<w-1(i+1).

(7.2) i* commutes with j for all i,j1.


We have

i*j𝔖w= { i*𝔖wsj= 𝔖siwsj if (siwsj)= (w)-2, 0 otherwise.


ji*𝔖w= { j𝔖siw= 𝔖siwsj if (siwsj)= (w)-2, 0 otherwise.

Hence i*j-ji* vanishes on each Schubert polynomial 𝔖w, and therefore vanishes identically.

(7.3) Let w0=w0(n) be the longest element of Sn. Then for r=1,2,,n-1 we have

(1+tn-r*) (1+tn-1*) 𝔖w0= (1+t1) (1+tr) 𝔖w0

as polynomials in t,x1,x2,.


The coefficient of tp (1pr) on the left-hand side is

(1) a1* ap*𝔖w0

summed over all reduced sequences (a1,,ap) satisfying

n-ra1 apn-1.

Let bi=n-ap+1-i for all 1ip, so that

(2) 1b1<< bpr.

Let w=sapsa1, so that w0ww0=sb1sbp. Then

a1* ap*𝔖w0 = 𝔖w-1w0= w0ww0 𝔖w0 = b1 bp𝔖w0.

Hence (1) is equal to

b1 bp𝔖w0

summed over all reduced sequences (b1,,bp) satisfying (2), which is the coefficient of tp on the right hand side of (7.2).

Next, we have

(7.4) 𝔖1×w0 ( t,x1,, xn-1 ) =(1+t1) (1+tn-1) 𝔖w0 (x1,,xn-1) .

By (4.22) we have to show that

(1+t1) (1+tn-1) sδ(X1,,Xn-1) =sδ ( t+X1,, t+Xn-1 )

where Xi=x1++xi for each i1, and δ=δn. For this it is enough to show that

(1) (1+ti)sδ ( X1,,Xi,t+ Xi+1,,t+ Xn-1 ) =sδ ( X1,,Xi-1, t+Xi,,t+ Xn-1 )

for i=1,2,,n-1.

Both sides of (1) are determinants with n-1 rows and columns which agree in all the ith row. On the left-hand side, the elements of the ith row are by (3.10)

hk(Xi)+t hk-1(Xi+1)

and on the right-hand side they are hk(t+Xi), where k runs form n-2i+1 to 2n-2i-1 in each case.

Now we have

hk(Xi)+t hk-1 (Xi+1) = hk(t+Xi)-t hk-1 (t+Xi)+t hk-1 (t+Xi+1)- t2hk-2 (t+Xi+1) = hk (t+Xi)-t (t-xi+1) hk-2 (t+Xi+1)

Hence if we add t(t-xi+1) times the (i+1)th row to the ith row in the determinant on the left-hand side, we shall obtain the right-hand side of (1).

For each r1, let

Φr(t)= tr (1+tr+1*) (1+tr+2*)

For each permutation w, we have (1+tj*)𝔖w=𝔖w for all sufficiently large j by (7.1), so that Φr(t)𝔖w is a polynomial in t (and x1,x2,). With this notation, we have

(7.5) 12 n-r+1 ( x1n x2n-1 xn ) =Φr-1 (x1) 𝔖w0(n) (x2,x3,)


Let s=n-r+1 and

a=x2s-1 x3s-2 xs, b=xs+2r-2 xs+3r-3 xn,c= (x2xs+1)r-1

so that abc=x2n-1x3n-2xn. Hence

1 2 n (x1nx2n-1xn) = x1r-1bc 1s (x1sx2s-1xs) = x1r-1bc 𝔖1×w0(s) (x1,,xs) by (4.21) = x1r-1bc (1+x12) (1+x1s)a by (7.4) = x1r-1 (1+x12) (1+x1s)abc = x1r-1 (1+x12) (1+x1s) 𝔖w0(n) (x1,,xn) = x1r-1 (1+x1r*) (1+x1n-1*) 𝔖w0(n) (x2,,xn) by (7.3).

Let w be any permutation. If w(1)=r, then s1sr-1w(1)=1, so that we may write

s1sr-1w= 1×w1

where w1 is defined by

w1(i)= { w(i+1) ifw(i+1)<r, w(i+1)-1 ifw(i+1)>r.

If the code of w is (c1,c2,) (so that c1=r-1), the code of w1 is (c2,c3,). With this notation we have

(7.6) 𝔖w(x1,x2,) =Φr-1(x1) 𝔖w1(x2,x3,)


Suppose that wSn+1. Then

w0(n+1)w = w0(n+1) sr-1s1 (1×w1) = sn-r+2 snw0(n+1) (1×w0(n)) (1×w0(n)w1) = sn-r+1s1 (1×w0(n)w1)

since w0(n+1) (1×w0(n)) = snsn-1 s1. Hence

𝔖w (x1,,xn) = w-1w0(n+1) ( x1n x2n-1 nx ) = 1×w1-1w0(n) 1n-r+1 (x1nxn) = 1×w1-1w0(n) Φr-1(x1) 𝔖w0(n) (x2,x3,,xn) by (7.5) = Φr-1(x1) 1×w1-1w0(n) 𝔖w0(n) (x2,x3,,xn) by (7.2) = Φr-1(x1) 𝔖w1 (x2,x3,).

Remark. The right-hand side of (7.6) is a sum of terms of the form x1p𝔖u(x2,x3,). By applying (7.6) to each 𝔖u, and so on, we can decompose 𝔖w into a sum of monomials, and thus we have another proof of the fact (4.17) that 𝔖w is a polynomial in x1,x2, with positive integer coefficients.

Next, let m1 and assume that the permutation w statisfies

w(1)>w(2)> >w(m).

Define a partition μ=μ(w,m) of length m by

μi=w(i)- (m+1-i) (1im).

If wSm+n we have μ1n, hence μ(nm).

Also let

Φμ (x1,,xm)= Φμm (xm) Φμ2 (x2) Φμ1 (x1)

and let wm be the permutation whose code is (cm+1,cm+2,), where (c1,c2,) is the code of w.

With this notation established, we have

(7.7) 𝔖w(x)= xδmΦμ (x1,,xm) 𝔖wm (xm+1,xm+2,).


We proceed by induction on m; the case m=1 is (7.6). From (7.6) we have

𝔖w(x) = Φμ1+m-1 (x1)𝔖w1 (x2,x3,) = ux1μ1+m+p-1 𝔖uw1 (x2,x3,)

summed over all u=sa1sap, where

c1(w)+1=μ1 +ma1<<ap

and (uw1)=(w1)-p. The code of uw1 satisfies ci(uw1)=ci(w1) for 1im-1, and hence

(uw1)m-1= sa1-m+1 sap-m+1wm.

It follows that

ux1μ1+m+p-1 𝔖(uw1)m-1 ( xm+1, xm+2, ) =x1m-1 Φμ1 (x1)𝔖wm ( xm+1, xm+2, )

and therefore, by the inductive hypothesis,

𝔖w(x) = u x1μ1+m+p-1 x2m-2 xm-1Φμm (xm)Φμ2 (x2) 𝔖(uw1)m-1 (xm+1,xm+2,) = x1m-1 x2m-2 xm-1Φμm (xm) Φμ1(x1) 𝔖wm (xm+1,xm+2,) .

Finally, for any permutation w, let v be the unique element of Sm such that wv(1)>>wv(m), and let μ=μ(wv,m). We have (wv)=(w)+(v) and (wv)m=wm, so that by (7.7)

𝔖wv(x)= xδmΦμ (x1,xm) 𝔖wm ( xm+1, xm+2, ) .


(7.8) 𝔖w(x) = v𝔖wv(x) = v ( xδmΦμ (x1,,xm) ) 𝔖wm (xm+1,xm+2,).

Now by (4.14), for any polynomial fPm, we have

f=uS(m) η(uf)Su

where S(m) consists of the permutations whose codes have length m, and η(uf) is the constant term of the polynomial uf. Applying this to (7.8), we obtain our final result:

(7.9) 𝔖w(x)=u 𝔖u(x1,,xm) η(uv(xδmΦμ(x1,,xm))) 𝔖wm(xm+1,xm+2,)

summed over all uS(m) such that (uv)=(u)+(v).

For each such u, the constant term η(uv(xδmΦμ(x1,,xm))) is a polynomial in the (non-commuting) operators i* with integer coefficients. Hence (7.9) gives a decomposition of the Schubert polynomial 𝔖w(x) of the form

(7.10) 𝔖w(x)=u,v duvw𝔖u(y) 𝔖v(z),

where y=(x1,,xm) and z=(xm+1,xm+2,). If wS(m+n), so that 𝔖w(x)Pm+n, then uS(m) and vS(n) in this sum. From (4.18) we know that the coefficients duvw in (7.10) are 0.

In particular, if we apply (7.7) to a permutation of the form w0(m)×w, we shall obtain

(1) 𝔖w0(m)×w (x)=xδmΦ0 (x1,,xm) 𝔖w(xm+1,xm+2,).

On the other hand, by (4.6) we have

(2) 𝔖w0(m)×w =𝔖w0(m) 𝔖1m×w

and comparison of (1) and (2) gives

(7.12) 𝔖1m×w(x)= Φ0 (x1,,xm)𝔖w (xm+1,xm+2,).

By (4.3), 𝔖1m×w is symmetrical in x1,,xm. Hence so is the operator Φ0(x1,,xm), and we may therefore write Φ0 in the form

(7.13) Φ0 (x1,,xm)= λ,vαm (λ,v)sλ (x1,,xm) v*

summed over partitions λ of length m and permutations v, with integral coefficients αm(λ,v). From (7.12) and (7.13) we have

(7.14) 𝔖1m×w=λ,v αm(λ,v)sλ (x1,,xm) 𝔖vm (xm+1,xm+2,)

summed over λ of length m and v such that (vw)=(w)-(v). The Schur functions occuring here are precisely the Schubert polynomials 𝔖u, where u is Grassmannian with descent at m. Hence, by (4.18),

(7.15) The coefficients αm(λ,v) in (7.13) are 0.

Since Φ0(x1,,xm,0)= Φ0(x1,,xm) and sλ(x1,,xm,0)= sλ(x1,,xm) if (λ)m, it follows from (7.13) that

(7.16) αm+1(λ,v)= αm(λ,v)=α (λ,v)say

for all partitions λ such that (λ)m.

We may also calculate the operator Φ0(x1,,xm) as follows. For each integer p1 and each subset D of {1,2,,p-1} let

QD,p (x1,,xm) =xu1 xup

summed over all sequences (u1,,up) such that 1u1upm and uiui+1 whenever iD. Then QD,p(x1,,xm) is a homogeneous polynomial of degree p, and is zero if mCard(D).

Now let a=(a1,,ap) be a reduced word, so that (sa1sap)=p. The descent set of a is

D(a)= {i:ai>ai+1}.

We now define, for each permutation w,

Fw(x1,,xm)= aR(w) QD(a),(w) (x1,,xm),

a homogeneous polynomial of degree (w).

With these definitions we have

(7.17) Φ0(x1,,xm) =wFw (x1,,xm) w*.


Let a=(a1,,ap) be a reduced word. Since

Φ0(xi)= (1+xi1*) (1+xi2*)

it is clear from the definitions that the coefficient of a*=a1*ap* in Φ0(x1,,xm)=i=1mΦ0(xi) is just QD(a),p(x1,,xm). Hence

Φ0 (x1,,xm) = a QD(a),p (x1,,xm) a* = wFw (x1,,xm) w*.

Comparison of (7.17) and (7.13) now shows that Fw(x1,,xm) is a symmetric polynomial in x1,,xm, and that

(7.18) Fw(x1,,xm) = λαm (λ,w)sλ (x1,,xm) = ρm (𝔖1m×w-1) .

The sum in (7.18) is over partitions λ such that (λ)m and |λ|=(w). By (7.16) we have

Fw(x1,,xm,0) =Fw(x1,,xm)

and therefore we have a well defined symmetric function FwΛ, such that ρm(Fw)=Fw(x1,,xm) for all m0: namely

(7.19) Fw=λα (λ,w)sλ

where the sum is over partitions λ of (w), and α(λ,w)=αm(λ,w) for any m(λ).

Since the coefficient of x1xp in QD,p(x1,,xm) is 1 if mp, it follows that the coefficient of x1xp (where p=(w)) in Fw(x1,,xm) is equal to Card(R(w)) whenever m(w). On the other hand, the coefficient of x1xp in a Schur function sλ, where |λ|=p, is equal to fλ, the number of standard tableaux of shape λ, or equivalently the degree of the irreducible representation χλ of Sp indexed by the partition λ ([Mac1979], Ch.I, §7). It follows therefore from (7.19) that

(7.20) CardR(w)= |λ|=(w) α(λ,w) fλ.

Remark. Since the coefficients α(λ,w) are 0 by (7.15), the number of reduced words for w is always equal to the degree of an (in general reducible) representation of the symmetric group S(w). It is therefore natural to ask whether there is a "natural" action of this symmetric group on the -span (or perhaps -span) of the set R(w), with character λα(λ,w)χλ.

We shall conclude with some properties of the symmetric functions Fw and the coefficients α(λ,w).

(7.21) Let uSm, vSn. Then

Fu×v(x)= Fu(x)Fv (x).


By (7.18), we have for any N,

Fu×v (x1,,xN) = ρN (S1N×u-1×v-1) = ρN ( S1N×u-1 S1m+N×v-1 ) by (4.6) = ρN (S1N×u-1) ρN (ρm+N(S1m+N×v-1)) = Fu(x1,,xN) Fv(x1,,xN).

(7.22) Let wSn and let w=w0ww0, where w0 is the longest element of Sn. Then

Fw-1= Fw=ω Fw

where ω is the involution that interchanges sλ and sλ. In other words

α(λ,w-1)= α(λ,w)= α(λ,w)

for all partitions λ.

For the proof of (7.22) we require a lemma. If t is a standard tableau of shape λ, the descent set D(t) of t is the set of i such that i+1 lies in a lower row than i in the tableau t. We have

(7.23) sλ=t QD(t),p

where the sum is over the standard tableaux of shape λ, and p=|λ|.


In the notation of [Mac1979, Ch. I, §5], sλ is the sum of monomials xT where T runs through the (column-strict) tableaux of shape λ. Each such tableau T determines a standard tableau t, as follows. If a square in the jth column of the diagram of λ is occupied by the number i, replace i by the pair (i,j). Since T is column-strict the pairs (i,j) so obtained are all distinct. If we now order them lexiographically, (so that (i,j) precedes λ(i,j) if and only if either i<i, or i=i and j<j) and relabel them as 1,2,,p, we have a standard tableau t: say Tt. It follows easily that TtxT=QD(t),p, which proves the lemma.

If D is any subset of {1,2,,p-1}, let D denote the complementary subset, and let D*={p-i:iD}. From the definition of QD,p we have

(1) QD,p (xm,xm-1,,x1)= QD*,p (x1,,xm).

If a=(a1,,ap)R(w), let a=(n-a1,,n-ap) and a*=(n-ap,,n-a1). Then we have

(2) aR(w), a*R(w*),

where w*=(w)-1=w0w-1w0. Also

(3) D(a)= D(a), D(a*)=D (a)*.

Moreover, it t is a standard tableau we have

(4) D(t)= D(t)

where t is the transpose of t, obtained by reflecting t in the main diagonal. For iD(t) if and only if i+1 does not lie in a later column than i in the tableau t, that is to say if and only if iD(t).

Since Fw is symmetric, it follows from (1),(2), and (3) that

Fw(x1,,xm)= Fw(xm,,x1)= Fw*(x1,,xm)

and hence by (7.16) that Fw=Fw*.

From (7.23) and (4) above we have

ωsλ=sλ= tSt()αλ QD(t),p

for all partitions λ of p, where St(λ) is the set of standard tableaux of shape λ, and hence it follows from (2) and (3) and the definition of Fw that ωFw=Fw. Hence

ωFw-1= Fw*=Fw,

which completes the proof of (7.22).

(i) α(μ,w)=0 unless λ(w-1)μ λ(w).
(ii) α(μ,w)=1 if μ=λ(w-1) or μ=λ(w).
(iii) w is vexillary if and only if Fw is a Schur function.


(i) Suppose α(μ,w)0. Then the monomial xμ occurs in Fw, and hence there is a reduced word (a1,,ap) for w such that

(1) a1<< aμ1, aμ1+1<< aμ1+μ1,.

By (1.14) the code of w is

(2) c(w)= i=1p sapsai+1 (εai).

If w(1)=sapsaμ1+1, the sum of the first μ1 terms of this series is

w(1) ( εaμ1+ saμ1 (εaμ1-1) ++saμ1 sa2 (εa1) ) ,

and since a1<<aμ1 this is equal to

(3) w(1) ( εaμ1+ εaμ1-1 ++εa1 ) =V1say,

where V1 is a (0,1) vector (i.e., a vector with each component 0 or 1) of weight μ1. Likewise the sum of the next block of μ2 terms of the series (2) is a (0,1) vector V2 of weight μ2, and so on. Hence

c(w)=V1++ Vm

where m=(λ), and each Vi is a (0,1) vector of weight μi. Let V be the (0,1) matrix whose ith row is Vi, for i=1,2,,m. Then V has row sums μ1,,μm and column sums c1(w),c2(w), As in the proof of (1.26) it follows that μλ(w). Since α(μ,w)=α(μ,w-1) by (7.23), the same argument applied to μ and w-1 gives μλ(w-1) i.e., λ(w-1)μ.

(ii) Suppose now that μ=λ(w). Then there is only one (0,1) matrix V with row sums μi and column sums ci. Its first row V1 is εj summed over j such that cj0, i.e. such that there exists k>j with w(k)<w(j). From (3) it follows that

wV1=i=1μ1 εai+1

and therefore a1+1,,aμ1+1 are the terms of the sequence w that have a smaller element somewhere to the right, in increasing order of magnitude. Hence a1 has no smaller elements to the right of it, and therefore lies to the right of a1+1, so that (sa1w)=(w)-1. The same argument shows that (sa2sa1w)=(sa1w)-1 and so on. Hence if w1=saμ1sa1w we have (w1)=(w)-μ1, and λ(w1)=(μ2,μ3,). It follows by induction on (μ) that the word (a1,,ap) determined by the matrix V is reduced, and hence α(μ,w)=1 when μ=λ(w). By (7.23) it follows that α(μ,w)=1 when μ=λ(w-1).

(iii) This follows immediately from (i) and (ii), and the characterization (1.27) of vexillary permutations.

Notes and References

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

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