## Notes on Schubert PolynomialsChapter 7

Last update: 2 July 2013

## Schubert Polynomials (2)

Recall the decomposition (4.17) of a Schubert polynomial ${𝔖}_{w}\text{:}$

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

Our first aim in this Chapter will be to give a method for calculating the coefficients ${d}_{uv}^{w}\text{.}$ We shall then apply our results to the calculation of $\text{Card} \left(R\left(w\right)\right),$ the number of reduced decompositions $w={s}_{{a}_{1}}\dots {s}_{{a}_{p}}$ (where $p=\ell \left(w\right)\text{)}$ of a permutation $w\text{.}$

For this purpose, we introduce the operators ${\partial }_{i}^{*},$ $i\ge 1,$ defined by

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

Remarks. 1. If $\omega$ is the (linear) involution defined by $\omega \left({𝔖}_{w}\right)={𝔖}_{{w}^{-1}}$ for each permutation $w,$ it follows from (4.2) that ${\partial }_{i}^{*}=\omega {\partial }_{i}\omega \text{.}$ Hence we may define ${\partial }_{w}^{*}=\omega {\partial }_{w}\omega$ for any permutation $w,$ and we have ${\partial }_{w}^{*}={\partial }_{{a}_{1}}^{*}\dots {\partial }_{{a}_{p}}^{*}\text{.}$ whenever $\left({a}_{1},\dots ,{a}_{p}\right)$ is a reduced word for $w\text{.}$

2. If $w\in {S}_{n}$ we have ${\partial }_{i}^{*}{𝔖}_{w}=0$ for all $i>n,$ because ${\partial }_{i}^{*}{𝔖}_{w}=\omega {\partial }_{i}{𝔖}_{{w}^{-1}},$ which is zero because ${w}^{-1}\left(i\right)<{w}^{-1}\left(i+1\right)\text{.}$

(7.2) ${\partial }_{i}^{*}$ commutes with ${\partial }_{j}$ for all $i,j\ge 1\text{.}$

 Proof. We have $∂i*∂j𝔖w= { ∂i*𝔖wsj= 𝔖siwsj if ℓ(siwsj)= ℓ(w)-2, 0 otherwise.$ Likewise $∂j∂i*𝔖w= { ∂j𝔖siw= 𝔖siwsj if ℓ(siwsj)= ℓ(w)-2, 0 otherwise.$ Hence ${\partial }_{i}^{*}{\partial }_{j}-{\partial }_{j}{\partial }_{i}^{*}$ vanishes on each Schubert polynomial ${𝔖}_{w},$ and therefore vanishes identically. $\square$

(7.3) Let ${w}_{0}={w}_{0}^{\left(n\right)}$ be the longest element of ${S}_{n}\text{.}$ Then for $r=1,2,\dots ,n-1$ we have

$(1+t∂n-r*)… (1+t∂n-1*) 𝔖w0= (1+t∂1)… (1+t∂r) 𝔖w0$

as polynomials in $t,{x}_{1},{x}_{2},\dots \text{.}$

 Proof. The coefficient of ${t}^{p}$ $\left(1\le p\le r\right)$ on the left-hand side is $(1) ∑∂a1*… ∂ap*𝔖w0$ summed over all reduced sequences $\left({a}_{1},\dots ,{a}_{p}\right)$ satisfying $n-r≤a1≤…≤ ap≤n-1.$ Let ${b}_{i}=n-{a}_{p+1-i}$ for all $1\le i\le p,$ so that $(2) 1≤b1<…< bp≤r.$ Let $w={s}_{{a}_{p}}\dots {s}_{{a}_{1}},$ so that ${w}_{0}w{w}_{0}={s}_{{b}_{1}}\dots {s}_{{b}_{p}}\text{.}$ Then $∂a1*… ∂ap*𝔖w0 = 𝔖w-1w0= ∂w0ww0 𝔖w0 = ∂b1… ∂bp𝔖w0.$ Hence (1) is equal to $∑∂b1… ∂bp𝔖w0$ summed over all reduced sequences $\left({b}_{1},\dots ,{b}_{p}\right)$ satisfying (2), which is the coefficient of ${t}^{p}$ on the right hand side of (7.2). $\square$

Next, we have

$(7.4) 𝔖1×w0 ( t,x1,…, xn-1 ) =(1+t∂1)… (1+t∂n-1) 𝔖w0 (x1,…,xn-1) .$

By (4.22) we have to show that

$(1+t∂1)… (1+t∂n-1) sδ(X1,…,Xn-1) =sδ ( t+X1,…, t+Xn-1 )$

where ${X}_{i}={x}_{1}+\dots +{x}_{i}$ for each $i\ge 1,$ and $\delta ={\delta }_{n}\text{.}$ For this it is enough to show that

$(1) (1+t∂i)sδ ( X1,…,Xi,t+ Xi+1,…,t+ Xn-1 ) =sδ ( X1,…,Xi-1, t+Xi,…,t+ Xn-1 )$

for $i=1,2,\dots ,n-1\text{.}$

Both sides of (1) are determinants with $n-1$ rows and columns which agree in all the ${i}^{\text{th}}$ row. On the left-hand side, the elements of the ${i}^{\text{th}}$ row are by (3.10)

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

and on the right-hand side they are ${h}_{k}\left(t+{X}_{i}\right),$ 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 $t\left(t-{x}_{i+1}\right)$ times the ${\left(i+1\right)}^{\text{th}}$ row to the ${i}^{\text{th}}$ row in the determinant on the left-hand side, we shall obtain the right-hand side of (1).

For each $r\ge 1,$ let

$Φr(t)= tr (1+t∂r+1*) (1+t∂r+2*) …$

For each permutation $w,$ we have $\left(1+t{\partial }_{j}^{*}\right){𝔖}_{w}={𝔖}_{w}$ for all sufficiently large $j$ by (7.1), so that ${\Phi }_{r}\left(t\right){𝔖}_{w}$ is a polynomial in $t$ (and ${x}_{1},{x}_{2},\dots \text{).}$ With this notation, we have

$(7.5) ∂1∂2… ∂n-r+1 ( x1n x2n-1… xn ) =Φr-1 (x1) 𝔖w0(n) (x2,x3,…)$

 Proof. Let $s=n-r+1$ and $a=x2s-1 x3s-2… xs, b=xs+2r-2 xs+3r-3… xn,c= (x2…xs+1)r-1$ so that $abc={x}_{2}^{n-1}{x}_{3}^{n-2}\dots {x}_{n}\text{.}$ Hence $∂1 ∂2… ∂n (x1nx2n-1…xn) = x1r-1bc ∂1…∂s (x1sx2s-1…xs) = x1r-1bc 𝔖1×w0(s) (x1,…,xs) by (4.21) = x1r-1bc (1+x1∂2)… (1+x1∂s)a by (7.4) = x1r-1 (1+x1∂2)… (1+x1∂s)abc = x1r-1 (1+x1∂2)… (1+x1∂s) 𝔖w0(n) (x1,…,xn) = x1r-1 (1+x1∂r*)… (1+x1∂n-1*) 𝔖w0(n) (x2,…,xn) by (7.3).$ $\square$

Let $w$ be any permutation. If $w\left(1\right)=r,$ then ${s}_{1}\dots {s}_{r-1}w\left(1\right)=1,$ so that we may write

$s1…sr-1w= 1×w1$

where ${w}_{1}$ is defined by

$w1(i)= { w(i+1) if w(i+1)r.$

If the code of $w$ is $\left({c}_{1},{c}_{2},\dots \right)$ (so that ${c}_{1}=r-1\text{),}$ the code of ${w}_{1}$ is $\left({c}_{2},{c}_{3},\dots \right)\text{.}$ With this notation we have

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

 Proof. Suppose that $w\in {S}_{n+1}\text{.}$ Then $w0(n+1)w = w0(n+1) sr-1…s1 (1×w1) = sn-r+2… snw0(n+1) (1×w0(n)) (1×w0(n)w1) = sn-r+1…s1 (1×w0(n)w1)$ since ${w}_{0}^{\left(n+1\right)}\left(1×{w}_{0}^{\left(n\right)}\right)={s}_{n}{s}_{n-1}\dots {s}_{1}\text{.}$ Hence $𝔖w (x1,…,xn) = ∂w-1w0(n+1) ( x1n x2n-1… nx ) = ∂1×w1-1w0(n) ∂1…∂n-r+1 (x1n…xn) = ∂1×w1-1w0(n) Φr-1(x1) 𝔖w0(n) (x2,x3,…,xn) by (7.5) = Φr-1(x1) ∂1×w1-1w0(n) 𝔖w0(n) (x2,x3,…,xn) by (7.2) = Φr-1(x1) 𝔖w1 (x2,x3,…).$ $\square$

Remark. The right-hand side of (7.6) is a sum of terms of the form ${x}_{1}^{p}{𝔖}_{u}\left({x}_{2},{x}_{3},\dots \right)\text{.}$ By applying (7.6) to each ${𝔖}_{u},$ and so on, we can decompose ${𝔖}_{w}$ into a sum of monomials, and thus we have another proof of the fact (4.17) that ${𝔖}_{w}$ is a polynomial in ${x}_{1},{x}_{2},\dots$ with positive integer coefficients.

Next, let $m\ge 1$ and assume that the permutation $w$ statisfies

$w(1)>w(2)>… >w(m).$

Define a partition $\mu =\mu \left(w,m\right)$ of length $\le m$ by

$μi=w(i)- (m+1-i) (1≤i≤m).$

If $w\in {S}_{m+n}$ we have ${\mu }_{1}\le n,$ hence $\mu \subset \left({n}^{m}\right)\text{.}$

Also let

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

and let ${w}_{m}$ be the permutation whose code is $\left({c}_{m+1},{c}_{m+2},\dots \right),$ where $\left({c}_{1},{c}_{2},\dots \right)$ is the code of $w\text{.}$

With this notation established, we have

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

 Proof. We proceed by induction on $m\text{;}$ 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 $u={s}_{{a}_{1}}\dots {s}_{{a}_{p}},$ where $c1(w)+1=μ1 +m≤a1<… and $\ell \left(u{w}_{1}\right)=\ell \left({w}_{1}\right)-p\text{.}$ The code of $u{w}_{1}$ satisfies ${c}_{i}\left(u{w}_{1}\right)={c}_{i}\left({w}_{1}\right)$ for $1\le i\le m-1,$ 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, $𝔖w(x) = ∑u x1μ1+m+p-1 x2m-2… xm-1Φμm (xm)…Φμ2 (x2) 𝔖(uw1)m-1 (xm+1,xm+2,…) = x1m-1 x2m-2… xm-1Φμm (xm)… Φμ1(x1) 𝔖wm (xm+1,xm+2,…) .$ $\square$

Finally, for any permutation $w,$ let $v$ be the unique element of ${S}_{m}$ such that $wv\left(1\right)>\dots >wv\left(m\right),$ and let $\mu =\mu \left(wv,m\right)\text{.}$ We have $\ell \left(wv\right)=\ell \left(w\right)+\ell \left(v\right)$ and ${\left(wv\right)}_{m}={w}_{m},$ so that by (7.7)

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

Hence

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

Now by (4.14), for any polynomial $f\in {P}_{m},$ we have

$f=∑u∈S(m) η(∂uf)Su$

where ${S}^{\left(m\right)}$ consists of the permutations whose codes have length $\le m,$ and $\eta \left({\partial }_{u}f\right)$ is the constant term of the polynomial ${\partial }_{u}f\text{.}$ 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 $u\in {S}^{\left(m\right)}$ such that $\ell \left(uv\right)=\ell \left(u\right)+\ell \left(v\right)\text{.}$

For each such $u,$ the constant term $\eta \left({\partial }_{uv}\left({x}^{{\delta }_{m}}{\Phi }_{\mu }\left({x}_{1},\dots ,{x}_{m}\right)\right)\right)$ is a polynomial in the (non-commuting) operators ${\partial }_{i}^{*}$ with integer coefficients. Hence (7.9) gives a decomposition of the Schubert polynomial ${𝔖}_{w}\left(x\right)$ of the form

$(7.10) 𝔖w(x)=∑u,v duvw𝔖u(y) 𝔖v(z),$

where $y=\left({x}_{1},\dots ,{x}_{m}\right)$ and $z=\left({x}_{m+1},{x}_{m+2},\dots \right)\text{.}$ If $w\in {S}^{\left(m+n\right)},$ so that ${𝔖}_{w}\left(x\right)\in {P}_{m+n},$ then $u\in {S}^{\left(m\right)}$ and $v\in {S}^{\left(n\right)}$ in this sum. From (4.18) we know that the coefficients ${d}_{uv}^{w}$ in (7.10) are $\ge 0\text{.}$

In particular, if we apply (7.7) to a permutation of the form ${w}_{0}^{\left(m\right)}×w,$ 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), ${𝔖}_{{1}_{m}×w}$ is symmetrical in ${x}_{1},\dots ,{x}_{m}\text{.}$ Hence so is the operator ${\Phi }_{0}\left({x}_{1},\dots ,{x}_{m}\right),$ and we may therefore write ${\Phi }_{0}$ in the form

$(7.13) Φ0 (x1,…,xm)= ∑λ,vαm (λ,v)sλ (x1,…,xm) ∂v*$

summed over partitions $\lambda$ of length $\le m$ and permutations $v,$ with integral coefficients ${\alpha }_{m}\left(\lambda ,v\right)\text{.}$ 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 $\lambda$ of length $\le m$ and $v$ such that $\ell \left(vw\right)=\ell \left(w\right)-\ell \left(v\right)\text{.}$ The Schur functions occuring here are precisely the Schubert polynomials ${𝔖}_{u},$ where $u$ is Grassmannian with descent at $m\text{.}$ Hence, by (4.18),

(7.15) The coefficients ${\alpha }_{m}\left(\lambda ,v\right)$ in (7.13) are $\ge 0\text{.}$

Since ${\Phi }_{0}\left({x}_{1},\dots ,{x}_{m},0\right)={\Phi }_{0}\left({x}_{1},\dots ,{x}_{m}\right)$ and ${s}_{\lambda }\left({x}_{1},\dots ,{x}_{m},0\right)={s}_{\lambda }\left({x}_{1},\dots ,{x}_{m}\right)$ if $\ell \left(\lambda \right)\le m,$ it follows from (7.13) that

$(7.16) αm+1(λ,v)= αm(λ,v)=α (λ,v) say$

for all partitions $\lambda$ such that $\ell \left(\lambda \right)\le m\text{.}$

We may also calculate the operator ${\Phi }_{0}\left({x}_{1},\dots ,{x}_{m}\right)$ as follows. For each integer $p\ge 1$ and each subset $D$ of $\left\{1,2,\dots ,p-1\right\}$ let

$QD,p (x1,…,xm) =∑xu1… xup$

summed over all sequences $\left({u}_{1},\dots ,{u}_{p}\right)$ such that $1\le {u}_{1}\le \dots \le {u}_{p}\le m$ and ${u}_{i}\le {u}_{i+1}$ whenever $i\in D\text{.}$ Then ${Q}_{D,p}\left({x}_{1},\dots ,{x}_{m}\right)$ is a homogeneous polynomial of degree $p,$ and is zero if $m\le \text{Card}\left(D\right)\text{.}$

Now let $a=\left({a}_{1},\dots ,{a}_{p}\right)$ be a reduced word, so that $\ell \left({s}_{{a}_{1}}\dots {s}_{{a}_{p}}\right)=p\text{.}$ The descent set of $a$ is

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

We now define, for each permutation $w,$

$Fw(x1,…,xm)= ∑a∈R(w) QD(a),ℓ(w) (x1,…,xm),$

a homogeneous polynomial of degree $\ell \left(w\right)\text{.}$

With these definitions we have

$(7.17) Φ0(x1,…,xm) =∑wFw (x1,…,xm) ∂w*.$

 Proof. Let $a=\left({a}_{1},\dots ,{a}_{p}\right)$ be a reduced word. Since $Φ0(xi)= (1+xi∂1*) (1+xi∂2*)…$ it is clear from the definitions that the coefficient of ${\partial }_{a}^{*}={\partial }_{{a}_{1}}^{*}\dots {\partial }_{{a}_{p}}^{*}$ in ${\Phi }_{0}\left({x}_{1},\dots ,{x}_{m}\right)={\prod }_{i=1}^{m}{\Phi }_{0}\left({x}_{i}\right)$ is just ${Q}_{D\left(a\right),p}\left({x}_{1},\dots ,{x}_{m}\right)\text{.}$ Hence $Φ0 (x1,…,xm) = ∑a QD(a),p (x1,…,xm) ∂a* = ∑wFw (x1,…,xm) ∂w*.$ $\square$

Comparison of (7.17) and (7.13) now shows that ${F}_{w}\left({x}_{1},\dots ,{x}_{m}\right)$ is a symmetric polynomial in ${x}_{1},\dots ,{x}_{m},$ 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 $\lambda$ such that $\ell \left(\lambda \right)\le m$ and $|\lambda |=\ell \left(w\right)\text{.}$ By (7.16) we have

$Fw(x1,…,xm,0) =Fw(x1,…,xm)$

and therefore we have a well defined symmetric function ${F}_{w}\in \Lambda ,$ such that ${\rho }_{m}\left({F}_{w}\right)={F}_{w}\left({x}_{1},\dots ,{x}_{m}\right)$ for all $m\ge 0\text{:}$ namely

$(7.19) Fw=∑λα (λ,w)sλ$

where the sum is over partitions $\lambda$ of $\ell \left(w\right),$ and $\alpha \left(\lambda ,w\right)={\alpha }_{m}\left(\lambda ,w\right)$ for any $m\ge \ell \left(\lambda \right)\text{.}$

Since the coefficient of ${x}_{1}\dots {x}_{p}$ in ${Q}_{D,p}\left({x}_{1},\dots ,{x}_{m}\right)$ is $1$ if $m\ge p,$ it follows that the coefficient of ${x}_{1}\dots {x}_{p}$ (where $p=\ell \left(w\right)\text{)}$ in ${F}_{w}\left({x}_{1},\dots ,{x}_{m}\right)$ is equal to $\text{Card}\left(R\left(w\right)\right)$ whenever $m\ge \ell \left(w\right)\text{.}$ On the other hand, the coefficient of ${x}_{1}\dots {x}_{p}$ in a Schur function ${s}_{\lambda },$ where $|\lambda |=p,$ is equal to ${f}^{\lambda },$ the number of standard tableaux of shape $\lambda ,$ or equivalently the degree of the irreducible representation ${\chi }^{\lambda }$ of ${S}_{p}$ indexed by the partition $\lambda$ ([Mac1979], Ch.I, §7). It follows therefore from (7.19) that

$(7.20) Card R(w)= ∑|λ|=ℓ(w) α(λ,w) fλ.$

Remark. Since the coefficients $\alpha \left(\lambda ,w\right)$ are $\ge 0$ 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 ${S}_{\ell \left(w\right)}\text{.}$ It is therefore natural to ask whether there is a "natural" action of this symmetric group on the $ℤ\text{-span}$ (or perhaps $ℚ\text{-span)}$ of the set $R\left(w\right),$ with character $\sum _{\lambda }\alpha \left(\lambda ,w\right){\chi }^{\lambda }\text{.}$

We shall conclude with some properties of the symmetric functions ${F}_{w}$ and the coefficients $\alpha \left(\lambda ,w\right)\text{.}$

(7.21) Let $u\in {S}_{m},$ $v\in {S}_{n}\text{.}$ Then

$Fu×v(x)= Fu(x)Fv (x).$

 Proof. By (7.18), we have for any $N,$ $Fu×v (x1,…,xN) = ρN (S1N×u-1×v-1) = ρN ( S1N×u-1 S1m+N×v-1 ) by (4.6) = ρN (S1N×u-1) ρN (ρm+N(S1m+N×v-1)) = Fu(x1,…,xN) Fv(x1,…,xN).$ $\square$

(7.22) Let $w\in {S}_{n}$ and let $\stackrel{‾}{w}={w}_{0}w{w}_{0},$ where ${w}_{0}$ is the longest element of ${S}_{n}\text{.}$ Then

$Fw-1= Fw‾=ω Fw$

where $\omega$ is the involution that interchanges ${s}_{\lambda }$ and ${s}_{\lambda \prime }\text{.}$ In other words

$α(λ,w-1)= α(λ,w‾)= α(λ′,w)$

for all partitions $\lambda \text{.}$

For the proof of (7.22) we require a lemma. If $t$ is a standard tableau of shape $\lambda ,$ the descent set $D\left(t\right)$ of $t$ is the set of $i$ such that $i+1$ lies in a lower row than $i$ in the tableau $t\text{.}$ We have

$(7.23) sλ=∑t QD(t),p$

where the sum is over the standard tableaux of shape $\lambda ,$ and $p=|\lambda |\text{.}$

 Proof. In the notation of [Mac1979, Ch. I, §5], ${s}_{\lambda }$ is the sum of monomials ${x}^{T}$ where $T$ runs through the (column-strict) tableaux of shape $\lambda \text{.}$ Each such tableau $T$ determines a standard tableau $t,$ as follows. If a square in the ${j}^{\text{th}}$ column of the diagram of $\lambda$ is occupied by the number $i,$ replace $i$ by the pair $\left(i,j\right)\text{.}$ Since $T$ is column-strict the pairs $\left(i,j\right)$ so obtained are all distinct. If we now order them lexiographically, (so that $\left(i,j\right)$ precedes $\lambda \left(i\prime ,j\prime \right)$ if and only if either $i or $i=i\prime$ and $j and relabel them as $1,2,\dots ,p,$ we have a standard tableau $t\text{:}$ say $T\to t\text{.}$ It follows easily that $\sum _{T\to t}{x}^{T}={Q}_{D\left(t\right),p},$ which proves the lemma. $\square$

If $D$ is any subset of $\left\{1,2,\dots ,p-1\right\},$ let $\stackrel{‾}{D}$ denote the complementary subset, and let ${D}^{*}=\left\{p-i:i\in D\right\}\text{.}$ From the definition of ${Q}_{D,p}$ we have

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

If $a=\left({a}_{1},\dots ,{a}_{p}\right)\in R\left(w\right),$ let $\stackrel{‾}{a}=\left(n-{a}_{1},\dots ,n-{a}_{p}\right)$ and ${a}^{*}=\left(n-{a}_{p},\dots ,n-{a}_{1}\right)\text{.}$ Then we have

$(2) a‾∈R(w‾), a*∈R(w*),$

where ${w}^{*}={\left(\stackrel{‾}{w}\right)}^{-1}={w}_{0}{w}^{-1}{w}_{0}\text{.}$ 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 $t\prime$ is the transpose of $t,$ obtained by reflecting $t$ in the main diagonal. For $i\in D\left(t\right)$ 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 $i\notin D\left(t\prime \right)\text{.}$

Since ${F}_{w}$ 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 ${F}_{w}={F}_{{w}^{}}\text{.}$

From (7.23) and (4) above we have

$ωsλ=sλ′= ∑t∈St()αℓλ QD(t)‾,p$

for all partitions $\lambda$ of $p,$ where $St\left(\lambda \right)$ is the set of standard tableaux of shape $\lambda ,$ and hence it follows from (2) and (3) and the definition of ${F}_{w}$ that $\omega {F}_{w}={F}_{\stackrel{‾}{w}}\text{.}$ Hence

$ωFw-1= Fw*=Fw,$

which completes the proof of (7.22).

(7.24)
 (i) $\alpha \left(\mu ,w\right)=0$ unless $\lambda \left({w}^{-1}\right)\le \mu \le \lambda \left(w\right)\prime \text{.}$ (ii) $\alpha \left(\mu ,w\right)=1$ if $\mu =\lambda \left({w}^{-1}\right)$ or $\mu =\lambda \left(w\right)\prime \text{.}$ (iii) $w$ is vexillary if and only if ${F}_{w}$ is a Schur function.

 Proof. (i) Suppose $\alpha \left(\mu ,w\right)\ne 0\text{.}$ Then the monomial ${x}^{\mu }$ occurs in ${F}_{w},$ and hence there is a reduced word $\left({a}_{1},\dots ,{a}_{p}\right)$ for $w$ such that $(1) a1<…< aμ1, aμ1+1<…< aμ1+μ1,….$ By (1.14) the code of $w$ is $(2) c(w)= ∑i=1p sap…sai+1 (εai).$ If ${w}^{\left(1\right)}={s}_{{a}_{p}}\dots {s}_{{a}_{{\mu }_{1}}+1},$ the sum of the first ${\mu }_{1}$ terms of this series is $w(1) ( εaμ1+ saμ1 (εaμ1-1) +…+saμ1… sa2 (εa1) ) ,$ and since ${a}_{1}<\dots <{a}_{{\mu }_{1}}$ this is equal to $(3) w(1) ( εaμ1+ εaμ1-1 +…+εa1 ) =V1 say,$ where ${V}_{1}$ is a $\left(0,1\right)$ vector (i.e., a vector with each component $0$ or $1\text{)}$ of weight ${\mu }_{1}\text{.}$ Likewise the sum of the next block of ${\mu }_{2}$ terms of the series (2) is a $\left(0,1\right)$ vector ${V}_{2}$ of weight ${\mu }_{2},$ and so on. Hence $c(w)=V1+…+ Vm$ where $m=\ell \left(\lambda \right),$ and each ${V}_{i}$ is a $\left(0,1\right)$ vector of weight ${\mu }_{i}\text{.}$ Let $V$ be the $\left(0,1\right)$ matrix whose ${i}^{\text{th}}$ row is ${V}_{i},$ for $i=1,2,\dots ,m\text{.}$ Then $V$ has row sums ${\mu }_{1},\dots ,{\mu }_{m}$ and column sums ${c}_{1}\left(w\right),{c}_{2}\left(w\right),\dots$ As in the proof of (1.26) it follows that $\mu \le \lambda \left(w\right)\prime \text{.}$ Since $\alpha \left(\mu ,w\right)=\alpha \left(\mu \prime ,{w}^{-1}\right)$ by (7.23), the same argument applied to $\mu \prime$ and ${w}^{-1}$ gives $\mu \prime \le \lambda \left({w}^{-1}\right)\prime$ i.e., $\lambda \left({w}^{-1}\right)\le \mu \text{.}$ (ii) Suppose now that $\mu =\lambda \left(w\right)\prime \text{.}$ Then there is only one $\left(0,1\right)$ matrix $V$ with row sums ${\mu }_{i}$ and column sums ${c}_{i}\text{.}$ Its first row ${V}_{1}$ is $\sum {\epsilon }_{j}$ summed over $j$ such that ${c}_{j}\ne 0,$ i.e. such that there exists $k>j$ with $w\left(k\right) From (3) it follows that $wV1=∑i=1μ1 εai+1$ and therefore ${a}_{1}+1,\dots ,{a}_{{\mu }_{1}}+1$ are the terms of the sequence $w$ that have a smaller element somewhere to the right, in increasing order of magnitude. Hence ${a}_{1}$ has no smaller elements to the right of it, and therefore lies to the right of ${a}_{1}+1,$ so that $\ell \left({s}_{{a}_{1}}w\right)=\ell \left(w\right)-1\text{.}$ The same argument shows that $\ell \left({s}_{{a}_{2}}{s}_{{a}_{1}}w\right)=\ell \left({s}_{{a}_{1}}w\right)-1$ and so on. Hence if ${w}_{1}={s}_{{a}_{{\mu }_{1}}}\dots {s}_{{a}_{1}}w$ we have $\ell \left({w}_{1}\right)=\ell \left(w\right)-{\mu }_{1},$ and $\lambda \left({w}_{1}^{\prime }\right)=\left({\mu }_{2},{\mu }_{3},\dots \right)\text{.}$ It follows by induction on $\ell \left(\mu \right)$ that the word $\left({a}_{1},\dots ,{a}_{p}\right)$ determined by the matrix $V$ is reduced, and hence $\alpha \left(\mu ,w\right)=1$ when $\mu =\lambda \left(w\right)\prime \text{.}$ By (7.23) it follows that $\alpha \left(\mu ,w\right)=1$ when $\mu =\lambda \left({w}^{-1}\right)\text{.}$ (iii) This follows immediately from (i) and (ii), and the characterization (1.27) of vexillary permutations. $\square$

## Notes and References

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