Patrimony of Hyrkoon
The Patrimony of Hyrkoon was an ancient nation that flourished to the east of the Bone Mountains in Essos. It was ruled by the Patriarchs of Hyrkoon. They founded the fortress cities of Kayakayanaya, Shamyriana and Bayasabhad.
The Patrimony is apparently named after the legendary figure "Hyrkoon the Hero" - the name that the local people have for the figure that in other legends is known as Azor Ahai, the hero who saved the world during the Long Night. As the "Patrimony" of Hyrkoon, it appears that its leaders claim descent from him.
It was centered around a fertile area of rivers and lakes, since desiccated to become the Great Sand Sea, a time known as the Dry Times. The fortified cities of Kayakayanaya, Samyriana and Bayasabhad were once part of this great nation, each guarding the Steel Road, the Stone Road, and the Sand Road, respectively, from brigands, outlaws, and wild men of the Bone Mountains, and the savages that dwelt beyond them.
Before the Dry Times, the Patrimony fought a savage border war with the Jogos Nhai, who poisoned rivers and wells, burned towns and cities, and carried off thousands into slavery on the plains. The Hyrkoon, for their part, were sacrificing tens of thousands of Jogos Nhai to their gods. This enmity survives into the present, with Jogos Nhai jhattars leading attacks on Kayakayanaya.
The custom of having warrior women, who fight bare-breasted and pierce their nipples with iron rings and cheeks with rubies, shared by the above cities, possibly originates in the Patrimony. West of the Bone Mountains, the city of Yinishar, now the ruins of Vaes Jini, may have been connected with Hyrkoon, as the people of Yinishar were akin to the people of Samyriana.
Behind the scenes
Hyrkoon appears to take its name from the Elric Saga by Michael Moorcock. The main character Elric of Melniboné is the ruler of his city, while his cousin and heir Yyrkoon plots against him. Indeed, in the same section of The World of Ice and Fire (on Yi Ti) that explains that the hero of the Long Night is known by many names, such as Azor Ahai or Hyrkoon the Hero, one of the alternate names provided for this figure is "Eldric Shadowchaser". Martin certainly seems familiar with the Elric Saga: in the episode of the TV adaptation "The Lion and the Rose", which was personally scripted by Martin, when the wedding guests shout out suggestions for Joffrey to name his sword (and he ultimately picks "Widow's Wail"), one of the alternate names that someone shouts is "Stormbringer" - which is the name of Elric's sword.