Vrftsjtryk

Let $V$ be a vector space and $W_1 subsetneq W_2 subsetneq ... subsetneq W_ell subsetneq V $ a flag of subspaces.

The associated Schubert variety is defined as :
$ Omega ( W_{ bullet } ) = { W in Gr_k ( V ) | mathrm{dim} ( W bigcap W_i ) geq i , i = 1 , dots , ell } $.

My question is to know what is the idea behind the fact to define this kind of varieties ?

Why did we choose to define a Shubert variety with this condition that :
$ mathrm{dim} ( W bigcap W_i ) geq i , i = 1 , dots , ell $

Thanks in advance for your help.

I get $G=mathrm{GL}(n,mathbb{C})$, let $B$ be a Borel subgroup of $G$, let $P$ be a parabolic subgroup of $G$ and let $T$ be a maximal torus in $G$; we know that:

• up to conjugation: $T$ is the group of diagonal matrices, $B$ is the group of upper triangular matrices; in particular: $B$ is $Tltimes U$, the semidirect product of $T$ with the group $U$ of unipotent upper triangular matrices;




• $G_{displaystyle/P}$ is the variety $mathcal{F}$ of flags of type $(1leq m_1<dots<m_rleq n)$ of $mathbb{V}$ (a complex vector space of dimension $n$); in particular $G_{displaystyle/B}$ is the variety $mathcal{B}$ of complete flags of $mathbb{V}$;




• without loss of generality, $T$ is contained in $P$;




• the Weyl group $W$ of $G$ is defined as $N_G(T)_{displaystyle/C_G(T)}=N_G(T)_{displaystyle/T}$ (the normalizer of $T$ in $G$ quotiented by the centralizer of $T$ in $G$), because $T$ is a maximal torus then $T=C_G(T)$ (ever);

and in particular, the Weyl group of $mathrm{GL}(n,mathbb{C})$ is the $n$-th symmetryc group $S_n$.

One can consider the action:
begin{gather}
alpha:(M,F_{bullet})in Gtimesmathcal{F}to Mcdot F_{bullet}inmathcal{F},\
F_{bullet}equiv{underline{0}}subsetneqqleftlangle e_1,dots, e_{m_1}rightranglesubsetneqqleftlangle e_1,dots, e_{m_2}rightranglesubsetneqqdotssubsetneqqmathbb{V},\
Mcdot F_{bullet}equiv{underline{0}}subsetneqqleftlangle Mcdot e_1,dots,Mcdot e_{m_1}rightranglesubsetneqqleftlangle Mcdot e_1,dots,Mcdot e_{m_2}rightranglesubsetneqqdotssubsetneqqmathbb{V}
end{gather}
where ${e_1,dots,e_n}$ is a basis of $mathbb{V}$.

One can prove that:
begin{equation}
mathrm{Fix}^{alpha}_T(mathcal{B})={E_wequiv wBinmathcal{B}mid win W},
end{equation}
where $E_{1_W}$ is the standard complete flag
begin{equation}
{underline{0}}subsetneqqlangle e_1ranglesubsetneqqlangle e_1,e_2ranglesubsetneqqdotssubsetneqqmathbb{V};
end{equation}
for example, see theorem 10.2.7 [BL].

Considered the set:
begin{gather}
mathrm{Orb}^{alpha}_B(E_w)={bcdot E_winmathcal{F}mid bin B}={ucdot E_winmathcal{F}mid uin U}=mathrm{Orb}^{alpha}_U(E_w),
end{gather}
it is called Schubert cell $C_w$ of $mathcal{F}$; one defines Schubert variety $X_w$ the Zariski closure of $C_w$ in $mathcal{F}$ (see definitions 1.1.2 and 1.2.2 from [B]).

Remarks.

1. Any Schubert cell and variety is parametrized by an element of $W$!


2. The Grassmannians are particular flag varieties!


3. Any complex Lie group is embeddable in some $mathrm{GL}(N,mathbb{C})$, therefore $alpha$ makes sense for any reductive, complex Lie group; and one can define again the flag varieties, the Grassmannians and the Schubert cells and varieties.

Choice $1leq k<n$, let $E_{w_k}$ be the $k$-dimensional space in the flag $E_w$; by the previous reasoning, $E_{w_k}$ is a $T$-fixed point in $G(k,n)$ (the Grassmannian of the $k$-plane of $mathbb{V}$); in particular, there exists $win W$ such that
begin{equation}
E_{w_k}=leftlangle e_{w(1)},dots,e_{w(k)}rightrangle
end{equation}
which corresponds to $left[e_{w(1)}wedgedotswedge e_{w(k)}right]=left[e_{underline w}right]in G(k,n)subseteqmathbb{P}left(bigwedge^kmathbb{V}right)$.

Let
begin{equation}
A_{w_k}=left(a_j^iright)inmathbb{C}_n^kmid e_{w(i)}=sum_{j=1}^na_i^je_jequiv a_i^je_j,
end{equation}
that is
begin{equation}
a_j^i=begin{cases}
1iff j=w(i)\
0iff text{otherwise}
end{cases};
end{equation}
because $C_w$ in $G(k,n)$ is the $U$-orbit of $E_{w_k}$, then $C_w$ is in bijection with set
begin{equation}
left{uA_{w_k}inmathbb{C}_n^kmid uin Uright}=left{M=left(m_i^jright)inmathbb{C}_n^kmidbegin{cases}
m_{w(i)}^j=delta_i^j\
m_i^j=0iff i>w(j)
end{cases}right},
end{equation}
that is $C_w$ is the set
begin{equation}
left{left[m_i^1e_{w(1)}wedgedotswedge m_i^ke_{w(k)}right]in G(k,n)midbegin{cases}
m_{w(i)}^j=delta_i^j\
m_i^j=0iff i>w(j)
end{cases}right};
end{equation}
in particular, $C_w$ is an affine agebraic variety in $G(k,n)$.

For these and other details, see lemmata 1.4.3 and 1.4.4 from [L].

Defined
begin{gather}
I_{k,n}=left{underline{i}=(i_1,dots,i_k)in{1,dots,n}^kmid i_1<dots<i_kright}\
underline{i},underline{j}in I_{k,n},,underline{i}lequnderline{j}iffforall hin{1,dots,k},i_hleq j_h;
end{gather}
by the previous reasoning and definition
begin{equation}
X_w=left{[v_1wedgedotswedge v_k]in G(k,n)midforallunderline{i}notlequnderline{w}in I_{k,n},,p_{underline{i}}([v_1wedgedotswedge v_k])=0right}
end{equation}
where the support of $underline{w}$ is ${w(1),dots,w(k)}$; in particular, $X_w$ is a projetive algebraic variety in $G(k,n)$.

As usual, $p_{underline{i}}$ is the $underline{i}$-th Plücker projection, that is the minor $detleft(a_h^{i_h}right)_{hin{1,dots,k}}=d_{underline i}$, where $v_i=a_i^je_j$ and $underline{i}=(i_1,dots,i_k)$.

Now, how can we interpret $X_w$?

First of all, let $underline{i}in I_{k,n}$, we can define
begin{gather}
win S_nmidforall h^{prime}in{1,dots,k},,w(h^{prime})=i_{h^{prime}},\
forall h^{primeprime}in{1,dots,n-k},,w(h^{primeprime}),text{is the},h^{primeprime}text{-th element of the ordered set},{1,dots,n}setminus{i_1,dots,i_k},
end{gather}
in this way there exists a bijection between $S_n$ and $I_{k,n}$. Considering the flag
begin{equation}
F_{underline{i}}equiv{underline0}<leftlangle e_1,dots,e_{i_1}rightrangle=F_{i_1}<leftlangle e_1,dots,e_{i_2}rightrangle=F_{i_2}<dots<leftlangle e_1,dots,e_{i_k}rightrangle=F_{i_k}<mathbb{V};
end{equation}
let $[W]in X=X_{underline i}$, by construction:
begin{equation}
[W]=[v_1wedgedotswedge v_k]=left[sum_{underline{j}lequnderline{i}}d_{underline j}e_{underline j}right]
end{equation}
in particular:
begin{equation}
forall hin{1,dots,k},,dimleft(Wcap F_{i_h}right)geq h
end{equation}
and vice versa; for other details, see lemma 1.4.5 from [L].

For other interpretation of the Schubert varieties, one can see the chapter 1, sections 1 and 2 from [B].

Bibliography

[B] Brion M. - Lectures on the geometry of flag varieties; available on arxiv.org

[BL] Brown J., Lakshimibai V. - Flag Varieties, Northeastern University

[L] Littelmann P. - Schubert varieties; available on personal web page