Show that $\sigma \in \mathfrak{S}_n$ and $\tau \in \mathfrak{S}_n$ are conjugate if and only if, for all $\ell \in \llbracket 1, n \rrbracket, c_\ell(\sigma) = c_\ell(\tau)$, where for $\ell \in \llbracket 2, n \rrbracket$, $c_\ell(\sigma)$ denotes the number of cycles of length $\ell$ in the decomposition of $\sigma$ into cycles with disjoint supports, and $c_1(\sigma) = \operatorname{Card}\{j \in \llbracket 1, n \rrbracket, \sigma(j) = j\}$.

For $\sigma \in \mathfrak{S}_n$ and $\ell \in \llbracket 2, n \rrbracket$, we denote $c_\ell(\sigma)$ the number of cycles of length $\ell$ in the decomposition of $\sigma$ into cycles with disjoint supports. We denote $c_1(\sigma)$ the number of fixed points of $\sigma$:
$$c_1(\sigma) = \operatorname{Card}\{j \in \llbracket 1, n \rrbracket, \sigma(j) = j\}.$$
Show that $\sigma \in \mathfrak{S}_n$ and $\tau \in \mathfrak{S}_n$ are conjugate if and only if, for all $\ell \in \llbracket 1, n \rrbracket, c_\ell(\sigma) = c_\ell(\tau)$.

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We choose $x$ in $\mathcal{V}^{\bullet} \backslash \{0\}$. We denote by $p$ the generic nilindex of $\mathcal{V}$ (with $p \geq n-1$), and we fix $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$. We have $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$.
Suppose that there exists $v_{0}$ in $\mathcal{V}$ such that $v_{0}(x) \neq 0$. Let $v \in \mathcal{V}$. By considering $v + tv_{0}$ for $t$ real, show that $\operatorname{Im} v^{p-1} \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$.

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We choose $x$ in $\mathcal{V}^{\bullet} \backslash \{0\}$. We denote by $p$ the generic nilindex of $\mathcal{V}$ (with $p \geq n-1$), and we fix $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$. We have established that $K(\mathcal{V}) = \operatorname{Vect}(\mathcal{V}^{\bullet}) \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$ (from question 23 applied to all $v \in \mathcal{V}$), and Lemma B states that if $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$ then $v(x) = 0$ for every $v \in \mathcal{V}$.
Conclude the proof of Gerstenhaber's theorem: if $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$ then there exists a basis of $E$ in which every element of $\mathcal{V}$ is represented by a strictly upper triangular matrix.

We identify $M_3(\mathbb{R})$ with the linear endomorphisms of $V$. Let $G$ be the set of endomorphisms $g$ such that $$B(gu,gv) = B(u,v)$$ for all $u,v\in V$.
Show that $G$ is a group under composition of linear maps.

Let $G$ be the group of endomorphisms $g$ of $V$ such that $B(gu,gv)=B(u,v)$ for all $u,v\in V$.
Show that for all $g\in G$, we have $g(\mathcal{H})=\mathcal{H}$ or $-g(\mathcal{H})=\mathcal{H}$.

For all $w\in V$ such that $B(w,w)>0$, define $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that $s_w \in G_0$.

For all $w\in V$ such that $B(w,w)>0$, define $$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$ Show that for all $u,v\in\mathcal{H}$, there exists $w\in V$ such that $B(w,w)>0$ and $s_w(u)=v$.

Let $F$ be a vector subspace of a symplectic space $(E,\omega)$, and let $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega(x,y) = 0 \}$. Show that the restriction $\omega _ { F }$ of $\omega$ to $F ^ { 2 }$ defines a symplectic form on $F$ if and only if $F \oplus F ^ { \omega } = E$.

Show that the restriction $\omega _ { F }$ of $\omega$ to $F ^ { 2 }$ defines a symplectic form on $F$ if and only if $F \oplus F ^ { \omega } = E$.

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
(a) Show that for all $\omega \in \mathscr{A}_p(E, \mathbb{R})$, we have $\omega = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \Omega_p(e_{\alpha})$.
(b) Deduce that $(\omega, \omega^{\prime}) \mapsto \langle\omega, \omega^{\prime}\rangle$ is an inner product on $\mathscr{A}_p(E, \mathbb{R})$ for which $(\Omega_p(e_{\alpha}))_{\alpha \in \mathcal{I}_p}$ is an orthonormal basis of $\mathscr{A}_p(E, \mathbb{R})$ and give the dimension of $\mathscr{A}_p(E, \mathbb{R})$.
(c) Construct in the case $p = d-1$ an isometry between $\mathscr{A}_p(E, \mathbb{R})$ and $E$.

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathcal{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha})$$
(a) Show that for all $\omega \in \mathcal{A}_p(E, \mathbb{R})$, we have $\omega = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \Omega_p(e_{\alpha})$.
(b) Deduce that $(\omega, \omega^{\prime}) \mapsto \langle\omega, \omega^{\prime}\rangle$ is an inner product on $\mathcal{A}_p(E, \mathbb{R})$ for which $(\Omega_p(e_{\alpha}))_{\alpha \in \mathcal{I}_p}$ is an orthonormal basis of $\mathcal{A}_p(E, \mathbb{R})$ and give the dimension of $\mathcal{A}_p(E, \mathbb{R})$.
(c) Construct in the case $p = d-1$ an isometry between $\mathcal{A}_p(E, \mathbb{R})$ and $E$.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ If $A$ and $B$ are two matrices in $\mathcal{M}_n(\mathbf{K})$, their Lie bracket is defined by $[A, B] = AB - BA$. In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{13}$ ▷ Deduce from question 12) that $\mathcal{A}_G$ is stable under the Lie bracket, i.e. $$\forall A \in \mathcal{A}_G, \forall B \in \mathcal{A}_G, [A, B] \in \mathcal{A}_G.$$

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ We recall that, if $M$ is a matrix in $\mathcal{M}_n(\mathbf{R})$, we say that $M$ is tangent to $G$ at $I_n$ if there exist $\varepsilon > 0$ and an application $\left.\gamma : \right]-\varepsilon, \varepsilon[ \rightarrow G$, differentiable, such that $\gamma(0) = I_n$ and $\gamma'(0) = M$. The set of matrices tangent to $G$ at $I_n$ is called the tangent space to $G$ at $I_n$, and is denoted $\mathcal{T}_{I_n}(G)$. In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{14}$ ▷ Prove the inclusion $\mathcal{A}_G \subset \mathcal{T}_{I_n}(G)$.

The set of real symplectic matrices is defined as $$\mathrm { Sp } _ { n } ( \mathbb { R } ) = \mathrm { Sp } _ { 2 m } ( \mathbb { R } ) = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid M ^ { \top } J M = J \right\}$$ where $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $\mathrm { Sp } _ { n } ( \mathbb { R } )$ is a subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$, stable under transposition and containing the matrix $J$.

We denote by $\mathrm { Sp } _ { n } ( \mathbb { R } ) = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid M ^ { \top } J M = J \right\}$ the set of real symplectic matrices. Show that $\mathrm { Sp } _ { n } ( \mathbb { R } )$ is a subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$, stable under transposition and containing the matrix $J$.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ The tangent space to $G$ at $I_n$ is denoted $\mathcal{T}_{I_n}(G)$.
$\mathbf{17}$ ▷ Show that, in the particular cases $G = \mathrm{SL}_n(\mathbf{R})$ and $G = \mathrm{O}_n(\mathbf{R})$, we have $\mathcal{T}_{I_n}(G) = \mathcal{A}_G$.

We denote by $\operatorname { OSp } _ { n } ( \mathbb { R } ) = \operatorname { Sp } _ { n } ( \mathbb { R } ) \cap \mathrm { O } _ { n } ( \mathbb { R } )$ the set of real symplectic and orthogonal matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$. We equip $\mathcal { M } _ { n } ( \mathbb { R } )$ with its topology as a normed vector space of finite dimension. Show that $\operatorname { OSp } _ { n } ( \mathbb { R } )$ is a compact subgroup of the symplectic group $\operatorname { Sp } _ { n } ( \mathbb { R } )$.

We denote by $\operatorname { OSp } _ { n } ( \mathbb { R } ) = \operatorname { Sp } _ { n } ( \mathbb { R } ) \cap \mathrm { O } _ { n } ( \mathbb { R } )$ the set of real symplectic and orthogonal matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$. We equip $\mathcal { M } _ { n } ( \mathbb { R } )$ with its topology as a normed vector space of finite dimension. Show that $\operatorname { OSp } _ { n } ( \mathbb { R } )$ is a compact subgroup of the symplectic group $\operatorname { Sp } _ { n } ( \mathbb { R } )$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and let $\lambda$ and $\mu$ be real numbers. The symplectic transvections are defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Show that $\tau _ { a } ^ { \mu } \circ \tau _ { a } ^ { \lambda } = \tau _ { a } ^ { \lambda + \mu }$.

Let $a \in E$ be a non-zero vector and let $\lambda$ and $\mu$ be real numbers. Show that $\tau _ { a } ^ { \mu } \circ \tau _ { a } ^ { \lambda } = \tau _ { a } ^ { \lambda + \mu }$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. The symplectic transvection is defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Show that $\operatorname { det } \left( \tau _ { a } ^ { \lambda } \right) > 0$.

Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. Show that $\operatorname { det } \left( \tau _ { a } ^ { \lambda } \right) > 0$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. The symplectic transvection is defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Is the inverse $\left( \tau _ { a } ^ { \lambda } \right) ^ { - 1 }$ still a symplectic transvection?

Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. Is the inverse $\left( \tau _ { a } ^ { \lambda } \right) ^ { - 1 }$ still a symplectic transvection?