We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$. Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$, $H^+$ the subset with $m_1 \geqslant m_2 \geqslant m_3$, and $\varphi : H \to E$ defined by $\varphi(m_1,m_2,m_3) = (m_1-m_2)\omega_1 + (m_2-m_3)\omega_2$. We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ and $\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\}$. Let $M \in S _ { 3 } ( \mathbb { R } )$ be a matrix with zero trace. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \geqslant \lambda _ { 3 } \right)$. We propose to describe $\varphi \left( \mathcal { D } _ { M } \right)$. (a) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$. Let $\sigma$ be a permutation of $\{ 1,2,3 \}$. Show that $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$ if and only if $\left( m _ { \sigma ( 1 ) } , m _ { \sigma ( 2 ) } , m _ { \sigma ( 3 ) } \right) \in \mathcal { D } _ { M }$. (b) Using question 13, show that the intersection $H ^ { + } \cap \mathcal { D } _ { M }$ is contained in $\mathcal { Q } _ { \widehat { \lambda } }$. (c) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$. Show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right)$ is contained in $\mathcal { D } _ { M }$. One may use question 12. Similarly, show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right)$ is contained in $\mathcal { D } _ { M }$. (d) Show that $\mathcal { D } _ { M }$ contains $\mathcal { Q } _ { \widehat { \lambda } }$. (e) Show that if $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$ then $\varphi \left( \mathcal { D } _ { M } \right)$ is a hexagon, whose vertices will be determined.
We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$. Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$, $H^+$ the subset with $m_1 \geqslant m_2 \geqslant m_3$, and $\varphi : H \to E$ defined by $\varphi(m_1,m_2,m_3) = (m_1-m_2)\omega_1 + (m_2-m_3)\omega_2$. We consider the application
$$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$
and $\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\}$.\\
Let $M \in S _ { 3 } ( \mathbb { R } )$ be a matrix with zero trace. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \geqslant \lambda _ { 3 } \right)$. We propose to describe $\varphi \left( \mathcal { D } _ { M } \right)$.\\
(a) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$. Let $\sigma$ be a permutation of $\{ 1,2,3 \}$. Show that $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$ if and only if $\left( m _ { \sigma ( 1 ) } , m _ { \sigma ( 2 ) } , m _ { \sigma ( 3 ) } \right) \in \mathcal { D } _ { M }$.\\
(b) Using question 13, show that the intersection $H ^ { + } \cap \mathcal { D } _ { M }$ is contained in $\mathcal { Q } _ { \widehat { \lambda } }$.\\
(c) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$. Show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right)$ is contained in $\mathcal { D } _ { M }$. One may use question 12.\\
Similarly, show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right)$ is contained in $\mathcal { D } _ { M }$.\\
(d) Show that $\mathcal { D } _ { M }$ contains $\mathcal { Q } _ { \widehat { \lambda } }$.\\
(e) Show that if $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$ then $\varphi \left( \mathcal { D } _ { M } \right)$ is a hexagon, whose vertices will be determined.