The Hexablock: a domain associated with the $\mu$-synthesis in $M_2(\mathbb C)$

We introduce a domain named \textit{hexablock} in $\mathbb C^4$ and show that its origin is a special case of $\mu$-synthesis in $M_2(\mathbb C)$, more precisely the $\mu_E$-unit ball with respect to the linear subspace $E$ consisting of $2 \times 2$ upper triangular matrices. The hexablock is denoted by $\mathbb H$ and is defined by \[ \mathbb{H}=\left\{(a, x_1, x_2, x_3) \,\in\, \mathbb{C} \times \mathbb{E}\,\,\big\vert\,\, \sup_{z_1,\, z_2 \,\in\, \mathbb D}\left|\frac{a\sqrt{(1-|z_1|^2)(1-|z_2|^2)}}{1-x_1z_1-x_2z_2+x_3z_1z_2}\right| <1\right\}, \] where $\mathbb{E}$ is the \textit{tetrablock}, another domain in $\mathbb C^3$ associated with a different case of $\mu$-synthesis, and is given by \[ \mathbb{E}=\{(x_1, x_2, x_3) \in \mathbb{C}^3 : 1-x_1z_1-x_2z_2+x_3z_1z_2 \ne 0 \ \text{for all } \, z_1, z_2 \in \overline{\mathbb D}\}. \] We show that two other objects in $\mathbb C^4$ namely, the $\mu$-hexablock $\mathbb H_{\mu}$ and the normed hexablock $\mathbb H_N$ naturally arise in the $\mu_E$-unit ball and the norm unit ball of $M_2(\mathbb C)$, respectively and pave the way to reach the domain $\mathbb H$. A set of independent characterizations for the points in $\mathbb H_{\mu}, \mathbb H_N$ and $\mathbb H$ are obtained. Geometric and function theoretic aspects of $\mathbb H$ are studied and its connections with the popular domains such as symmetrized bidisc $\mathbb G_2$, tetrablock $\mathbb E$ and pentablock $\mathbb P$ are explored.