Fixed points of an involution

Question

Let $V=\mathbb C^{2n}$ with the standard basis $\{e_1,e_2, \cdots , e_{2n}\}$ and let $\sigma$ be the involution $e_i \mapsto -e_{2n+1-i}$. This induces an involution of the Grassmannian $G(n,2n)$ of $n$ dimensional subspaces of $\mathbb C^{2n}$. Then what are the fixed points of this involution ? Does it have a nice structure as a projective variety ?

Answer 1

In general, the fixed points of any map of the Grassmannian induced by a diagonalizable linear transformation is just a union of products of Grassmannians in the eigenspaces. Any subspace $V$ fixed by a diagonalizable transformation $A$ is the sum of the intersections of $V$ with the eigenspaces of $A$ (since the projection to each eigenspace is a polynomial in $A$). If we fix the dimension of each of these intersections, we get a map to the product of Grassmannians of the eigenspaces, which is obviously an isomorphism.

In this case, $\mathbb{R}^{2n}$ is the sum of the 1 and -1 eigenspaces, both having dimension $n$. Thus, the fixed points are a disjoint union of $Gr(k,n)\times Gr(n-k,n)$ for all $0\leq k\leq n$.

Answer 2

Let us analyze in details the simplest non-trivial case, namely $n=2$. Let $$x_{ij} :=x_i \wedge x_j, \quad i <j$$ be the Plücker coordinates of $\mathbb{P}^5$. Since your involution $\sigma$ exchanges $e_1$ with $e_4$ and $e_2$ with $e_3$, it is easy to check that the action on the Plücker coordinates is \begin{equation*} \begin{split} [x_{12}: \, x_{13}: \, x_{14}: x_{23}: \, x_{24}: \, x_{34}] \mapsto & [-x_{34}: \, -x_{24}: \, -x_{14}: \, -x_{23}: \, -x_{13}: \, -x_{12}] \\ =& [x_{34}: \, x_{24}: \, x_{14}:\, x_{23}: \, x_{13}: \, x_{12}]. \end{split} \end{equation*} The fixed locus $\Sigma$ of such an involution in $\mathbb{P}^5$ is given by a disjoint union $$\Sigma = \Sigma_1 \sqcup \Sigma_2,$$ where $\Sigma_1$ is the line of equation $$x_{12}+x_{34}=x_{13}+x_{24}=x_{14}=x_{23}=0,$$ whereas $\Sigma_2$ is the $2$-plane of equation $$x_{12}-x_{34}=x_{13}-x_{24}=0.$$ Now recall that the Grasmannian $\mathbb{G}(1, \, 3)=G(2, \, 4)$ of lines in $\mathbb{P}^3$ (or, equivalently, of $2$-planes in $\mathbb{C}^4$) is the quadric hypersurface $Q \subset \mathbb{P}^5$ whose Plücker equation is $$x_{12}x_{34}-x_{13}x_{24}+x_{23}x_{14}=0.$$ Such a quadric is $\sigma$-invariant, as expected, and the fixed locus $\Sigma_Q$ of the involution $\sigma \colon Q \to Q$ is given by intersecting $Q$ with $\Sigma$. Then we obtain a disjoint union
$$\Sigma_Q = \Sigma_{Q1} \sqcup \Sigma_{Q2},$$ where $\Sigma_{Q1}:=Q \cap \Sigma_1$ consists of the two points $$[1: \, -1: \, 0: \, 0: \, 1: \, -1], \quad [1: \, 1: \, 0: \, 0: \, -1: \, -1],$$ whereas $\Sigma_{Q2}:=Q \cap \Sigma_2$ is a smooth linear section of dimension $2$, namely a smooth quadric surface.

Summing up, the fixed locus of the involution induced by $\sigma$ on the Grasmannian $\mathbb{G}(1, \, 3)$ consists of the disjoint union of two distinct points and a copy of $\mathbb{P}^1 \times \mathbb{P}^1$, and this agrees with Ben Webster's answer.