Let $\mathcal{F}\colon V_1\subseteq\dotsb\subseteq V_n$ and $\mathcal{F}'\colon U_1\subseteq\dotsb\subseteq U_n$ be two complete flags. Then $\mathcal{F}$ and $\mathcal{F}'$ are said to be in relative position $w$, where $w\in S_n$ is a permutation, if $$\dim V_i\cap U_j=\#(\{ 1,\dotsc,i \}\cap \{ w(1),\dots,w(j) \})$$ for any $i,j\in\{1,\dotsc,n\}$.

In particular, the variety $X((1\ 2))$ consists of the flags $V_1\subseteq V_2$ in which $V_2$ is an $F$-stable plane and $V_1$ is an $F$-unstable line. So it is $$\operatorname{Gr}(1,3)^F\times \mathbb{P}^1\backslash\mathbb{P}^1(\mathbb{F}_q),$$ a disjoint union of copies of the Drinfeld curve.

Computations like this are very useful for making explicit connections with combinatorics. For a reference, you may want to take a look at Steinberg's paper An occurrence of the Robinson–Schensted correspondence.

Let $\mathcal{F}\colon V_1\subseteq\dotsb\subseteq V_n$ and $\mathcal{F}'\colon U_1\subseteq\dotsb\subseteq U_n$ be two complete flags. Then $\mathcal{F}$ and $\mathcal{F}'$ are said to be in relative position $w$, where $w\in S_n$ is a permutation, if $$\dim V_i\cap U_j=\#(\{ 1,\dotsc,i \}\cap \{ w(1),\dots,w(j) \})$$ for any $i,j\in\{1,\dotsc,n\}$.

In particular, the variety $X((1\ 2))$ consists of the flags $V_1\subseteq V_2$ in which $V_2$ is an $F$-stable plane and $V_1$ is an $F$-unstable line. So it is $$\operatorname{Gr}(1,3)^F\times \mathbb{P}^1\setminus\mathbb{P}^1(\mathbb{F}_q),$$ a disjoint union of copies of the Drinfeld curve.

Computations like this are very useful for making explicit connections with combinatorics. For a reference, you may want to take a look at Steinberg's paper An occurrence of the Robinson–Schensted correspondence.

Let $\mathcal{F}\colon V_1\subseteq...\subseteq V_n$ and $\mathcal{F}'\colon U_1\subseteq...\subseteq U_n$ be two complete flags. Then $\mathcal{F}$ and $\mathcal{F}'$ are said to be in relative position $w$, where $w\in S_n$ is a permutation, if $$\dim V_i\cap U_j=\#(\{ 1,...,i \}\cap \{ w(1),...,w(j) \})$$ for any $i,j\in\{1,...,n\}$.

In particular, the variety $X((12))$ consists of the flags $V_1\subseteq V_2$ in which $V_2$ is an $F$-stable plane and $V_1$ is an $F$-unstable line. So it is $$\mathrm{Gr}(1,3)^F\times \mathbb{P}^1\backslash\mathbb{P}^1(\mathbb{F}_q),$$ a disjoint union of copies of the Drinfeld curve.

Computations like this are very useful for making explicit connections with combinatorics. For a reference, you may want to take a look at Steinberg's paper An occurrence of the Robinson-Schensted correspondence.

Let $\mathcal{F}\colon V_1\subseteq\dotsb\subseteq V_n$ and $\mathcal{F}'\colon U_1\subseteq\dotsb\subseteq U_n$ be two complete flags. Then $\mathcal{F}$ and $\mathcal{F}'$ are said to be in relative position $w$, where $w\in S_n$ is a permutation, if $$\dim V_i\cap U_j=\#(\{ 1,\dotsc,i \}\cap \{ w(1),\dots,w(j) \})$$ for any $i,j\in\{1,\dotsc,n\}$.

In particular, the variety $X((1\ 2))$ consists of the flags $V_1\subseteq V_2$ in which $V_2$ is an $F$-stable plane and $V_1$ is an $F$-unstable line. So it is $$\operatorname{Gr}(1,3)^F\times \mathbb{P}^1\backslash\mathbb{P}^1(\mathbb{F}_q),$$ a disjoint union of copies of the Drinfeld curve.

Computations like this are very useful for making explicit connections with combinatorics. For a reference, you may want to take a look at Steinberg's paper An occurrence of the Robinson–Schensted correspondence.