Let $G=SL\_3(\mathbb{C})$ and $X=G/B$ be the associated full flag variety. Fix a non-degenerate symmetric quadratic form $Q$ on $\mathbb{C}^3$. This gives an order $2$ automorphism $F\_Q$ of $X$, mapping a flag $(L,P)$ to $(P^{\perp},L^{\perp})$. Is there any explicit description of the quotient $X/F\_Q$ ?



Quotient of Flag varieties
Let's coordinatize $G/B = Proj\ \mathbb C[m\_{11}\ldots m\_{33}]/\langle trace=0, rank \leq 1\rangle$, with $F\_Q(M) = M^T$, the image of $(L,P) \mapsto L P^T$ (a column vector times row vector).


Then the $F\_Q$-invariants are generated by $n\_i := m\_{ii}, q\_{i < j} := m\_{ij}m\_{ji}, s\_{i < j} := s\_{ij}s\_{ji}$. They satisfy
$${n}\_{1}+{n}\_{2}+{n}\_{3}, \qquad
{p}\_{12} {p}\_{13}+{p}\_{12} {p}\_{23}+{p}\_{13} {p}\_{23},\qquad
{n}\_{2} {p}\_{13}+{n}\_{2} {p}\_{23}+{n}\_{3} {p}\_{23}$$
$${n}\_{3} {p}\_{12}+{n}\_{2} {p}\_{23}+{n}\_{3} {p}\_{23},\qquad {n}\_{3}^{2}+{p}\_{13}+{p}\_{23},\qquad {n}\_{2} {n}\_{3}-{p}\_{23},\qquad {n}\_{2}^{2}+{p}\_{12}+{p}\_{23}$$
$${n}\_{3} {s}\_{12}^{2}+{n}\_{2} {s}\_{13}^{2}-{s}\_{12} {s}\_{13} {s}\_{23}-{n}\_{2} {s}\_{23}^{2}-{n}\_{3} {s}\_{23}^{2}+4 {n}\_{2} {p}\_{23}+4 {n}\_{3} {p}\_{23}$$
$${n}\_{3} {s}\_{12} {s}\_{13} {s}\_{23}+{p}\_{13} {s}\_{12}^{2}+{p}\_{23} {s}\_{12}^{2}-{p}\_{23} {s}\_{13}^{2}-{p}\_{13} {s}\_{23}^{2}+4 {p}\_{13} {p}\_{23}$$
$${n}\_{2} {s}\_{12} {s}\_{13} {s}\_{23}-{p}\_{23} {s}\_{12}^{2}+{p}\_{12} {s}\_{13}^{2}+{p}\_{23} {s}\_{13}^{2}-{p}\_{12} {s}\_{23}^{2}+4 {p}\_{12} {p}\_{23}$$
$${n}\_{2} {p}\_{23} {s}\_{12}^{2}-{n}\_{2} {p}\_{12} {s}\_{13}^{2}-{n}\_{2} {p}\_{23} {s}\_{13}^{2}+{p}\_{12} {s}\_{12} {s}\_{13} {s}\_{23}+{p}\_{23} {s}\_{12} {s}\_{13} {s}\_{23}+{n}\_{2} {p}\_{12} {s}\_{23}^{2}-4 {n}\_{2} {p}\_{12} {p}\_{23}$$
$${n}\_{2} {s}\_{13}^{3} {s}\_{23}-{s}\_{12} {s}\_{13}^{2} {s}\_{23}^{2}-{n}\_{2} {s}\_{13} {s}\_{23}^{3}-{n}\_{3} {s}\_{13} {s}\_{23}^{3}-{p}\_{13} {s}\_{12}^{3}-{p}\_{23} {s}\_{12}^{3}+{p}\_{23} {s}\_{12} {s}\_{13}^{2}+4 {n}\_{2} {p}\_{23} {s}\_{13} {s}\_{23}+4 {n}\_{3} {p}\_{23} {s}\_{13} {s}\_{23}+{p}\_{13} {s}\_{12} {s}\_{23}^{2}-4 {p}\_{13} {p}\_{23} {s}\_{12}$$
$${s}\_{12}^{2} {s}\_{13}^{2} {s}\_{23}^{2}+{p}\_{13} {s}\_{12}^{4}+{p}\_{23} {s}\_{12}^{4}-2 {p}\_{23} {s}\_{12}^{2} {s}\_{13}^{2}+{p}\_{12} {s}\_{13}^{4}+{p}\_{23} {s}\_{13}^{4}-2 {p}\_{13} {s}\_{12}^{2} {s}\_{23}^{2}-2 {p}\_{12} {s}\_{13}^{2} {s}\_{23}^{2}+{p}\_{12} {s}\_{23}^{4}+{p}\_{13} {s}\_{23}^{4}+8 {p}\_{13} {p}\_{23} {s}\_{12}^{2}+8 {p}\_{12} {p}\_{23} {s}\_{13}^{2}-8 {p}\_{12} {p}\_{23} {s}\_{23}^{2}-8 {p}\_{13} {p}\_{23} {s}\_{23}^{2}+16 {p}\_{12} {p}\_{23}^{2}+16 {p}\_{13} {p}\_{23}^{2}$$
$${n}\_{3} {p}\_{23} {s}\_{13}^{3} {s}\_{23}+{p}\_{13} {s}\_{12} {s}\_{13}^{2} {s}\_{23}^{2}+{p}\_{23} {s}\_{12} {s}\_{13}^{2} {s}\_{23}^{2}+{n}\_{3} {p}\_{13} {s}\_{13} {s}\_{23}^{3}+{p}\_{13}^{2} {s}\_{12}^{3}+2 {p}\_{13} {p}\_{23} {s}\_{12}^{3}+{p}\_{23}^{2} {s}\_{12}^{3}-{p}\_{13} {p}\_{23} {s}\_{12} {s}\_{13}^{2}-{p}\_{23}^{2} {s}\_{12} {s}\_{13}^{2}-4 {n}\_{3} {p}\_{13} {p}\_{23} {s}\_{13} {s}\_{23}-{p}\_{13}^{2} {s}\_{12} {s}\_{23}^{2}-{p}\_{13} {p}\_{23} {s}\_{12} {s}\_{23}^{2}+4 {p}\_{13}^{2} {p}\_{23} {s}\_{12}+4 {p}\_{13} {p}\_{23}^{2} {s}\_{12}$$
a quick, if useless, Macaulay 2 computation:



```
R = QQ[m_(1,1)..m_(3,3)]
M = matrix apply(3,i->apply(3,j->(m_(i+1,j+1))))
I = ideal trace M + ideal exteriorPower(2,M)
S = QQ[n_1..n_3, p_(1,2), p_(1,3), p_(2,3), s_(1,2), s_(1,3), s_(2,3)]

f = map(R/I,S, {m_(1,1),m_(2,2),m_(3,3), 
m_(1,2)*m_(2,1), m_(1,3)*m_(3,1), m_(2,3)*m_(3,2), 
m_(1,2)+m_(2,1), m_(1,3)+m_(3,1), m_(2,3)+m_(3,2)})
ker f

```

