representation of symplectic group

Symplectic Group

Every symplectic form can be put into a canonical form by finding a symplectic basis . So, up to conjugation, there is only one symplectic group, in contrast to the orthogonal group which preserves a nondegenerate symmetric bilinear form . As with the orthogonal group , the columns of a symplectic matrix form a symplectic basis .

The matrices

In fact, both of these examples are 1-parameter subgroups.

A matrix can be tested to see if it is symplectic using the Wolfram Language code:

Portions of this entry contributed by Todd Rowland

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Rowland, Todd and Weisstein, Eric W. "Symplectic Group." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/SymplecticGroup.html

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Symplectic group

2020 Mathematics Subject Classification: Primary: 20-XX [ MSN ][ ZBL ]

One of the classical groups, defined as the group of automorphisms of a skew-symmetric bilinear form $\Phi$ on a left $K$-module $E$, where $K$ is a commutative ring (cf. Classical group ). In the case when $E=K^{2m}$ and the matrix of $\Phi$ with respect to the canonical basis $\{e_i\}$ of $E$ has the form

$$J_m = \begin{pmatrix}0 & I_m\\ -I_m & 0\end{pmatrix},$$ where $I_m$ is the identity matrix of order $m$, the corresponding symplectic group is called the symplectic group of $2m$ variables over the ring $K$ and is denoted by $\def\Sp{ {\rm Sp}}\Sp(m,K)$ or $\Sp_{2m}(K)$. The matrix of any automorphism in $\Sp_{2m}(K)$ with respect to $\{e_i\}$ is called a symplectic matrix.

Let $K$ be a field and $\Phi$ a non-degenerate skew-symmetric bilinear form on an $n$-dimensional vector space $E$ over $K$. If $n$ is even, then the symplectic group associated with $\Phi$ is isomorphic to $\Sp_{n}(K)$ and is generated by all linear transformations of $E$ of the form $\def\a{\alpha}\def\s{\sigma}\s_{e,\a}$, given by

$$x\mapsto \s_{e,\a}(x) = x+\a\Phi(e,x)e,$$ where $e\in E$, $\a\in K$. Linear transformations of the form $\s_{e,\a}$ are called symplectic transvections, or translations in the direction of the line $Ke$. The centre $Z$ of $\Sp_{n}(K)$ consists of the matrices $I_n$ and $-I_n$ if ${\rm char}\; K \ne 2$, and $Z=\{I_n\}$ if ${\rm char}\; K = 2$. The quotient group $\Sp_{n}(K)/Z$ is called the projective symplectic group and is denoted by $\def\PSp{ {\rm PSp}}\PSp_{n}(K)$. All projective symplectic groups are simple, except

$$\PSp_2(\F_2) = \Sp_2(\F_2),\quad \PSp_4(\F_2) = \Sp_4(\F_2) \textrm{ and }\PSp_2(\F_3)$$ (here $\F_q$ denotes the field of $q$ elements) and these are isomorphic to the symmetric groups $S_3$, $S_6$ (cf. Symmetric group ) and the alternating group $A_4$, respectively. The order of $\Sp_{2m}(\F_q)$ is

$$q^{m^2}(q^2-1)\cdots(q^{2m-2}-1)(q^{2m}-1).$$ The symplectic group $\Sp_2(K)$ coincides with the special linear group ${\rm SL}_2(K)$. If ${\rm char}\; K \ne 2$, $\PSp_4(K)$ is isomorphic to the quotient group of $\def\Om{\Omega}\Om_5(K,f)$ by its centre, where $\Om_5(K,f)$ is the commutator subgroup of (index 2 in) the orthogonal group associated with a symmetric bilinear form $f$ in five variables.

Except when $m=2$ and ${\rm char}\; K = 2$, every automorphism $\def\phi{\varphi}\Phi$ of $\Sp_{2m}(K)$ can be written as

$$\phi(g)=h_1h_2g^\tau h_2^{-1}h_1^{-1},$$ where $\tau$ is an automorphism of the field $K$, $h_1\in\Sp_{2m}(K)$ and $h_2$ is a linear transformation of the space $E$, represented on the basis $\{e_i\}$ by a matrix of the form

$$\begin{pmatrix}I_m & 0 \\ 0 & \beta I_m\end{pmatrix}$$ ($\beta$ is a non-zero element of $K$).

$\Sp_{2m}(K)$ coincides with the group of $K$-points of the linear algebraic group $\Sp_{2m}$ defined by the equation $A^tJ_m A = J_m$. This algebraic group, also called a symplectic group, is a simple simply-connected linear algebraic group of type $C_m$ of dimension $2m^2+m$.

In the case when $K=\C$ or $\R$, $\Sp_{2m}(K)$ is a connected simple complex (respectively, real) Lie group. $\Sp_{2m}(\R)$ is one of the real forms of the complex symplectic group $\Sp_{2m}(\C)$. The other real forms of this group are also sometimes called symplectic groups. These are the subgroups $\Sp(p,q)$ of $\Sp_{2m}(\C)$, $p,q\ge 0$, $p+q=m$, consisting of those elements of $\Sp_{2m}(\C)$ that preserve the Hermitian form

$$\def\e{\epsilon}\sum_{i=1}^{2m} \e_i z_i\bar z_i,$$ where $\e_i=1$ for $1\le i\le p$ and $m+1\le i \le m+p$, and $\e_i=-1$ otherwise. The group $\Sp(0,m)$ is a compact real form of the complex symplectic group $\Sp_{2m}(\C)$. The symplectic group $\Sp(p,q)$ is isomorphic to the group of all linear transformations of the right vector space $\def\H{ {\mathbb H}}\H^m$ of dimension $m=p+q$ over the division ring $\H$ of quaternions that preserve the quaternionic Hermitian form of index $\min(p,q)$, that is, the form

$$(x,y) = \sum_{i=1}^p x_i\bar y_i - \sum_{i=p+1}^m x_i\bar y_i,$$ where

$$x=(x_1,\dots,x_m,\ y = (y_1,\dots,y_m) \in \H^m,$$ and the bar denotes conjugation of quaternions.

$\Sp_{2m}(\C)$ is also simply connected. But $\Sp_{2m}(\R)$ has the homotopy type of $S^1\times {\rm SU}_n$, so that $\pi_1(\Sp_{2m}(\R)) = \Z$. Here $S^1$ is the circle and ${\rm SU}_n$ is the special unitary group. The unitary symplectic group ${\rm USp}_{2m}(\C)$ is the intersection (in ${\rm GL}_{2m}(\C)$) of the unitary group ${\rm U}_{2m}$ and $\Sp_{2m}(\C)$. Topologically, $\Sp_{2m}(\C) \simeq {\rm USp}_{2m}(\C)\times \R^{2n^2+n}.$.

In Hamiltonian mechanics (cf. Hamilton equations ) the phase space is a symplectic manifold, a manifold $M$ provided with a symplectic form (a closed differential form $\omega$ of degree $2$ which is non-degenerate at each point). If $M=T^* Q$, the cotangent bundle of a configuration space $Q$, with local coordinates $(q_1,\dots,q_n;p1,\dots,p_n)$, then the symplectic form $\sum_{j=1}^n dp_j\wedge dq_j$ is called canonical. The flow of a Hamiltonian system leaves the symplectic form invariant. As a consequence, its tangent mapping at a fixed point belongs to the symplectic group of the tangent space.

Cf. Symplectic homogeneous space ; Symplectic structure .

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M. D. Gould; Representation theory of the symplectic groups. I. J. Math. Phys. 1 June 1989; 30 (6): 1205–1218. https://doi.org/10.1063/1.528346

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Certain aspects of symplectic group representation theory are investigated. In particular, it is shown that every irreducible representation of Sp( n ) admits a relatively large class of states, referred to herein as canonical states, which possess properties analogous to the Gelfand–Tsetlin states appearing in the theory of the orthogonal and unitary groups. The properties of canonical states are investigated and some matrix element formulas are derived.

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On Representations of the Symplectic Group

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One can consider the compact group Sp(n) as a subgroup of the unitary group U (2 n ). The imbedding of Sp(n) into U (2 n ) is given by the isomorphic imbedding of the skew field of quaternions into the linear algebra of 2 × 2 matrices,

\(q = a + bi + cj + dk \to \left( {\frac{z}{{ - w}}\frac{w}{z}} \right),z = a + b;w = c + di\)

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W. Rudin, “Function Theory in the Unit Ball of C n ,” Springer, Berlin (1980).

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A. U. Klimyk, A. M. Gavrilik, J. Math. Phys. 20 :1624 (1979).

N. Ja. Vilenkin, “Special Functions and the Theory of Group Representations,” Transl. Math. Monogr. 22 , Amer. Math. Soc., Providence, R.I. (1968).

M. K. F. Wong, Hsin-Yang Yeh, J. Math. Phys. 21 :630 (1980).

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Groza, V.A. (1989). On Representations of the Symplectic Group. In: Gruber, B., Iachello, F. (eds) Symmetries in Science III. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0787-7_34

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As it has been in the past 40 years, Mimsa believe in providing competitive prices without compromising their principles of quality. We have managed to create lasting relationships based on honesty and cooperation while adding new customers each year.

Nothing is more important for us than Customer satisfaction!

Mimsa prioritizes customer satisfaction in the services they provide, and strives to understand the customers’ requests thoroughly in order to fulfil their needs and expectations. According to Mimsa Aluminium, every single customer should always be provided with the quality and services above expectations.

Every single completed project is the beginning of a lasting relationship for us.

Mimsa executes every project with experience and knowledge, while continuously improving itself and its high-quality production. Therefore, Mimsa never regards a project as a completed business. Every single project is a successful representation of lasting relationships. Thus, Mimsa pay great attention to post-sale support and keep on supplying uninterrupted support to their customers after completion.

It is very important for us that every single project we execute creates value to our workers, community and environment!

Aiming to create value for the community, environment and humankind in each project. Mimsa perceive that the occupational training of its employees and the new entrants to the workforce gets these individuals well equipped for the industry and community, and so does whatever needed without second thoughts.