📂Representation Theory

Compact Symplectic Group

Definition¹

The intersection of the symplectic group $\operatorname{Sp}(n, \mathbb{C})$ and the unitary group $\operatorname{U}(2n)$ is called the compact symplectic group.

$$ \operatorname{Sp}(n) := \operatorname{Sp}(n, \mathbb{C}) \cap \operatorname{U}(2n) $$

Explanation

The symplectic group $\operatorname{Sp}(n, \mathbb{C})$ is the set of matrices $Q$ that preserve the following skew-symmetric bilinear form $\omega$. For $\mathbf{x} = \begin{bmatrix} x_{1} & \cdots & x_{2n}\end{bmatrix}$ and $\mathbf{y} = \begin{bmatrix} y_{1} & \cdots & y_{2n}\end{bmatrix}$,

$$ \omega(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^{n} (x_{i}y_{n+i} - x_{n+i}y_{i}) $$

$$ \operatorname{Sp}(n, \mathbb{C}) = \left\{ Q \in M_{2n \times 2n}(\mathbb{C}) : \omega(Q \mathbf{x}, Q \mathbf{y}) = \omega(\mathbf{x}, \mathbf{y}), \quad \forall \mathbf{x}, \mathbf{y} \in \mathbb{C}^{2n} \right\} $$

The unitary group is the set of matrices $Q$ that preserve the inner product.

$$ \braket{\mathbf{x}, \mathbf{y}} = \sum_{i=1}^{2n} \overline{x_{i}} y_{i} $$

$$ \operatorname{U}(2n) = \left\{ Q \in M_{2n \times 2n}(\mathbb{C}) : \braket{Q \mathbf{x}, Q \mathbf{y}} = \braket{\mathbf{x}, \mathbf{y}}, \quad \forall \mathbf{x}, \mathbf{y} \in \mathbb{C}^{2n} \right\} $$

Hence the compact symplectic group is the set of matrices that preserve both $\omega$ and $\braket{\cdot, \cdot}$. It is convenient to express $\omega$ in terms of the inner product by defining the conjugate-linear map $J$ as follows.

$$ \begin{align*} J: \mathbb{C}^{2n} &\to \mathbb{C}^{2n} \\ \begin{bmatrix} \alpha \\ \beta \end{bmatrix} &\mapsto \begin{bmatrix} -\overline{\beta} \ \\ \overline{\alpha} \end{bmatrix} \end{align*} $$

For the above $J$, the following hold.

$$ \omega (\mathbf{z}, \mathbf{w}) = \braket{J \mathbf{z}, \mathbf{w}}, \quad \mathbf{z}, \mathbf{w} \in \mathbb{C}^{2n} $$

$$ \braket{J \mathbf{z}, \mathbf{w}} = - \overline{\braket{\mathbf{z}, J\mathbf{w}}} = - \braket{J \mathbf{w}, \mathbf{z}} $$

$$ J^{2} = - I $$

Properties

(a) $\operatorname{Sp}(1) = \operatorname{SU}(2)$

(b) $\operatorname{Sp}(n)$ is a compact Lie group.

(c) $\operatorname{Sp}(n)$ is a connected Lie group.

(d) $\forall U \in \operatorname{Sp}(n)$, $\det U = 1$

Theorem

For $U \in \operatorname{U}(2n)$, a necessary and sufficient condition for $U$ to belong to $\operatorname{Sp}(n)$ is that $U$ and $J$ commute (commutative).

$$ \forall U \in \operatorname{U}(2n),\quad U \in \operatorname{Sp}(n) \iff UJ = JU $$

Proof

Let $U \in \operatorname{U}(2n)$ and $\mathbf{z}, \mathbf{w} \in \mathbb{C}^{2}$. The following holds.

$$ \omega(U \mathbf{z}, U \mathbf{w}) = \braket{J U \mathbf{z}, U \mathbf{w}} = \braket{U^{\ast} J U \mathbf{z}, \mathbf{w}} = \braket{U^{-1} J U \mathbf{z}, \mathbf{w}} $$

Then we obtain the following result.

$$ \begin{align*} && U &\in \operatorname{Sp}(n) \\ \iff && \omega(U \mathbf{z}, U \mathbf{w}) &= \omega(\mathbf{z}, \mathbf{w}) \quad \forall \mathbf{z}, \mathbf{w} \\ \iff && \braket{U^{-1} J U \mathbf{z}, \mathbf{w}} &= \braket{J\mathbf{z}, \mathbf{w}}, \quad \forall \mathbf{z}, \mathbf{w} \\ \iff && U^{-1} J U &= J \\ \iff && J U &= U J \end{align*} $$

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