Notes on Schubert Polynomials
Appendix

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

Last update: 3 July 2013

Schubert varieties

Let $V$ be a vector space of dimension $n$ over a field $K,$ and let $(e_1,\dots ,e_n)$ be a basis of $V,$ fixed once and for all. A flag in $V$ is a sequence $U={\left(U_i\right)}_{0\le i\le n}$ of subspaces of $V$ such that

$$0=U_0\subset U_1\subset \dots \subset U_n=V$$

with strict inclusions at each stage, so that $\text{dim}\hspace{0.17em}{U}_i=i$ for each $i\text{.}$ In particular, if $V_i$ is the subspace of $V$ spanned by $e_1,\dots ,e_i,$ then $V={\left(V_i\right)}_{0\le i\le n}$ is a flag in $V,$ called the standard flag.

The set $F=F\left(V\right)$ of flags in $V$ is called the flag manifold of $V\text{.}$

Let $G$ be the group of all automorphisims of the vector space $V\text{.}$ Since we have fixed a basis of $V,$ we may identify $G$ with the general linear group ${GL}_n\left(k\right)\text{:}$ if $g\in G$ and

$$g{e}_j=\sum _{i=1}^n{g}_{ij}{e}_i(1\le j\le n)$$

then $g$ is identified with the matrix $\left(g_{ij}\right)\text{.}$

The group $G$ acts on $F\text{:}$ if $U=\left(U_i\right)$ and $g\in G,$ then $gU$ is the flag $\left(g{U}_i\right)\text{.}$ Let $B$ be the subgroup of $G$ that fixes the standard flag $V\text{.}$ Then $g\in B$ if and only if $g{e}_j$ is a linear combination of $e_1,\dots ,e_j,$ for $1\le j\le n,$ that is to say if and only if $g_{ij}=0$ whenever $i>j,$ so that $B$ is the group of upper triangular matrices in ${GL}_n\left(k\right)\text{.}$

A basis of a flag $U=\left(U_i\right)$ is a sequence $(u_1,\dots ,u_n)$ in $V$ such that $u_i\in U_i-U_{i-1}$ $1\le i\le n,$ or equivalently such that $u_1,\dots ,u_i$ is a basis of $U_i$ for each $i\text{.}$ Given such a basis of $U,$ there is a unique $g\in G$ such that $g{e}_i=u_i$ for each $i,$ and we have $U=gV\text{.}$ Hence $G$ acts transitively on the flag manifold $F,$ and the mapping $gV\mapsto gB$ is a bijection of $F$ onto the coset space $G/B\text{.}$

For a flag $U=\left(U_i\right),$ let

$$E_i=E_i\left(U\right)=\{j:1\le j\le n\hspace{0.17em}\text{and}\hspace{0.17em}{U}_i\cap V_j\ne U_i\cap V_{j-1}\}$$

for $0\le i\le n\text{.}$ Then $(E_0,\dots ,E_n)$ is a 'flag of sets', i.e. we have

(A.1)

(i) $\text{Card}\left(E_i\right)=i$ for $0\le i\le n,$ (ii) $E_{i-1}\subset E_i$ for $1\le i\le n\text{.}$

		Proof.
	(i) Fix $i$ and let $d_j=\text{dim}\hspace{0.17em}(U_i\cap V_j)\text{.}$ Since $$\frac{V_i\cap V_j}{U_i\cap V_{j-1}}=\frac{U_i\cap V_j}{(U_i\cap V_j)\cap V_{j-1}}\cong \frac{(U_i\cap V_j)+V_{j-1}}{V_{j-1}}\subset \frac{V_j}{V_{j-1}}$$ it follows that $d_j-d_{j-1}=0$ or $1\text{.}$ Since $d_0=0$ and $d_n=i,$ there are therefore $i$ jumps in the sequence $(d_0,d_1,\dots ,d_n),$ which proves (i). (ii) Suppose that $j\notin E_i,$ so that $U_i\cap V_j=U_i\cap V_{j-1}\text{.}$ Intersecting with $U_{i-1},$ we see that $j\notin E_{i-1}\text{.}$ Hence $E_{i-1}\subset E_i\text{.}$ $\square$

From (A.1) it follows that that each $U\in F$ determines a permutation $w\in S_n$ as follows: $w\left(i\right)$ is the unique element of $E_i-E_{i-1},$ for $i=1,2,\dots ,n\text{.}$ Let $\varphi :F\to S_n$ denote the mapping so defined.

The symmetric group acts on $V$ by permuting the basis elements $e_i:$

$$w\left(e_i\right)=e_{w\left(i\right)}$$

for $w\in S_n$ and $1\le i\le n\text{.}$ Hence we may regard $S_n$ as a subgroup of $G\text{.}$

(A.2) Let $U\in F,$ $w\in S_n\text{.}$ Then $\varphi \left(U\right)=w$ if and only if $U=bwV$ for some $b\in B\text{.}$

		Proof.
	Suppose $\varphi \left(U\right)=w\text{.}$ Then for $i-1,\dots ,n$ we have $$\begin{array}{cc}\text{(1)}& U_i\cap V_{w\left(i\right)}\supset U_i\cap V_{w\left(i\right)-1}\end{array}$$ and $$\begin{array}{cc}\text{(2)}& U_{i-1}\cap V_{w\left(i\right)}=U_{i-1}\cap V_{w\left(i\right)-1}\end{array}$$ By virtue of (1) we can choose $u_i\in U_i$ of the form $$\begin{array}{cc}\text{(3)}& u_i=e_{w\left(i\right)}+\text{lower terms}\end{array}$$ where by 'lower terms' is meant a linear combination of $e_1,\dots ,e_{w\left(i\right)-1}\text{;}$ and $u_i\notin U_{i-1}$ by virtue of (2). By rewriting (3) in the form $$u_{w^{-1}\left(j\right)}=e_j+\text{lower terms}(1\le j\le n)$$ we see that there exists $b\in B$ such that $u_{w^{-1}\left(j\right)}=b{e}_j$ for all $j,$ or equivalently $$u_i=b{e}_{w\left(i\right)}=bw{e}_i\text{.}$$ Hence $U-bwV$ as required. For the converse it is enough to show that (i) $\varphi \left(wV\right)=w$ and (ii) $\varphi \left(bU\right)=\varphi \left(U\right)$ for all $b\in B$ and $U\in F\text{.}$ As to (i), $w{V}_i\cap V_j$ is spanned by the basis vectors $e_{w\left(k\right)}$ such that $k\le i$ and $w\left(k\right)\le j,$ and therefore $w{V}_i\cap V_j\ne w{V}_i\cap V_{j-1}$ if and only if $j=w\left(k\right)$ for some $k\le i\text{.}$ Thus the set $E_i\left(wV\right)$ consists of $w\left(1\right),\dots ,w\left(i\right),$ which establishes (i). Finally as to (ii), we have $b{U}_i\cap V_j=b(U_i\cap V_j)$ if $b\in B,$ so that $E_i\left(bU\right)=E_i\left(U\right)$ and hence $\varphi \left(bV\right)=\varphi \left(U\right)$ as required. $\square$

From (A2) we have immediately

(A3) (Bruhat decomposition) $G$ is the disjoint union of the double cosets $BwB,$ $w\in S_n\text{.}$

For each $w\in S_n,$ let

$$C_w=\left(BwB\right)/B\subset G/B=F\text{.}$$

The subsets $C_w$ are the Schubert cells in the flag manifold $F\text{.}$ By (A.3), $F$ is the disjoint union of the $C_w\text{.}$

Let $U\in F\text{.}$ Then $U\in C_w$ if and only if $U$ has a basis $(u_1,\dots ,u_n)$ such that $u_i\in V_{w\left(i\right)}-V_{w\left(i\right)-1}$ for each $i\text{.}$ We may normalize the $u_i$ by taking

$$u_i=e_{w\left(i\right)}+\text{lower terms.}$$

We can then subtract from $u_i$ suitable multiples of the $u_k$ for which $k<i$ and $w\left(k\right)<w\left(i\right),$ so as to make the coefficient of $e_{w\left(k\right)}$ in $u_i$ zero for each such $k\text{.}$ Then $u_i$ is replaced by a vector of the form

$$e_{w\left(i\right)}+\sum _j{a}_{ij}{e}_j$$

where the sum is over $j<w\left(i\right)$ such that $j\ne w\left(k\right)$ for any $k<i,$ i.e., such that $j<w\left(i\right)$ and $w^{-1}\left(j\right)>i,$ or equivalently $(i,j)\in D\left(w\right),$ the diagram of $w\text{.}$

(A.4) Let $U\in F\text{.}$ Then $U\in C_w$ if and only if $U$ has a basis $(u_1,\dots ,u_n)$ of the form

$$u_i=e_{w\left(i\right)}+\sum _j{a}_{ij}{e}_j$$

where the sum is over all $j$ in the $i^{\text{th}}$ row of the diagram of $w,$ and the coefficients $a_{ij}$ are arbitrary elements of the field $K\text{.}$ Moreover, the $a_{ij}$ are uniquely determined by the flag $U,$ and the mapping $C_w\to K^{D\left(w\right)}$ so defined is a bijection.

		Proof.
	Clearly each "matrix" $a=\left(a_{ij}\right)$ of shape $D\left(w\right)$ determines a basis $(u_1,\dots ,u_n)$ of $V$ as above, and hence a flag $U\in C_w\text{.}$ If $a^{*}=\left(a_{ij}^{*}\right)$ determines $(u_1^{*},\dots ,u_n^{*})$ and the same flag $U,$ then each $u_i^{*}$ must be expressible as $$u_i^{*}=u_i+\sum _{j<i}{c}_{ij}{u}_j,$$ and from the form of $u_i^{*}$ and the $u_j$ it follows that $u_i^{*}=u_i$ for each $i,$ and hence $a^{*}=a\text{.}$ $\square$

Since $\text{Card}\hspace{0.17em}D\left(w\right)=\ell \left(w\right)$ it follows from (A.4) that the Schubert cell $C_w$ is isomorphic to affine space of dimension $\ell \left(w\right)\text{.}$

Let $U\in F$ and let $(u_1,\dots ,u_n)$ be any basis of $U\text{.}$ Since $u_1,\dots ,u_i$ is a basis for $U_i$ for each $i=1,\dots ,n-1,$ the flag $U$ determines each of the exterior products $u_1\wedge \dots \wedge u_i\in {\Lambda }^i\left(V\right)$ up to a nonzero scalar multiple, and hence $U$ determines the vector

$$\begin{array}{cc}\text{(1)}& u_1\otimes (u_1\wedge u_2)\otimes \dots \otimes (u_1\wedge \dots \wedge u_{n-1})\in E\end{array}$$

up to a nonzero scalar multiple, where $E=V\otimes {\Lambda }^{2}V\otimes \dots \otimes {\Lambda }^{n-1}V\text{.}$ If $P\left(E\right)$ denotes the projective space of $E$ (i.e. the space whose points are the lines in $E\text{),}$ we have an injective mapping

$$\pi :F\mapsto P\left(E\right)$$

(the Plücker embedding) for which $\pi \left(U\right)$ is the line in $E$ generated by the vector (1).

Assume from now on that the field $K$ is the field of complex numbers. Then the embedding $\pi$ realizes the flag manifold $F$ as a complex projective algebraic variety, which is smooth because $F$ has a transitive group of automorphisms (namely $G\text{).}$ Each Schubert cell $C_w$ is a locally closed subvariety of $F,$ isomorphic to affine space of dimension $\ell \left(w\right)\text{.}$

For each $w\in S_n$ let

$$X_w={\bar{C}}_w$$

be the closure of $C_w$ in $F\text{.}$ The $X_w$ are the Schubert varieties in $F,$ and a flag $U$ lies in $X_w$ if and only if $U$ has a basis $(u_1,\dots ,u_n)$ such that $u_i\in V_{w\left(i\right)}$ for each $i\text{.}$ Each $X_w$ is in fact a union of Schubert cells $C_v\text{:}$ if $(a_1,\dots ,a_p)$ is a reduced word for $w,$ then $C_v\subset X_w$ if and only if $v$ is of the form $s_{b_1}\dots s_{b_q}$ where $(b_1,\dots ,b_q)$ is a subsequence of $(a_1,\dots ,a_p),$ that is to say if and only if $v\le w$ in the Bruhat order. In particular, $X_1=C_1$ is the single point $V\in F\text{.}$ At the other extreme, if $w_0$ is the longest element of $S_n,$ then $X_{w_0}$ is the whole of $F,$ and the dimension of $F$ is $\ell \left(w_0\right)=\frac{1}{2}n(n-1)\text{.}$

Let $H^{*}(F;\mathbb{Z})$ be the cohomology ring (with integral coefficients) of the projective variety $F\text{.}$ Each closed subvariety $X$ of $F$ determines an element $\left[X\right]\in H^{*}(F;\mathbb{Z}),$ and cup-product in $H^{*}(F;\mathbb{Z})$ corresponds, roughly speaking, to intersection of subvarieties. In particular, for each $w\in S_n,$ we have a cohomology class $\left[X_w\right]\in H^{*}(F;\mathbb{Z}),$ and it is a consequence of the cell decomposition (A.3) of $F$ that the $\left[X_w\right]$ form a $\mathbb{Z}\text{-basis}$ of $H^{*}(F;\mathbb{Z})\text{.}$ In particular, $\left[X_{w_0}\right]$ is the identity element.

The connection between the classes $\left[X_w\right]$ and the Schubert polynomials ${𝔖}_w(w\in S_n)$ is given by

(A.5) There is a surjective ring homomorphism

$$\alpha :\mathbb{Z}[x_1,\dots ,x_n]\to H^{*}(F;\mathbb{Z})$$

such that

$$\alpha \left({𝔖}_w\right)=\left[X_{w_{0}w}\right]$$

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

		Proof.
	Let us temporarily write $${\sigma }_w=\left[X_{w_{0}w}\right]$$ for $w\in S_n\text{.}$ Monk [Mon1959] proved that for all $w\in S_n$ and $r=1,\dots ,n-1$ $$\begin{array}{cc}\text{(1)}& {\sigma }_w·{\sigma }_{s_r}=\sum _t{\sigma }_{wt}\end{array}$$ where the sum on the right hand side is over all transpositions $t=t_{ij}$ such that $i\le r<j\le n$ and $\ell \left(wt\right)=\ell \left(w\right)+1,$ as in (4.15''), Define ${\xi }_1,\dots ,{\xi }_n\in H^{*}(F,\mathbb{Z})$ by $$\begin{array}{ccc}{\xi }_1& =& {\sigma }_1\\ {\xi }_i& =& {\sigma }_i-{\sigma }_{i-1}(2\le i\le n-1)\\ {\xi }_n& =& -{\sigma }_{n-1}\end{array}$$ From (1) we deduce the counterpart of (4.16): if $r$ is the last descent of $w$ (so that $r\le n-1\text{),}$ then we have $$\begin{array}{cc}\text{(2)}& {\sigma }_w={\sigma }_v{\xi }_r+\sum _{w\prime }{\sigma }_{w\prime }\end{array}$$ where $v,w\prime$ are as in (4.16). Now iteration of (4.16) will ultimately express ${𝔖}_w$ as a sum of monomiais, i.e. as a polynomial in $x_1,\dots ,x_{n-1}\text{;}$ and iteration of (2) will express ${\sigma }_w$ as the same polynomial in ${\xi }_1,\dots ,{\xi }_{n-1}\text{.}$ Hence if we define $\alpha :P_n\mapsto H^{*}(F;\mathbb{Z})$ by $\alpha \left(x_i\right)={\xi }_i$ $(1\le i\le n),$ we have ${\sigma }_w=\alpha \left({𝔖}_w\right)$ for all $w\in S_n,$ and the proof of (A.5) is complete. $\square$

In fact the kernel of the homomorphism $\alpha$ is generated by the elementary symmetric functions $e_1,\dots ,e_n$ of the $x\text{'s.}$

We shall draw one consequence of (A.5) that we have not succeeded in deriving directly from the definition (4.1) of the Schubert polynomials. Since the ${\sigma }_w,w\in S_n,$ form a $\mathbb{Z}\text{-basis}$ of $H^{*}(F;\mathbb{Z}),$ any product ${\sigma }_u{\sigma }_v(u,v\in S_n)$ is uniquely a linear combination of the ${\sigma }_w,$ and it follows from intersection theory on $F$ that the coefficient of ${\sigma }_w$ in ${\sigma }_u{\sigma }_v$ is a non-negative integer. From this we deduce

(A.6) Let $u,v$ be permutations, and write ${𝔖}_u{𝔖}_v$ as an integral linear combination of the ${𝔖}_w,$ say

$$\begin{array}{cc}\text{(1)}& {𝔖}_u{𝔖}_v=\sum _w{c}_{uv}^w{𝔖}_w\text{.}\end{array}$$

Then the coefficients $c_{uv}^w$ are non-negative.

We have only to choose $n$ sufficiently large so that $u,v$ and all the permutations $w$ such that $c_{uv}^w\ne 0$ lie in $S_n,$ and then apply the homomorphism $\alpha$ of (A,5).

Remark. The coefficients $c_{uv}^w$ in (A.6) are zero unless

(a)	$\ell \left(w\right)=\ell \left(u\right)+\ell \left(v\right),$
(b)	$u\le w$ and $v\le w\text{.}$

For ${𝔖}_u{𝔖}_v$ is homogeneous of degree $\ell \left(u\right)+\ell \left(v\right),$ which gives condition (a). Also we have

$$\begin{array}{ccc}{c}_{uv}^w& =& {\partial }_w\left({𝔖}_u{𝔖}_v\right)\\ & =& \sum _{v_1\le w}{v}_1{\partial }_{w/v_1}\left({𝔖}_u\right){\partial }_{v_1}\left({𝔖}_v\right)\end{array}$$

by (2.17), and the only possible nonzero term in this sum is that corresponding to $v_1=v\text{.}$ Hence if $c_{uv}^w\ne 0$ we must have $v\le w,$ and by symmetry also $u\le w\text{.}$

Notes and References

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

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