Notes on Schubert Polynomials
Chapter 6

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 2 July 2013

Double Schubert Polynomials

Let $x=(x_1,\dots ,x_n),$ $y=(y_1,\dots ,y_n)$ be two sequences of independent indeterminates, and recall (5.8) that

$$\Delta (x,y)=\prod _{i+j\le n}(x_i-y_j)\text{.}$$

For each $w\in S_n,$ we define the double Schubert polynomial ${𝔖}_w(x,y)$ to be

$$\begin{array}{cc}\text{(6.1)}& {𝔖}_w(x,y)={\partial }_{w^{-1}{w}_0}\Delta (x,y)\end{array}$$

where ${\partial }_{w^{-1}{w}_0}$ acts on the $x$ variables.

Since $\Delta (x,0)=x^{\delta }$ we have

$$\begin{array}{cc}\text{(6.2)}& {𝔖}_w(x,0)={𝔖}_w\left(x\right),\end{array}$$

the (single) Schubert polynomial indexed by $w\text{.}$

From the Cauchy formula (5.10) we have

$${𝔖}_w(x,y)=\sum _{v\in S_n}{\partial }_{w^{-1}{w}_0}{𝔖}_{v{w}_0}\left(x\right){𝔖}_v(-y)$$

and by (4.2)

$${\partial }_{w^{-1}{w}_0}{𝔖}_{v{w}_0}\left(x\right)={𝔖}_{vw}\left(x\right)$$

if $\ell \left(vw\right)=\ell \left(v{w}_0\right)-\ell \left(w^{-1}{w}_0\right),$ i.e. if $\ell \left(vw\right)=\ell \left(w\right)-\ell \left(v\right),$ and

$${\partial }_{w^{-1}{w}_0}{𝔖}_{v{w}_0}\left(x\right)=0$$

otherwise. Hence

$$\begin{array}{cc}\text{(6.3)}& {𝔖}_w(x,y)=\sum _{u,v}{𝔖}_u\left(x\right){𝔖}_v(-y)\end{array}$$

summed over all $u,v\in S_n$ such that $w=v^{-1}u$ and $\ell \left(w\right)=\ell \left(u\right)+\ell \left(v\right)\text{.}$

From (6.3) it follows that ${𝔖}_w(x,y)$ is a homogeneous polynomial of degree $\ell \left(w\right)$ in $x_1,\dots ,x_{n-1},$ $y_1,\dots ,y_{n-1}\text{.}$ We have

(6.4)

(i) ${𝔖}_{w_0}(x,y)=\Delta (x,y),$ (ii) ${𝔖}_1(x,y)=1,$ (iii) ${𝔖}_{w^{-1}}(x,y)={𝔖}_w(-y,-x)=\epsilon \left(w\right){𝔖}_w(y,x)$ for all $w\in S_n,$ (iv) ${𝔖}_w(x,x)=0$ for all $w\in S_n$ except $w=1\text{.}$

		Proof.
	(i) is immediate from the definition (6.1). (ii) and (iii) follow from (6.3). (iv) follows from (5.20), since ${𝔖}_w(x,x)=\theta \left({\partial }_{w^{-1}{w}_0}\Delta \right)=0$ if $w\ne 1\text{.}$ $\square$

(6.5) (Stability) If $m>n$ and $i$ is the embedding of $S_n$ in $S_m,$ then

$${𝔖}_{i\left(w\right)}(x,y)={𝔖}_w(x,y)$$

for all $w\in S_n\text{.}$

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

From (6.5) it follows that the double Schubert polynomials ${𝔖}_w(x,y)$ are well defined for all permutations $w\in S_{\infty }\text{.}$

For any commutative ring $K,$ let $K\left(S_{\infty }\right)$ denote the $K\text{-module}$ of all functions on $S_{\infty }$ with values in $K\text{.}$ We define a multiplication in $K\left(S_{\infty }\right)$ as follows: for $f,g\in K\left(S_{\infty }\right),$

$$\left(fg\right)\left(w\right)=\sum _{u,v}f\left(u\right)g\left(v\right)$$

summed over all $u,v\in S_{\infty }$ such that $uv=w$ and $\ell \left(u\right)+\ell \left(v\right)=\ell \left(w\right)\text{.}$ For this multiplication, $K\left(S_{\infty }\right)$ is an associative (but not commutative) ring, with identity element $\underset{\underset{\_}{\_}}{1},$ the characteristic function of the identity permutation $1\text{.}$ It carries an involution $f\mapsto f^{*},$ defined by

$$f^{*}\left(w\right)=f\left(w^{-1}\right)$$

which satisfies

$${\left(fg\right)}^{*}=g^{*}{f}^{*}$$

for all $f,g\in K\left(S_{\infty }\right)\text{.}$

(6.6) Let $f,g\in K\left(S_{\infty }\right)\text{.}$

(i)	If $fg=f$ and $f\left(1\right)$ is not a zero divisor in $K,$ then $g=\underset{\underset{\_}{\_}}{1}\text{.}$
(ii)	If $fg=\underset{\underset{\_}{\_}}{1},$ then $gf=\underset{\underset{\_}{\_}}{1}\text{.}$
(iii)	$f$ is a unit (i.e. invertible) in $K\left(S_{\infty }\right)$ if and only if $f\left(1\right)$ is a unit in $K\text{.}$

		Proof.
	(i) We have $f\left(1\right)=f\left(1\right)g\left(1\right)$ and hence $g\left(1\right)=1\text{.}$ We shall show by induction on $\ell \left(w\right)$ that $g\left(w\right)=0$ for all $w\ne 1\text{.}$ So let $r>0$ and assume that $g\left(v\right)=0$ for all $v\in S_{\infty }$ such that $1\le \ell \left(v\right)\le r-1\text{.}$ Let $w$ be a permutation of length $r\text{.}$ We have $$\begin{array}{cc}\text{(1)}& f\left(w\right)=\left(fg\right)\left(w\right)=f\left(w\right)g\left(1\right)+f\left(1\right)g\left(w\right)+\sum _{u,v}f\left(u\right)g\left(v\right)\end{array}$$ where the sum on the right is over $u,v\in S_{\infty }$ such that $u\ne 1,$ $v\ne 1,$ $uv=w$ and $\ell \left(u\right)+\ell \left(v\right)=\ell \left(w\right),$ so that $1\le \ell \left(v\right)\le r-1$ and therefore $g\left(v\right)=0\text{.}$ Hence (1) reduces to $f\left(1\right)g\left(w\right)=0$ and therefore $g\left(w\right)=0$ as required. (ii) We have $f\left(1\right)g\left(1\right)=1$ so that $f\left(1\right)$ is a unit in $K\text{.}$ Also $f\left(gf\right)=\left(fg\right)f=f,$ whence $gf=\underset{\underset{\_}{\_}}{1}$ by (i) above. (iii) Suppose $f$ is a unit in $K\left(S_{\infty }\right),$ with inverse $g\text{.}$ Since is $fg=\underset{\underset{\_}{\_}}{1}$ we have $f\left(1\right)g\left(1\right)=1,$ whence $f\left(1\right)$ is an unit in $K\text{.}$ Conversely, if $f\left(1\right)$ is an unit in $K$ we construct an inverse $g$ of $f$ as follows. We define $g\left(1\right)=f{\left(1\right)}^{-1}$ and proceed to define $g\left(w\right)$ by induction on $\ell \left(w\right)\text{.}$ Assume that $g\left(v\right)$ has been defined for all $v$ such that $\ell \left(v\right)<\ell \left(w\right)$ and set $$g\left(w\right)=-f{\left(1\right)}^{-1}\sum _{u,v}f\left(u\right)g\left(v\right)$$ summed over $u,v$ such that $uv=w,$ $v\ne w$ and $\ell \left(u\right)+\ell \left(v\right)=\ell \left(w\right)\text{.}$ This definition gives $\left(fg\right)\left(w\right)=0$ as required. $\square$

Now let $𝔖\left(x\right)$ (resp. $𝔖(x,y)\text{)}$ be the function on $S_{\infty }$ whose value at a permutation $w$ is ${𝔖}_w\left(x\right)$ (resp. ${𝔖}_w(x,y)\text{).}$ (The coefficient ring $K$ is now the ring $\mathbb{Z}[x,y]$ of polynomials in the $x\text{'s}$ and $y\text{'s.)}$ Since ${𝔖}_1\left(x\right)={𝔖}_1(x,y)=1,$ it follows from (6.6)(iii) that $𝔖\left(x\right)$ and $𝔖(x,y)$ are units in $K\left(S_{\infty }\right)\text{.}$

(6.7)

(i) $𝔖(x,0)=𝔖\left(x\right),$ (ii) $𝔖(x,x)=\underset{\underset{\_}{\_}}{1},$ (iii) $𝔖{(x,y)}^{*}=𝔖(-y,-x),$ (iv) $𝔖{\left(x\right)}^{-1}=𝔖(0,x),$ (v) $𝔖{\left(x\right)}^{*}=𝔖{(-x)}^{-1},$ (vi) $𝔖(x,y)=𝔖{\left(y\right)}^{-1}𝔖\left(x\right)=𝔖{(y,x)}^{-1}\text{.}$

		Proof.
	(i)-(iii) follow directly from (6.2) and (6.4). From (6.3) and (6.4) we have $${𝔖}_w(x,y)=\sum _{u,v}{𝔖}_{u^{-1}}(-y){𝔖}_v\left(x\right)=\sum _{u,v}{𝔖}_u(0,y){𝔖}_v\left(x\right)$$ summed over $u,v\in S_{\infty }$ such that $uv=w$ and $\ell \left(u\right)+\ell \left(v\right)=\ell \left(w\right)\text{.}$ In other words, $$\begin{array}{cc}\text{(1)}& 𝔖(x,y)=𝔖(0,y)𝔖\left(x\right)\text{.}\end{array}$$ In particular, when $y=x$ we obtain $𝔖(0,x)𝔖\left(x\right)=𝔖(x,x)=\underset{\underset{\_}{\_}}{1}$ by (ii) above, and hence $𝔖(0,x)=𝔖{\left(x\right)}^{-1}\text{.}$ This establishes (iv); part (v) now follows from (iv) and (iii), and (vi) from (iv) and (1) above. $\square$

From (6.7) (vi) we have

$$𝔖\left(x\right)=𝔖\left(y\right)𝔖(x,y)$$

or explicitly

$${𝔖}_w\left(x\right)=\sum _{u,v}{𝔖}_u\left(y\right){𝔖}_v(x,y)$$

summed over $u,v$ such that $uv=w$ and $\ell \left(u\right)+\ell \left(v\right)=\ell \left(w\right),$ so that $u=w{v}^{-1}$ and ${𝔖}_u={\partial }_v{𝔖}_w$ by (4.2). Hence

$${𝔖}_w\left(x\right)=\sum _v{𝔖}_v(x,y){\partial }_v{𝔖}_w\left(y\right)$$

(where the operators ${\partial }_v$ act on the $y$ variables). The sum here may be taken over all permutations $v,$ since ${\partial }_v{𝔖}_w=0$ unless $\ell \left(w{v}^{-1}\right)=\ell \left(w\right)-\ell \left(v\right)\text{.}$ By linearity and (4.13) it follows that

(6.8) (Interpolation Formula) For all $f\in P_n=\mathbb{Z}[x_1,\dots ,x_n]$ we have

$$f\left(x\right)=\sum _w{𝔖}_w(x,y){\partial }_{w}f\left(y\right)$$

summed over permutations $w\in S^{\left(n\right)}\text{.}$

(The reason for the restriction to $S^{\left(n\right)}$ in the summation is that if $w\notin S^{\left(n\right)}$ we shall have $w\left(m\right)>w(m+1)$ for some $m>n,$ and hence ${\partial }_w={\partial }_v{\partial }_m$ where $v=w{s}_m\text{;}$ but ${\partial }_{m}f=0$ for all $f\in P_n,$ since $m>n,$ and therefore ${\partial }_{w}f=0\text{.)}$

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

2. When $n=1,$ the sum is over $S^{\left(1\right)},$ which consists of the permutations $w_p=s_p{s}_{p-1}\dots s_1$ $(p\ge 0)\text{;}$ $w_p$ is dominant, of shape $\left(p\right),$ so that (see (6.15) below) ${𝔖}_{w_p}(x,y)=(x-y_1)\dots (x-y_p)\text{.}$ Hence the case $n=1$ of (6.8) is Newton's interpolation formula

$$f\left(x\right)=\sum _{p\ge 0}(x-y_1)\dots (x-y_p)f_p(y_1,\dots ,y_{p+1})$$

where $f_p={\partial }_p{\partial }_{p-1}\dots {\partial }_{1}f,$ or explicitly

$$f_p(y_1,\dots ,y_{p+1})=\sum _{i=1}^{p+1}\frac{f\left(y_i\right)}{{\prod }_{j\ne i}(y_i-y_j)}\text{.}$$

For any integer $r,$ let ${𝔖}_w(x,r)$ denote the polynomial obtained from ${𝔖}_w(x,y)$ by setting $y_1=y_2=\dots =r\text{.}$ Since

$$\begin{array}{ccc}{𝔖}_{w_0}(x,r)& =& \Delta (x,r)=\prod _{i=1}^{n-1}{(x_i-r)}^{n-i}\\ & =& {𝔖}_{w_0}(x-r)\end{array}$$

where $x-r$ means $(x_1-r,x_2-r,\dots ),$ it follows from the definitions (6.1) and (4.1) that

$${𝔖}_w(x,r)={𝔖}_w(x-r)$$

for all permutations $w\text{.}$ Hence, by (6.7)(vi),

$$𝔖(x-r)=𝔖{\left(r\right)}^{-1}𝔖\left(x\right)$$

and in particular, for all integers $q,$

$$𝔖(q-r)=𝔖{\left(r\right)}^{-1}𝔖\left(q\right)$$

from which it follows that

$$\begin{array}{cc}\text{(6.9)}& 𝔖\left(r\right)=𝔖{\left(1\right)}^r\end{array}$$

for all $r\in \mathbb{Z}\text{.}$

Since ${𝔖}_w\left(x\right)$ is a sum of monomials with positive integral coefficients (4.17), ${𝔖}_w\left(1\right)$ is the number of monomials in ${𝔖}_w\left(x\right)$ (each monomial counted the number of times it occurs). By homogeneity, we have

$$\begin{array}{cc}\text{(6.10)}& {𝔖}_w\left(r\right)=r^{\ell \left(w\right)}{𝔖}_w\left(1\right)\text{.}\end{array}$$

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

$$𝔖{\left(1\right)}^{*}=𝔖{(-1)}^{-1}=𝔖\left(1\right)$$

so that we have another proof of the fact (4.30) that ${𝔖}_w\left(1\right)={𝔖}_{w^{-1}}\left(1\right)\text{.}$

Now consider the function $F=𝔖\left(1\right)-\underset{\underset{\_}{\_}}{1},$ whose value at $w\in S_{\infty }$ is

$$F\left(w\right)=\{\begin{array}{cc}\text{numbers of monomials in}\hspace{0.17em}{𝔖}_w,& \text{if}\hspace{0.17em}w\ne 1,\\ 0,& \text{if}\hspace{0.17em}w=1\text{.}\end{array}$$

For each positive integer $p$ we have

$$\begin{array}{cc}\text{(1)}& \begin{array}{ccc}{F}^p& =& {(𝔖\left(1\right)-\underset{\underset{\_}{\_}}{1})}^p\\ & =& \sum _{r=0}^p{(-1)}^r\left(\genfrac{}{}{0ex}{}{p}{r}\right)𝔖{\left(1\right)}^r\\ & =& \sum _{r=0}^p{(-1)}^r\left(\genfrac{}{}{0ex}{}{p}{r}\right)𝔖\left(1\right)\end{array}\end{array}$$

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

$$\left(\sum _{r=0}^p{(-1)}^r\left(\genfrac{}{}{0ex}{}{p}{r}\right)r^p\right){𝔖}_w\left(1\right)$$

which is equal to $p!{𝔖}_w\left(1\right)$ (consider the coefficient of $t^p$ in ${(e^t-1)}^p\text{).}$ On the other hand, $F^p\left(w\right)$ is by definition equal to

$$\begin{array}{cc}\text{(2)}& \sum _{w_1,\dots ,w_p}F\left(w_1\right)\dots F\left(w_p\right)\end{array}$$

summed over all sequences $(w_1,\dots ,w_p)$ of permutations such that $w_1\dots w_p=w,\ell \left(w_1\right)+\dots +\ell \left(w_p\right)=\ell \left(w\right)=p,$ and $w_i\ne 1$ for $1\le i\le p\text{.}$ It follows that each $w_i$ has length $1,$ hence $w_i=s_{a_i}$ say, and that $(a_1,\dots ,a_p)$ is a reduced word for $w\text{.}$ Since

$${𝔖}_{s_a}=x_1+\dots +x_a$$

by (4.4). we have $F\left(w_i\right)={𝔖}_{s_{a_i}}$ $\text{(1)}=a_i,$ and hence the sum (2) is equal to $\sum a_1{a}_2\dots a_p$ summed over all $(a_1,\dots ,a_p)\in R\left(w\right)\text{.}$

We have therefore proved that

(6.11) The number of monomials in ${𝔖}_w$ is

$${𝔖}_w\left(1\right)=\frac{1}{p!}\sum a_1{a}_2\dots a_p$$

summed over all $(a_1,\dots ,a_p)\in R\left(w\right),$ where $p=\ell \left(w\right)\text{.}$

Remarks. 1. The reduced words for $1_m\times w$ $(m\ge 1)$ are $(m+a_1,\dots ,m+a_p)$ where $(a_1,\dots ,a_p)\in R\left(w\right)\text{.}$ Hence from (6.11) and homogeneity we have

$${𝔖}_{1_m\times w}\left(\frac{1}{m}\right)=\frac{1}{p!}\sum (1+\frac{a_1}{m})\dots (1+\frac{a_p}{m})$$

summed over $R\left(w\right)$ as before. Letting $m\to \infty ,$ we deduce that

$$\begin{array}{cc}\text{(6.12)}& \text{Card}\hspace{0.17em}R\left(w\right)=p!\underset{m\to \infty }{\text{lim}}{𝔖}_{1_m\times w}\left(\frac{1}{m}\right)\text{.}\end{array}$$

2. If $w$ is dominant of length $p,$ then ${𝔖}_w$ is a monomial by (4.7). and hence in this case

$$\sum _{R\left(w\right)}{a}_1\dots a_p=p!$$

3. Suppose that $w$ is vexillary of length $p\text{.}$ Then by (4.9) we have

$${𝔖}_w=s_{\lambda }(X_{{\varphi }_1},\dots ,X_{{\varphi }_r})$$

where $\lambda$ is the shape of $w$ and $\varphi =({\varphi }_1,\dots ,{\varphi }_r)$ the flag of $w\text{.}$ Hence

$${𝔖}_{1_m\times w}=s_{\lambda }(X_{{\varphi }_1+m},\dots ,X_{{\varphi }_r+m})$$

for each $m\ge 1\text{.}$ If we now set each $x_i=\frac{1}{m}$ and then let $m\to \infty ,$ we shall obtain in the limit the Schur function $s_{\lambda }$ for the series $e^t$ ([Mac1979], Ch. I. §3. Ex. 5). which is equal to $h{\left(\lambda \right)}^{-1},$ where $h\left(\lambda \right)$ is the product of the hook-lengths of $\lambda \text{.}$ Hence it follows from (6.12) that if $w$ is vexillary of length $p,$ then

$$\begin{array}{cc}\text{(6.13)}& \text{Card}\hspace{0.17em}R\left(w\right)=\frac{p!}{h\left(\lambda \right)}\end{array}$$

where $\lambda$ is the shape of $w\text{.}$ In other words. the number of reduced words for a vexillary permutation of length $p$ and shape $\lambda ⊢p$ is equal to the degree of the irreducible representation of $S_p$ indexed by $\lambda \text{.}$

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

$$\begin{array}{cc}\text{(}{6.11}_q\text{?)}& {𝔖}_w(1,q,q^2,\dots )=\sum q^{\varphi \left(a\right)}\frac{(1-q^{a_1})\dots (1-q^{a_p})}{(1-q)\dots (1-q^p)}\end{array}$$

summed as in (6.11) over all reduced words $a=(a_1,\dots ,a_p)$ for $w,$ where

$$\varphi \left(a\right)=\sum \{i:a_i<a_{i+1}\}\text{.}$$

When $w$ is vexillary the double Schubert polynomial ${𝔖}_w(x,y)$ 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 $\lambda ,$ then

$$\begin{array}{ccc}{𝔖}_w(x,y)& =& \prod _{(i,j)\in \lambda }(x_i-y_j)\\ & =& s_{\lambda }(X_1-Y_{{\lambda }_1},\dots ,X_m-Y_{{\lambda }_m})\end{array}$$

where $m=\ell \left(\lambda \right)$ and $X_i=x_1+\dots +x_i,$ $Y_i=y_1+\dots +y_i$ for all $i\ge 1\text{.}$

		Proof.
	As in (4.6) we proceed by descending induction on $\ell \left(w\right),w\in S_n\text{.}$ The result is true for $w=w_0,$ since $w_0$ is dominant of shape $\delta$ and $${𝔖}_{w_0}(x,y)=\Delta (x,y)=\prod _{(i,j)\in \delta }(x_i-y_j)\text{.}$$ Suppose $w\ne w_0$ is dominant of shape $\lambda \text{.}$ Then $\lambda \subset \delta$ (and $\lambda \ne \delta \text{).}$ Let $r\ge 0$ be the largest integer such that ${\lambda }_i^{\prime }=n-i$ for $1\le i\le r,$ and let $a={\lambda }_{r+1}^{\prime }+1\le n-r-1\text{.}$ Then $w{s}_a$ is dominant, $\ell \left(w{s}_a\right)=\ell \left(w\right)+1,$ and $\lambda \left(w{s}_a\right)=\lambda \left(w\right)+{\epsilon }_a,$ and therefore $$\begin{array}{ccc}{𝔖}_w(x,y)& =& {\partial }_a{𝔖}_{w{s}_a}(x,y)\\ & =& {\partial }_a\left((x_a-y_{r+1})\prod _{(i,j)\in \lambda }(x_i-y_j)\right)\end{array}$$ by the inductive hypothesis; since ${\lambda }_a={\lambda }_{a+1}$ it follows that $${𝔖}_w(x,y)=\prod _{(i,j)\in \lambda }(x_i-y_j)$$ which is equal to $s_{\lambda }(X_1-Y_{{\lambda }_1},\dots ,X_m-Y_{{\lambda }_m})$ by (3.5). $\square$

(6.15) If $w$ is Grassmannian of shape $\lambda$ then

$${𝔖}_w(x,y)=s_{\lambda }(X_m-Y_{{\lambda }_1+m-1},\dots ,X_m-Y_{{\lambda }_m})\text{.}$$

		Proof.
	This follows from (6.14) just as (4.8) follows from (4.7). $\square$

Finally, let $w$ be vexillary with shape

$$\lambda \left(w\right)=(p_1^{m_1},\dots ,p_k^{m_k})$$

and flag

$$\varphi \left(w\right)=(f_1^{m_1},\dots ,f_k^{m_k})$$

as in Chapter IV. Then $w^{-1}$ is also vexillary, with shape

$$\lambda \left(w^{-1}\right)=\lambda \left(w\right)\prime =(q_1^{n_1},\dots ,q_k^{n_k})$$

the conjugate of $\lambda \left(w\right),$ and flag

$$\varphi \left(w^{-1}\right)=(g_1^{n_1},\dots ,g_k^{n_k})$$

where by (1.41)

$$g_i+q_i=f_{k+1-i}+p_{k+1-i}(1\le i\le k)\text{.}$$

With this notation recalled, we have

$$\begin{array}{cc}\text{(6.16)}& {𝔖}_w(x,y)=s_{\lambda }({(X_{f_1}-Y_{g_k})}^{m_1},\dots ,{(X_{f_k}-Y_{g_1})}^{m_k})\text{.}\end{array}$$

		Proof.
	The proof is essentially the same as that of (4.9) (which is the case $y=0\text{).}$ By (4.10) the dominant permutation $w_k$ constructed from $w$ in the proof of (4.9) has shape $$\mu =(g_k^{m_1},g_{k-1}^{m_2},\dots ,g_1^{m_k})$$ and therefore by (6.15) we have $${𝔖}_{w_k}(x,y)=s_{\mu }(X_1^{\prime },\dots ,X_m^{\prime })$$ where $m=m_1+\dots +m_k=\ell \left(\lambda \right)$ and the sequence $(X_1^{\prime },\dots ,X_m^{\prime })$ is obtained by subtracting the sequence $({\left(Y_{g_k}\right)}^{m_1},\dots ,{\left(Y_{g_1}\right)}^{m_k})$ term by term from the sequence $(X_1,\dots ,X_m)\text{.}$ Hence the same argument as in (4.9) establishes (6.17). $\square$

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

$$s_{\lambda }(Z_1^{m_1},\dots ,Z_k^{m_k})={(-1)}^{\left|\lambda \right|}{s}_{\lambda \prime }({(-Z_k)}^{n_1},\dots ,{(-Z_1)}^{n_k})$$

where $Z_i=X_{f_i}-Y_{g_{k+i-1}}$ so that (if $rk\hspace{0.17em}\left(x_i\right)=rk\hspace{0.17em}\left(y_i\right)=1$ for each $i\ge 1\text{)}$

$$\begin{array}{ccc}rk\hspace{0.17em}(Z_{i+1}-Z_i)& =& f_{i+1}-f_i+g_{k+1-i}-g_{k-i}\\ & =& m_{i+1}-n_{k+1-i}\end{array}$$

by (1.41). Hence (6.4)(iii) reduces to the duality theorem (3.8'') (with $\mu =0\text{)}$ when $w$ is vexillary.

Let ${\tau }_x$ (resp. ${\tau }_y\text{)}$ be the shift operator (4.21) acting on the $x$ (resp. $y\text{)}$ variables. Then we have

$$\begin{array}{cc}\text{(6.17)}& {\tau }_x^r{\tau }_y^r{𝔖}_w(x,y)={𝔖}_{1_r\times w}(x,y)\end{array}$$

for all $r\ge 1$ and all permutations $w\text{.}$

		Proof.
	By (6.3) and (4.21) we have $${\tau }_x^r{\tau }_y^r{𝔖}_w(x,y)=\sum _{u,v}\epsilon \left(v\right){𝔖}_{1_r\times u}\left(x\right){𝔖}_{1_r\times v}\left(y\right)$$ summed over $u,v$ such that $v^{-1}u=w$ and $\ell \left(u\right)+\ell \left(v\right)=\ell \left(w\right)\text{.}$ By (6.3) again, the right-hand side is equal to ${𝔖}_{1_r\times w}(x,y)\text{.}$ $\square$

In particular, suppose that $w$ is vexillary. With the notation of (6.16), the flag of $1_r\times w$ (resp. $1_r\times w^{-1}\text{)}$ is obtained from that of $w$ (resp. $w^{-1}\text{)}$ by replacing each $f_i$ by $f_i+r$ (resp. each $g_i$ by $g_i+r\text{).}$ Hence by (6.16) we have

$${𝔖}_{1_r\times w}(x,y)=s_{\lambda }({(X_{f_1+r}-Y_{g_k+r})}^{m_1},\dots ,{(X_{f_k+r}-Y_{g_1+r})}^{m_k})$$

and hence

$$\begin{array}{cc}\text{(6.18)}& {\rho }_r^{\left(x\right)}{\rho }_r^{\left(y\right)}{𝔖}_{1_r\times w}(x,y)=s_{\lambda }(X_r-Y_r)\end{array}$$

for all $r\ge 1,$ where ${\rho }_r^{\left(x\right)}$ (resp. ${\rho }_r^{\left(y\right)}\text{)}$ is the homomorphism ${\rho }_r$ of (4.25) acting on the $x$ (resp. $y\text{)}$ variables.

(6.19) Let ${\pi }_x$ (resp. ${\pi }_y\text{)}$ denote ${\pi }_{w_0^{\left(r\right)}}$ acting on the $x$ (resp. $y\text{)}$ variables. Then if $w$ is vexillary of shape $\lambda ,$ we have

$${\pi }_x{\pi }_y{𝔖}_w(x,y)=s_{\lambda }(X_r-Y_r)\text{.}$$

		Proof.
	By (4.24) we have ${\pi }_x={\rho }_r^{\left(x\right)}{\tau }_x^r$ and ${\pi }_y={\rho }_y^{\left(r\right)}{\tau }_y^r\text{.}$ Hence (6.19) follows from (6.17) and (6.18). $\square$

In particular, suppose that $w$ is dominant of shape $\lambda ,$ so that by (6.14)

$${𝔖}_w(x,y)=\prod _{(i,j)\in \lambda }(x_i-y_j)=f_{\lambda }(x,y)\hspace{0.17em}\text{say.}$$

In this case (6.19) gives

$${\pi }_x{\pi }_y{f}_{\lambda }(x,y)=s_{\lambda }(X_r-Y_r)$$

for all $r\ge 1,$ 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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