Notes on Schubert Polynomials
Chapter 2

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

Last update: 27 June 2013

Divided differences

If f is a function of x and y (and possibly other variables), let

xyf= f(x,y)-f(y,x) x-y

("divided difference"). Equivalently

xyf= (x-y)-1 (1-sxy)

where sxy interchanges x and y. The operator xy takes polynomials to polynomials, and has degree -1 (i.e., if f is homogeneous of degree d, then xyf is homogeneous of degree d-1). Explicitly, if f=xrys we have

(2.1) xy(xrys) = xrys-xsyr x-y = σ(r-s) xpyq

where the sum is over (p,q) such that p+q=r+s-1 and max(p,q)<max(r,s), and σ(r-s) is +1,0 or -1 according as r-s is positive, zero or negative.

On a product fg,xy acts according to the rule

(2.2) xy(fg)= (xyf)g+ (sxyf) (xyg).

In particular we have

(2.2') xy(fg)= fxyg

if f(x,y)=f(y,x).


(i) xysxy= -xy, sxyxy= xy,
(ii) xy2=0,
(iii) xyyzxy= yzxyyz.


(i) and (ii) are immediate from the definitions, and (iii) is verified by direct calculation: each side is equal to

(x-y)-1 (x-z)-1 (y-z)-1 wS3 ε(w)w,

where the symmetric group S3 permutes x,y and z, and ε(w) is the sign of the permutation w.

Let x1,x2,,xn, be independent variables, and let

Pn= [ x1,x2,, xn ]

for each n1, and

P = [x1,x2,] = n=1Pn.

For each i1 let


Each i is a linear operator on P (and on Pn for n>i) of degree -1. From (2.3) we have (compare with (1.1))

(2.4) { i2=0, ij= ji if|i-j|>1, ii+1 i=i+1 ii+1

For any sequence a=(a1,,ap) of positive integers, we define

a=a1 ap.

Recall that if w is any permutation, R(w) denotes the set of reduced words for w, i.e. sequences (a1,,ap) such that w=sa1sap and p=(w).

(2.5) If a,bR(w) then a=b.


We proceed by induction on p=(w). Let us write ab to mean that a=b. The inductive hypothesis then implies that

(*) ab if eithera1=b1 orap= bp.

By the exchange lemma (1.8) we have

ci= ( b1,a1,, aˆi,,ap ) R(w)

for some i=1,,p. If ip then bcia by virtue of (*), so that ab. If i=p and |b1-a1|>1 then by (2.4) and (1.1)

cp= ( a1,b1,a2, ,ap-1 ) R(w)

and acpcpb, so that again ab.

Finally, if i=p and |b1-a1|=1, we apply the exchange lemma again, this time to cp and a; this shows that

di= ( a1,b1,a1,, aˆi,,ap-1 ) R(w)

for some i=2,,p-1. But then by (2.4) and (1.1) we have

di= ( b1,a1,b1, a2,,aˆi, ,ap-1 ) R(w)

and adidib. Hence ab in all cases.

Remark. For any permutation w, let GR(w) denote the graph whose vertices are the reduced words for w, and in which a reduced word a is joined by an edge to each of the words obtained from a by either interchanging two consecutive terms i,j such that |i-j|>1, or by replacing three consecutive terms i,j,i such that |i-j|=1 by j,i,j. Then the proof of (2.5) shows that

(2.5') The graph GR(w) is connected.

From (2.5) it follows that we may define


unambiguously, where a is any reduced word for w. By (2.2'), the operators w for wSn are Λn linear, where

Λn= [x1,,xn] Sn Pn

is the ring of symmetric polynomials in x1,,xn.

A sequence a=(a1,,ap) will be said to be reduced if aR(w) for some permutation w.

(2.6) If a=(a1,,ap) is not reduced, then a=0.


By induction on p. If a=(a1,,ap-1) is not reduced, then a=0 and hence a=aap=0. So we may assume that a is reduced. Let v=sa1sap-1, w=sa1sap. We have (v)=p-1 and (w)p-1, hence by (1.3) (w)=p-2, so that (v)=(wsap)=(w)+1. Consequently v=wap and therefore a= vap= wap2= 0.

(2.7) Let u,v be permutations. Then

uv= { uv if(uv)= (u)+(v) , 0 otherwise.


(2.5), (2.6).

(2.8) Let w be a permutation, i1. Then

siw=w (siw)=(w) -1.


We have siw=wiw=0, hence the result follows from (2.7).

As before let w0=(n,n-1,,2,1) be the longest element of Sn. One element of R(w0) is the sequence

(2.9) ( 1,2,,n-1,1, 2,,n-2,,1, 2,3,1,2,1 ) .

(2.10) We have

w0=aδ-1 wSnε(w) w

where a=1i<jn(xi-xj), and ε(w)=±1 is the sign of w.


From the definition it follows that w0 is of the form

(1) w0=wSn cww

with coefficients cw rational functions of x1,,xn. By (2.8) we have siw0=w0 for 1in-1, so that vw0=w0 for all vSn, and therefore

(2) w0=wSn v(cw)vw.

Comparison of (1) and (2) shows that

(3) cvw=v(cw) (v,wSn).

Hence all the coefficients cw are determined by one of them, say cw0. From the sequence (2.9) for w0 it is easily checked that the coefficient of w0 in 0 is

cw0=ε (w0)aδ-1.

Hence from (3) we have

cw=ww0 (cw0)=ε (w)aδ-1

which proves (2.10).

From (2.10) it follows that, for any α=(α1,,αn)n,

(2.11) w0xα= sα-δ (x1,,xn)

where xα means x1α1xnαn, δ=(n-1,n-2,,1,0) and sα-δ is the Schur function indexed by α-δ. Thus w0 is a Λn-linear mapping of Pn onto Λn.

For wSn, let w=w0ww0. Then

(2.12) w=ε (w)w0w w0.


From the definition of i we have

w0iw0=- n-i

from which (2.12) follows easily, since w02=1.

If f and g are polynomials in x1,x2,, the expression of w(fg) as a sum of polynomials uf·vg (i.e. the "Leibnitz formula" for w) is in general rather complicated. However, there is one case in which it is reasonably simple, namely when one of the factors f,g is linear:

(2.13) If f=αixi then

w(fg)=w(f) wg+(αi-αj) wtijg

summed over all pairs i<j such that (wtij)=(w)-1, where tij is the transposition that interchanges i and j.


Let (a1,,ap) be a reduced word for w. Since f is linear it follows from (2.2) that

w(fg) = a1ap (fg) = sa1sap (f)a1 apg+r=1p sa1ar sap(f)a1 ˆar apg.

Now a1ˆarap=0 unless (a1,,aˆr,,ap) is reduced, and then by (1.11) it is equal to wt, where wt=saps^arsap has length p-1=(w)-1, and t=sapsarsap=tij where (i,j)=sapsar+1(ar,ar+1), so that

sa1 sar-1 arsar+1 sap(f)= αi-αj.

We also introduce the operators πi(i1) defined by


In place of (2.4) we have

(2.14) { πi2=πi, πiπj= πjπi if|i-j|>1, πiπi+1 πi=πi+1 πiπi+1.

If we define πa to be πa1πap for any sequence a=(a1,,ap) of positive integers, then corresponding to (2.5) we have

(2.15) If a,bR(w) then πa=πb.

The proof is the same as that of (2.5), and rests only on the second and third of the relations (2.14). From (2.15) it follows that we may define


unambiguously, where a is any reduced word for w.

In place of (2.10) we have

(2.16) For any fPn,

πw0f=aδ-1 wSnε(w) w(xδf)=w0 (xδf).

In particular, if αn,

(2.16') πw0xα=sα (x1,,xn).


We have

π1f=1 (x1f), π1π2f= 1 (x12(x2f)) =12(x1x2f)

and generally

π1πrf=1 r(x1xrf)

for each r1. From this and (2.10) it follows easily that πw0f=w0(xδf).

Let (a1,,ap) be a reduced word for w. Then

w = a1ap = (xa1-xa1+1)-1 (1-sa1) (xa2-xa2+1)-1 (1-sa2)

which shows on expansion that w is of the form

w=vw fvwv

where fvw are rational functions of x1,x2,, and in particular (by (1.7))

fww=(-1)p (i,j)I(w-1) (xi-xj)-1

and thus is 0. It follows that the w are linearly independent over the field of rational functions =(x1,x2,).

Now from (2.2) we have

a(fg)= (af)g+ (saf) (ag)

or equivalently, if μ:PPP is the multiplication map,

aμ=μ ( a1+saa ) .

From this it follows that

wμ=μ ( a11+sa1 a1 ) ( ap1+sap ap )

On expansion this is a sum over subsequences b of a=(a1,,ap), say

(1) wμ=μ ba ϕ(a,b) b


ϕ(a,b)= ϕ1(a,b) ϕp (a,b)


ϕi(a,b)= { sa1 ifaib, a1 ifaib.

Since b=0 if b is not reduced (2.6), the sum is over reduced subsequences b of a, and by (1.17) we can write

(2) wμ=μ vwvw/v v

where for vw

(3) w/v=v-1 ϕ(a,b)

summed over subsequences ba such that b is a reduced word for v.

So for each pair of permutations w,v such that wv we have a well-defined operator w/v on P, defined by (3). Since the v are linearly independent, the definition (3) is independent of the reduced word aR(w).

(2.17) For each pair w,vS such that wv there is a linear operator w/v on P such that

w(fg)=vw v(w/vf)· vg.

w/v has degree -(w)+(v).



Let v=w, then

w/w=w-1 ϕ(a,a)= w-1sa1 sap=1.

Let v=1, then

w/1=ϕ (a,)=a1 ap=w.

Suppose that vw, so that v=sa1ŝarsap for an unique r[1,p]. Then b=(a1,,aˆr,,ap) and

w/v = v-1ϕ (a,b) = v-1sa1 sar-1 ar sar+1 sap = sapsar+1 arsar+1 sap

Now w=vt where t is the transposition

t=tij=sap sarsap (i<j)

so that (i,j)=sapsar+1(ar,ar+1) and therefore

w/v = sap sar+1 (xar-xar+1)-1 (1-sar) sar+1 sap = (xi-xj)-1 (1-tij)

is the divided difference operator xi,xj.

The product formula for w/u is

(2.18) w/u(fg)= uvw u-1v (w/vf) w/ug.


We have

w(fgh)= uwu w/u(fg) uh (1)

and on the other hand

w(fgh) = uwv w/u(f) v(gh) = uvwv w/v(f) ·uv/u(g) ·uh.

Comparison of (1) and (2) gives

uw/u(fg)= uvwv w/u(f)· uv/u(g)

which gives the result.

When u=1, this reduces to (2.17).

Notes and References

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

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