The Hexablock: a domain associated with the $μ$-synthesis in M₂( C)

We study a new domain in C⁴, namely the hexablock H that arises in connection with a special case of the $μ$-synthesis problem in M₂( C). Previous attempts to study a few instances of the $μ$-synthesis problem led to domains such as symmetrized bidisc G₂, tetrablock E and pentablock P. Throughout, the relations of H with these three domains are explored. Several characterizations of the hexablock are established. We determine the distinguished boundary b H of H and obtain a subgroup of the automorphism group of H. The rational H̄-inner functions, i.e., the rational functions from the unit disc D to H that map the boundary of D to bH are characterized. We obtain a Schwarz lemma for H. Finally, it is shown that G₂, E, and P are analytic retracts of H. By an application of the theory of the hexablock, we find alternative proofs to a few existing results on geometry and function theory of the pentablock.