how does the "lord of the flies" offering symbolize jacks authority over his tribe?

The pig's head on the stick validates Jack's function on the island. As long as the kids believe in a Beast, Jack is able to offer himself as protector of all things scary. Jack puts on the facade as being tough and violent. He organizes his hunters like a quasi-militia. The "Lord of the Flies" is a tangible symbol of the boys' fear and validates Jack as their necessary to their survival.