The Triumvirate, or "Salasari" in the Qunlat language, is the governing body of all of Qunari society. They are three individuals that represent a perfect trinity, a whole that the three complete. Arishok the body, Arigena the mind, and Ariqun the soul, are the three pillars of Qunari society.
Three individuals that are the head of their respective paths, they divide and govern all Qunari under the "nation that must be". Since the Triumvirate represents the whole of the Qunari, they govern those who have also been assigned to their respective paths. The Arishok governs the soldiers, the Arigena governs the craftsmen, and the Ariqun governs the priesthood.
This difference of government to that of nobility backed monarchies (the norm in Thedas) would be fairly striking to a human, elf, and dwarf. However, as it can be said that the brain governs the body (in the case of monarchies) the Qunari believe that other parts of the body do so as well, such as the heart and the lungs governing their respective spheres. They are simply part of the whole.