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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.

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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 $T\ltimes U$, 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 $(1\leq m_1<\dots<m_r\leq n)$ 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 G\times\mathcal{F}\to M\cdot F_{\bullet}\in\mathcal{F},\\ F_{\bullet}\equiv\{\underline{0}\}\subsetneqq\left\langle e_1,\dots, e_{m_1}\right\rangle\subsetneqq\left\langle e_1,\dots, e_{m_2}\right\rangle\subsetneqq\dots\subsetneqq\mathbb{V},\\ M\cdot F_{\bullet}\equiv\{\underline{0}\}\subsetneqq\left\langle M\cdot e_1,\dots,M\cdot e_{m_1}\right\rangle\subsetneqq\left\langle M\cdot e_1,\dots,M\cdot e_{m_2}\right\rangle\subsetneqq\dots\subsetneqq\mathbb{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_w\equiv wB\in\mathcal{B}\mid w\in W\}, \end{equation} where $E_{1_W}$ is the standard complete flag \begin{equation} \{\underline{0}\}\subsetneqq\langle e_1\rangle\subsetneqq\langle e_1,e_2\rangle\subsetneqq\dots\subsetneqq\mathbb{V}; \end{equation} for example, see theorem 10.2.7 [BL].

Considered the set: \begin{gather} \mathrm{Orb}^{\alpha}_B(E_w)=\{b\cdot E_w\in\mathcal{F}\mid b\in B\}=\{u\cdot E_w\in\mathcal{F}\mid u\in 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 $1\leq k<n$, 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 $w\in W$ such that \begin{equation} E_{w_k}=\left\langle e_{w(1)},\dots,e_{w(k)}\right\rangle \end{equation} which corresponds to $\left[e_{w(1)}\wedge\dots\wedge e_{w(k)}\right]=\left[e_{\underline w}\right]\in G(k,n)\subseteq\mathbb{P}\left(\bigwedge^k\mathbb{V}\right)$.

Let \begin{equation} A_{w_k}=\left(a_j^i\right)\in\mathbb{C}_n^k\mid e_{w(i)}=\sum_{j=1}^na_i^je_j\equiv a_i^je_j, \end{equation} that is \begin{equation} a_j^i=\begin{cases} 1\iff j=w(i)\\ 0\iff \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}\in\mathbb{C}_n^k\mid u\in U\right\}=\left\{M=\left(m_i^j\right)\in\mathbb{C}_n^k\mid\begin{cases} m_{w(i)}^j=\delta_i^j\\ m_i^j=0\iff i>w(j) \end{cases}\right\}, \end{equation} that is $C_w$ is the set \begin{equation} \left\{\left[m_i^1e_{w(1)}\wedge\dots\wedge m_i^ke_{w(k)}\right]\in G(k,n)\mid\begin{cases} m_{w(i)}^j=\delta_i^j\\ m_i^j=0\iff 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\}^k\mid i_1<\dots<i_k\right\}\\ \underline{i},\underline{j}\in I_{k,n},\,\underline{i}\leq\underline{j}\iff\forall h\in\{1,\dots,k\},i_h\leq j_h; \end{gather} by the previous reasoning and definition \begin{equation} X_w=\left\{[v_1\wedge\dots\wedge v_k]\in G(k,n)\mid\forall\underline{i}\not\leq\underline{w}\in I_{k,n},\,p_{\underline{i}}([v_1\wedge\dots\wedge v_k])=0\right\} \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 $\det\left(a_h^{i_h}\right)_{h\in\{1,\dots,k\}}=d_{\underline i}$, 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} w\in S_n\mid\forall h^{\prime}\in\{1,\dots,k\},\,w(h^{\prime})=i_{h^{\prime}},\\ \forall h^{\prime\prime}\in\{1,\dots,n-k\},\,w(h^{\prime\prime})\,\text{is the}\,h^{\prime\prime}\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\}<\left\langle e_1,\dots,e_{i_1}\right\rangle=F_{i_1}<\left\langle e_1,\dots,e_{i_2}\right\rangle=F_{i_2}<\dots<\left\langle e_1,\dots,e_{i_k}\right\rangle=F_{i_k}<\mathbb{V}; \end{equation} let $[W]\in X=X_{\underline i}$, by construction: \begin{equation} [W]=[v_1\wedge\dots\wedge v_k]=\left[\sum_{\underline{j}\leq\underline{i}}d_{\underline j}e_{\underline j}\right] \end{equation} in particular: \begin{equation} \forall h\in\{1,\dots,k\},\,\dim\left(W\cap 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

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