Notes on Schubert Polynomials
Chapter 6

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

Last update: 2 July 2013

Double Schubert Polynomials

Let x=(x1,,xn), y=(y1,,yn) be two sequences of independent indeterminates, and recall (5.8) that

Δ(x,y)= i+jn (xi-yj).

For each wSn, we define the double Schubert polynomial 𝔖w(x,y) to be

(6.1) 𝔖w(x,y)= w-1w0 Δ(x,y)

where w-1w0 acts on the x variables.

Since Δ(x,0)=xδ we have

(6.2) 𝔖w(x,0)= 𝔖w(x),

the (single) Schubert polynomial indexed by w.

From the Cauchy formula (5.10) we have

𝔖w(x,y)= vSn w-1w0 𝔖vw0(x) 𝔖v(-y)

and by (4.2)

w-1w0 𝔖vw0(x)= 𝔖vw(x)

if (vw)=(vw0)-(w-1w0), i.e. if (vw)=(w)-(v), and

w-1w0 𝔖vw0(x)=0

otherwise. Hence

(6.3) 𝔖w(x,y)= u,v𝔖u (x)𝔖v(-y)

summed over all u,vSn such that w=v-1u and (w)=(u)+(v).

From (6.3) it follows that 𝔖w(x,y) is a homogeneous polynomial of degree (w) in x1,,xn-1, y1,,yn-1. We have

(i) 𝔖w0(x,y)= Δ(x,y),
(ii) 𝔖1(x,y)=1,
(iii) 𝔖w-1(x,y)= 𝔖w(-y,-x)= ε(w)𝔖w(y,x) for all wSn,
(iv) 𝔖w(x,x)=0 for all wSn except w=1.


(i) is immediate from the definition (6.1).

(ii) and (iii) follow from (6.3).

(iv) follows from (5.20), since 𝔖w(x,x)= θ(w-1w0Δ) =0 if w1.

(6.5) (Stability) If m>n and i is the embedding of Sn in Sm, then

𝔖i(w)(x,y) =𝔖w(x,y)

for all wSn.


This again follows from (6.3) and the stability of the single Schubert polynomials (4.5).

From (6.5) it follows that the double Schubert polynomials 𝔖w(x,y) are well defined for all permutations wS.

For any commutative ring K, let K(S) denote the K-module of all functions on S with values in K. We define a multiplication in K(S) as follows: for f,gK(S),

(fg)(w)= u,vf(u) g(v)

summed over all u,vS such that uv=w and (u)+(v)=(w). For this multiplication, K(S) is an associative (but not commutative) ring, with identity element 1__, the characteristic function of the identity permutation 1. It carries an involution ff*, defined by

f*(w)= f(w-1)

which satisfies

(fg)*= g*f*

for all f,gK(S).

(6.6) Let f,gK(S).

(i) If fg=f and f(1) is not a zero divisor in K, then g=1__.
(ii) If fg=1__, then gf=1__.
(iii) f is a unit (i.e. invertible) in K(S) if and only if f(1) is a unit in K.


(i) We have f(1)=f(1)g(1) and hence g(1)=1. We shall show by induction on (w) that g(w)=0 for all w1. So let r>0 and assume that g(v)=0 for all vS such that 1(v)r-1. Let w be a permutation of length r. We have

(1) f(w)=(fg) (w)=f(w) g(1)+f(1) g(w)+u,v f(u)g(v)

where the sum on the right is over u,vS such that u1, v1, uv=w and (u)+(v)=(w), so that 1(v)r-1 and therefore g(v)=0. Hence (1) reduces to f(1)g(w)=0 and therefore g(w)=0 as required.

(ii) We have f(1)g(1)=1 so that f(1) is a unit in K. Also f(gf)=(fg)f=f, whence gf=1__ by (i) above.

(iii) Suppose f is a unit in K(S), with inverse g. Since is fg=1__ we have f(1)g(1)=1, whence f(1) is an unit in K.

Conversely, if f(1) is an unit in K we construct an inverse g of f as follows. We define g(1)=f(1)-1 and proceed to define g(w) by induction on (w). Assume that g(v) has been defined for all v such that (v)<(w) and set

g(w)=-f (1)-1 u,vf(u) g(v)

summed over u,v such that uv=w, vw and (u)+(v)=(w). This definition gives (fg)(w)=0 as required.

Now let 𝔖(x) (resp. 𝔖(x,y)) be the function on S whose value at a permutation w is 𝔖w(x) (resp. 𝔖w(x,y)). (The coefficient ring K is now the ring [x,y] of polynomials in the x's and y's.) Since 𝔖1(x)=𝔖1(x,y)=1, it follows from (6.6)(iii) that 𝔖(x) and 𝔖(x,y) are units in K(S).

(i) 𝔖(x,0)= 𝔖(x),
(ii) 𝔖(x,x)= 1__,
(iii) 𝔖(x,y)*= 𝔖(-y,-x),
(iv) 𝔖(x)-1= 𝔖(0,x),
(v) 𝔖(x)*= 𝔖(-x)-1,
(vi) 𝔖(x,y)= 𝔖(y)-1𝔖(x) =𝔖(y,x)-1.


(i)-(iii) follow directly from (6.2) and (6.4).

From (6.3) and (6.4) we have

𝔖w(x,y)= u,v 𝔖u-1 (-y)𝔖v(x)= u,v𝔖u (0,y)𝔖v(x)

summed over u,vS such that uv=w and (u)+(v)=(w). In other words,

(1) 𝔖(x,y)=𝔖 (0,y)𝔖(x).

In particular, when y=x we obtain 𝔖(0,x)𝔖(x)=𝔖(x,x)=1__ by (ii) above, and hence 𝔖(0,x)=𝔖(x)-1. This establishes (iv); part (v) now follows from (iv) and (iii), and (vi) from (iv) and (1) above.

From (6.7) (vi) we have

𝔖(x)=𝔖(y) 𝔖(x,y)

or explicitly

𝔖w(x)=u,v 𝔖u(y)𝔖v (x,y)

summed over u,v such that uv=w and (u)+(v)=(w), so that u=wv-1 and 𝔖u=v𝔖w by (4.2). Hence

𝔖w(x)=v 𝔖v(x,y)v 𝔖w(y)

(where the operators v act on the y variables). The sum here may be taken over all permutations v, since v𝔖w=0 unless (wv-1)=(w)-(v). By linearity and (4.13) it follows that

(6.8) (Interpolation Formula) For all fPn=[x1,,xn] we have

f(x)=w𝔖w (x,y)wf(y)

summed over permutations wS(n).

(The reason for the restriction to S(n) in the summation is that if wS(n) we shall have w(m)>w(m+1) for some m>n, and hence w=vm where v=wsm; but mf=0 for all fPn, since m>n, and therefore wf=0.)

Remarks. 1. By setting each yi=0 in (6.8) we regain (4.14).

2. When n=1, the sum is over S(1), which consists of the permutations wp=spsp-1s1 (p0); wp is dominant, of shape (p), so that (see (6.15) below) 𝔖wp(x,y)=(x-y1)(x-yp). Hence the case n=1 of (6.8) is Newton's interpolation formula

f(x)=p0 (x-y1) (x-yp)fp (y1,,yp+1)

where fp=pp-11f, or explicitly

fp(y1,,yp+1)= i=1p+1 f(yi) ji (yi-yj) .

For any integer r, let 𝔖w(x,r) denote the polynomial obtained from 𝔖w(x,y) by setting y1=y2==r. Since

𝔖w0(x,r) = Δ(x,r)= i=1n-1 (xi-r)n-i = 𝔖w0(x-r)

where x-r means (x1-r,x2-r,), it follows from the definitions (6.1) and (4.1) that

𝔖w(x,r)=𝔖w (x-r)

for all permutations w. Hence, by (6.7)(vi),

𝔖(x-r)=𝔖 (r)-1𝔖 (x)

and in particular, for all integers q,

𝔖(q-r)=𝔖 (r)-1𝔖(q)

from which it follows that

(6.9) 𝔖(r)=𝔖(1)r

for all r.

Since 𝔖w(x) is a sum of monomials with positive integral coefficients (4.17), 𝔖w(1) is the number of monomials in 𝔖w(x) (each monomial counted the number of times it occurs). By homogeneity, we have

(6.10) 𝔖w(r)= r(w) 𝔖w(1).

From (6.7)(v) and (6.9) we obtain

𝔖(1)*=𝔖 (-1)-1= 𝔖(1)

so that we have another proof of the fact (4.30) that 𝔖w(1)=𝔖w-1(1).

Now consider the function F=𝔖(1)-1__, whose value at wS is

F(w)= { numbers of monomials in𝔖w, ifw1, 0, ifw=1.

For each positive integer p we have

(1) Fp = (𝔖(1)-1__)p = r=0p (-1)r (pr) 𝔖(1)r = r=0p (-1)r (pr) 𝔖(1)

by (6.9). The value of (1) at a permutation w of length p is by (6.10) equal to

( r=0p (-1)r (pr) rp ) 𝔖w(1)

which is equal to p!𝔖w(1) (consider the coefficient of tp in (et-1)p). On the other hand, Fp(w) is by definition equal to

(2) w1,,wp F(w1)F(wp)

summed over all sequences (w1,,wp) of permutations such that w1wp=w,(w1)++(wp)=(w)=p, and wi1 for 1ip. It follows that each wi has length 1, hence wi=sai say, and that (a1,,ap) is a reduced word for w. Since

𝔖sa=x1++ xa

by (4.4). we have F(wi)=𝔖sai (1)=ai, and hence the sum (2) is equal to a1a2ap summed over all (a1,,ap)R(w).

We have therefore proved that

(6.11) The number of monomials in 𝔖w is

𝔖w(1)=1p! a1a2ap

summed over all (a1,,ap)R(w), where p=(w).

Remarks. 1. The reduced words for 1m×w (m1) are (m+a1,,m+ap) where (a1,,ap)R(w). Hence from (6.11) and homogeneity we have

𝔖1m×w(1m) =1p! (1+a1m) (1+apm)

summed over R(w) as before. Letting m, we deduce that

(6.12) CardR(w)=p! limm 𝔖1m×w (1m).

2. If w is dominant of length p, then 𝔖w is a monomial by (4.7). and hence in this case

R(w)a1 ap=p!

3. Suppose that w is vexillary of length p. Then by (4.9) we have

𝔖w=sλ (Xϕ1,,Xϕr)

where λ is the shape of w and ϕ=(ϕ1,,ϕr) the flag of w. Hence

𝔖1m×w=sλ ( Xϕ1+m,, Xϕr+m )

for each m1. If we now set each xi=1m and then let m, we shall obtain in the limit the Schur function sλ for the series et ([Mac1979], Ch. I. §3. Ex. 5). which is equal to h(λ)-1, where h(λ) is the product of the hook-lengths of λ. Hence it follows from (6.12) that if w is vexillary of length p, then

(6.13) CardR(w)= p!h(λ)

where λ is the shape of w. In other words. the number of reduced words for a vexillary permutation of length p and shape λp is equal to the degree of the irreducible representation of Sp indexed by λ.

4. It seems likely that there is a q-analogue of (6.11). Some experimental evidence suggests the following conjecture:

(6.11q?) 𝔖w(1,q,q2,) =qϕ(a) (1-qa1) (1-qap) (1-q) (1-qp)

summed as in (6.11) over all reduced words a=(a1,,ap) for w, where

ϕ(a)= {i:ai<ai+1}.

When w is vexillary the double Schubert polynomial 𝔖w(x,y) can be expressed as a multi-Schur function, just as in the case of (single) Schubert polynomials (Chap. IV). We consider first the case of a dominant permutation:

(6.14) If w is dominant of shape λ, then

𝔖w(x,y) = (i,j)λ (xi-yj) = sλ ( X1-Yλ1,, Xm-Yλm )

where m=(λ) and Xi=x1++xi, Yi=y1++yi for all i1.


As in (4.6) we proceed by descending induction on (w),wSn. The result is true for w=w0, since w0 is dominant of shape δ and

𝔖w0(x,y)= Δ(x,y)= (i,j)δ (xi-yj).

Suppose ww0 is dominant of shape λ. Then λδ (and λδ). Let r0 be the largest integer such that λi=n-i for 1ir, and let a=λr+1+1n-r-1. Then wsa is dominant, (wsa)=(w)+1, and λ(wsa)=λ(w)+εa, and therefore

𝔖w(x,y) = a𝔖wsa (x,y) = a ( (xa-yr+1) (i,j)λ (xi-yj) )

by the inductive hypothesis; since λa=λa+1 it follows that

𝔖w(x,y)= (i,j)λ (xi-yj)

which is equal to sλ(X1-Yλ1,,Xm-Yλm) by (3.5).

(6.15) If w is Grassmannian of shape λ then

𝔖w(x,y)=sλ ( Xm-Yλ1+m-1, , Xm-Yλm ) .


This follows from (6.14) just as (4.8) follows from (4.7).

Finally, let w be vexillary with shape

λ(w)= ( p1m1,, pkmk )

and flag

ϕ(w)= ( f1m1,, fkmk )

as in Chapter IV. Then w-1 is also vexillary, with shape

λ(w-1)=λ (w)= ( q1n1,, qknk )

the conjugate of λ(w), and flag

ϕ(w-1)= ( g1n1,, gknk )

where by (1.41)

gi+qi= fk+1-i+ pk+1-i (1ik).

With this notation recalled, we have

(6.16) 𝔖w(x,y)=sλ ( (Xf1-Ygk)m1 ,, (Xfk-Yg1)mk ) .


The proof is essentially the same as that of (4.9) (which is the case y=0). By (4.10) the dominant permutation wk constructed from w in the proof of (4.9) has shape

μ= ( gkm1, gk-1m2,, g1mk )

and therefore by (6.15) we have

𝔖wk(x,y)=sμ ( X1,, Xm )

where m=m1++mk=(λ) and the sequence (X1,,Xm) is obtained by subtracting the sequence ( (Ygk)m1,, (Yg1)mk ) term by term from the sequence (X1,,Xm). Hence the same argument as in (4.9) establishes (6.17).

Remark. From (6.16) and (6.4)(iii) we obtain

sλ ( Z1m1,, Zkmk ) = (-1)|λ| sλ ( (-Zk)n1,, (-Z1)nk )

where Zi=Xfi-Ygk+i-1 so that (if rk(xi)=rk(yi)=1 for each i1)

rk (Zi+1-Zi) = fi+1-fi+ gk+1-i- gk-i = mi+1- nk+1-i

by (1.41). Hence (6.4)(iii) reduces to the duality theorem (3.8'') (with μ=0) when w is vexillary.

Let τx (resp. τy) be the shift operator (4.21) acting on the x (resp. y) variables. Then we have

(6.17) τxrτyr𝔖w (x,y)= 𝔖1r×w(x,y)

for all r1 and all permutations w.


By (6.3) and (4.21) we have

τxrτyr𝔖w (x,y)=u,v ε(v)𝔖1r×u (x)𝔖1r×v(y)

summed over u,v such that v-1u=w and (u)+(v)=(w). By (6.3) again, the right-hand side is equal to 𝔖1r×w(x,y).

In particular, suppose that w is vexillary. With the notation of (6.16), the flag of 1r×w (resp. 1r×w-1) is obtained from that of w (resp. w-1) by replacing each fi by fi+r (resp. each gi by gi+r). Hence by (6.16) we have

𝔖1r×w(x,y) =sλ ( (Xf1+r-Ygk+r)m1,, (Xfk+r-Yg1+r)mk )

and hence

(6.18) ρr(x) ρr(y) 𝔖1r×w (x,y)=sλ (Xr-Yr)

for all r1, where ρr(x) (resp. ρr(y)) is the homomorphism ρr of (4.25) acting on the x (resp. y) variables.

(6.19) Let πx (resp. πy) denote πw0(r) acting on the x (resp. y) variables. Then if w is vexillary of shape λ, we have

πxπy𝔖w (x,y)=sλ (Xr-Yr).


By (4.24) we have πx=ρr(x)τxr and πy=ρy(r)τyr. Hence (6.19) follows from (6.17) and (6.18).

In particular, suppose that w is dominant of shape λ, so that by (6.14)

𝔖w(x,y)= (i,j)λ (xi-yj)= fλ(x,y) say.

In this case (6.19) gives

πxπyfλ (x,y)=sλ (Xr-Yr)

for all r1, which is Sergeev's formula (3.12').

Notes and References

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

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