# (Real) Symplectic Grassmann

Manifolds.SymplecticGrassmannType
SymplecticGrassmann{T,𝔽} <: AbstractEmbeddedManifold{𝔽, DefaultIsometricEmbeddingType}

The symplectic Grassmann manifold consists of all symplectic subspaces of $ℝ^{2n}$ of dimension $2k$, $n ≥ k$.

Points on this manifold can be represented as corresponding representers on the SymplecticStiefel

$$$\operatorname{SpGr}(2n,2k) = \bigl\{ \operatorname{span}(p)\ \big| \ p ∈ \operatorname{SpSt}(2n, 2k, ℝ)\},$$$

or as projectors

$$$\operatorname{SpGr}(2n, 2k, ℝ) = \bigl\{ p ∈ ℝ^{2n×2n} \ \big| \ p^2 = p, \operatorname{rank}(p) = 2k, p^+=p \bigr\},$$$

where $⋅^+$ is the symplectic_inverse. See also ProjectorPoint and StiefelPoint for these two representations, where arrays are interpreted as those on the Stiefel manifold.

With respect to the quotient structure, the canonical projection $π = π_{\mathrm{SpSt},\mathrm{SpGr}}$ is given by

$$$π: \mathrm{SpSt}(2n2k) → \mathrm{SpGr}(2n,2k), p ↦ π(p) = pp^+.$$$

The tangent space is either the tangent space from the symplectic Stiefel manifold, where tangent vectors are representers of their corresponding congruence classes, or for the representation as projectors, using a ProjectorTVector as

$$$T_p\operatorname{SpGr}(2n, 2k, ℝ) = \bigl\{ [X,p] \ \mid\ X ∈ \mathfrak{sp}(2n,ℝ), Xp+pX = X \bigr\},$$$

where $[X,p] = Xp-pX$ denotes the matrix commutator and $\mathfrak{sp}(2n,ℝ)$ is the Lie algebra of the symplectic group consisting of HamiltonianMatrices.

The first representation is in StiefelPoints and StiefelTVectors, which both represent their symplectic Grassmann equivalence class. Arrays are interpreted in this representation as well

For the representation in ProjectorPoint and ProjectorTVectors, we use the representation from the surjective submersion

$$$ρ: \mathrm{SpSt}(2n,2k) → \mathrm{SpGr}(2n,2k), \qquad ρ(p) = pp^+$$$

and its differential

$$$\mathrm{d}ρ(p,X) = Xp^+ + pX^+,$$$

respectively. The manifold was first introduced in [BZ21]

Constructor

SymplecticGrassmann(2n::Int, 2k::Int, field::AbstractNumbers=ℝ; parameter::Symbol=:type)

Generate the (real-valued) symplectic Grassmann manifold. of $2k$ dimensional symplectic subspace of $ℝ^{2n}$. Note that both dimensions passed to this constructor have to be even.

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## The (default) symplectic Stiefel representation

ManifoldDiff.riemannian_gradientMethod
riemannian_gradient(M::SymplecticGrassmann, p, Y)

Given a gradient $Y = \operatorname{grad} \tilde f(p)$ in the embedding $ℝ^{2n×2k}$ or at least around the SymplecticGrassmann M where p (the embedding of) a point on M, and the restriction $\tilde f$ to the SymplecticStiefel be invariant for the equivalence classes. In other words $f(p) = f(qp)$ for $q \in \mathrm{Sp}(2k, ℝ)$, where $\mathrm{Sp}(2k, ℝ)$ denotes the SymplecticMatrices manifold. Then the Riemannian gradient $X = \operatorname{grad} f(p)$ is given by

$$$X = J_{2n}^THJ_{2k}p^{\mathrm{T}}p - J_{2n}^TpJ_{2k}H^{\mathrm{T}}p,$$$

where $J_{2n}$ denotes the SymplecticElement, and $H = (I_{2n} - pp^+)J_{2n}^{\mathrm{T}}YJ$.

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ManifoldsBase.innerMethod
inner(::SymplecticGrassmann, p, X, Y)

Compute the Riemannian inner product $g^{\mathrm{SpGr}}_p(X,Y)$ on the SymplecticGrassmann manifold \mathrm{SpGr}.

For the case where $p$ is represented by a point on the SymplecticStiefel manifold acting as a representant of its equivalence class $[p] \in \mathrm{SpGr}$ and the tangent vectors $X,Y \in \mathrm{Hor}_p^π\operatorname{SpSt}(2n,2k)$ are horizontal tangent vectors.

Then the inner product reads according to Proposition Lemma 4.8 [BZ21].

$$$g^{\mathrm{SpGr}}_p(X,Y) = \operatorname{tr}\bigl( (p^{\mathrm{T}}p)^{-1}X^{\mathrm{T}}(I_{2n} - pp^+)Y \bigr),$$$

where $I_{2n}$ denotes the identity matrix and $(⋅)^+$ the symplectic_inverse.

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ManifoldsBase.inverse_retractMethod
inverse_retract(::SymplecticGrassmann, p, q, ::CayleyInverseRetraction)
inverse_retract!(::SymplecticGrassmann, q, p, X, ::CayleyInverseRetraction)

Compute the Cayley Inverse Retraction on the Symplectic Grassmann manifold, when the points are represented as symplectic bases, i.e. on the SymplecticStiefel.

Here we can directly employ the CaleyInverseRetraction on the symplectic Stiefel manifold.

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## The symplectic projector representation

ManifoldsBase.check_pointMethod
check_point(M::SymplecticGrassmann, p::ProjectorPoint; kwargs...)

Check whether p is a valid point on the SymplecticGrassmann, $\operatorname{SpGr}(2n, 2k)$, that is a proper symplectic projection:

• $p^2 = p$, that is $p$ is a projection
• $\operatorname{rank}(p) = 2k$, that is, the supspace projected onto is of right dimension
• $p^+ = p$ the projection is symplectic.
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ManifoldsBase.check_vectorMethod
check_vector(M::SymplecticGrassmann, p::ProjectorPoint, X::ProjectorTVector; kwargs...)

Check whether X is a valid tangent vector at p on the SymplecticGrassmann`, $\operatorname{SpGr}(2n, 2k)$ manifold by verifying that it

• $X^+ = X$
• $X = Xp + pX$

For details see Proposition 4.2 in [BZ21] and the definition of $\mathfrak{sp}_P(2n)$ before, especially the $\bar{Ω}$, which is the representation for $X$ used here.

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## Literature

[BZ21]
T. Bendokat and R. Zimmermann. The real symplectic Stiefel and Grassmann manifolds: metrics, geodesics and applications, arXiv Preprint, 2108.12447 (2021), arXiv:2108.12447.