## Notes on Schubert PolynomialsAppendix

Last update: 3 July 2013

## Schubert varieties

Let $V$ be a vector space of dimension $n$ over a field $K,$ and let $\left({e}_{1},\dots ,{e}_{n}\right)$ 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

$0=U0⊂U1⊂…⊂ Un=V$

with strict inclusions at each stage, so that $\text{dim} {U}_{i}=i$ 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

$gej=∑i=1n gijei (1≤j≤n)$

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 $\left({u}_{1},\dots ,{u}_{n}\right)$ 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 $gV↦gB$ is a bijection of $F$ onto the coset space $G/B\text{.}$

For a flag $U=\left({U}_{i}\right),$ let

$Ei=Ei(U)= { j:1≤j≤n and Ui∩Vj≠Ui ∩Vj-1 }$

for $0\le i\le n\text{.}$ Then $\left({E}_{0},\dots ,{E}_{n}\right)$ 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 ${d}_{j}=\text{dim} \left({U}_{i}\cap {V}_{j}\right)\text{.}$ Since $Vi∩Vj Ui∩Vj-1 = Ui∩Vj (Ui∩Vj)∩Vj-1 ≅ (Ui∩Vj)+Vj-1 Vj-1 ⊂VjVj-1$ 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 $\left({d}_{0},{d}_{1},\dots ,{d}_{n}\right),$ 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 ${e}_{i}:$

$w(ei)= ew(i)$

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 $(1) Ui∩Vw(i)⊃ Ui∩Vw(i)-1$ and $(2) Ui-1∩ Vw(i)= Ui-1∩ Vw(i)-1$ By virtue of (1) we can choose ${u}_{i}\in {U}_{i}$ of the form $(3) ui=ew(i)+ lower terms$ 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 $uw-1(j)= ej+lower terms (1≤j≤n)$ we see that there exists $b\in B$ such that ${u}_{{w}^{-1}\left(j\right)}=b{e}_{j}$ for all $j,$ or equivalently $ui=bew(i)= bwei.$ 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 $b{U}_{i}\cap {V}_{j}=b\left({U}_{i}\cap {V}_{j}\right)$ 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

$Cw=(BwB)/B ⊂G/B=F.$

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 $\left({u}_{1},\dots ,{u}_{n}\right)$ 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

$ui=ew(i)+ lower terms.$

We can then subtract from ${u}_{i}$ suitable multiples of the ${u}_{k}$ for which $k and $w\left(k\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

$ew(i)+∑j aijej$

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

$ui=ew(i)+ ∑jaijej$

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 $\left({u}_{1},\dots ,{u}_{n}\right)$ of $V$ as above, and hence a flag $U\in {C}_{w}\text{.}$ If ${a}^{*}=\left({a}_{ij}^{*}\right)$ determines $\left({u}_{1}^{*},\dots ,{u}_{n}^{*}\right)$ and the same flag $U,$ then each ${u}_{i}^{*}$ must be expressible as $ui*=ui+ ∑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 $\text{Card} D\left(w\right)=\ell \left(w\right)$ 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 $\left({u}_{1},\dots ,{u}_{n}\right)$ 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

$(1) u1⊗ (u1∧u2)⊗ …⊗ (u1∧…∧un-1) ∈E$

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

$π:F↦P(E)$

(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

$Xw=C‾w$

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 $\left({u}_{1},\dots ,{u}_{n}\right)$ 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 $\left({a}_{1},\dots ,{a}_{p}\right)$ 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 $\left({b}_{1},\dots ,{b}_{q}\right)$ is a subsequence of $\left({a}_{1},\dots ,{a}_{p}\right),$ 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 $\ell \left({w}_{0}\right)=\frac{1}{2}n\left(n-1\right)\text{.}$

Let ${H}^{*}\left(F;ℤ\right)$ be the cohomology ring (with integral coefficients) of the projective variety $F\text{.}$ Each closed subvariety $X$ of $F$ determines an element $\left[X\right]\in {H}^{*}\left(F;ℤ\right),$ and cup-product in ${H}^{*}\left(F;ℤ\right)$ corresponds, roughly speaking, to intersection of subvarieties. In particular, for each $w\in {S}_{n},$ we have a cohomology class $\left[{X}_{w}\right]\in {H}^{*}\left(F;ℤ\right),$ and it is a consequence of the cell decomposition (A.3) of $F$ that the $\left[{X}_{w}\right]$ form a $ℤ\text{-basis}$ of ${H}^{*}\left(F;ℤ\right)\text{.}$ In particular, $\left[{X}_{{w}_{0}}\right]$ is the identity element.

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

(A.5) There is a surjective ring homomorphism

$α:ℤ[x1,…,xn] →H*(F;ℤ)$

such that

$α(𝔖w)= [Xw0w]$

for each $w\in {S}_{n}\text{.}$

 Proof. Let us temporarily write $σw= [Xw0w]$ for $w\in {S}_{n}\text{.}$ Monk [Mon1959] proved that for all $w\in {S}_{n}$ and $r=1,\dots ,n-1$ $(1) σw· σsr= ∑tσwt$ where the sum on the right hand side is over all transpositions $t={t}_{ij}$ such that $i\le r and $\ell \left(wt\right)=\ell \left(w\right)+1,$ as in (4.15''), Define ${\xi }_{1},\dots ,{\xi }_{n}\in {H}^{*}\left(F,ℤ\right)$ by $ξ1 = σ1 ξi = σi-σi-1 (2≤i≤n-1) ξn = -σn-1$ 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 $(2) σw=σvξr+ ∑w′σw′$ 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 $\alpha :{P}_{n}↦{H}^{*}\left(F;ℤ\right)$ by $\alpha \left({x}_{i}\right)={\xi }_{i}$ $\left(1\le i\le n\right),$ 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 $ℤ\text{-basis}$ of ${H}^{*}\left(F;ℤ\right),$ any product ${\sigma }_{u}{\sigma }_{v}\left(u,v\in {S}_{n}\right)$ 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

$(1) 𝔖u𝔖v=∑w cuvw𝔖w.$

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

$cuvw = ∂w(𝔖u𝔖v) = ∑v1≤wv1 ∂w/v1 (𝔖u)∂v1 (𝔖v)$

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.