I need to find types of power,character archytypes and gender norm in the short story "The Lottery" by Shirley Jackson.

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From the point of view of power, women in Jackson's short story "The Lottery" are weak and not significant. The people who have power and significance are men. They run the town and the lottery - Mr. Summers and Mr. Graves. The women are portrayed as subservient and gossips. Right from the beginning this is made clear.

Soon the men began to gather. surveying their own children, speaking of planting and rain, tractors and taxes. They stood together, away from the pile of stones in the corner, and their jokes were quiet and they smiled rather than laughed. The women, wearing faded house dresses and sweaters, came shortly after their menfolk. They greeted one another and exchanged bits of gossip as they went to join their husbands. Soon the women, standing by their husbands, began to call to their children, and the children came reluctantly, having to be called four or five times. Bobby Martin ducked under his mother's grasping hand and ran, laughing, back to the pile of stones. His father spoke up sharply, and Bobby came quickly and took his place between his father and his oldest brother.

Notice that the men come first and then the women. Also notice that women are gossips and cannot even control their children. Later on Tessie comes to the fore and she, too, is portrayed as a weakling. In fact, she is like a child. She keeps repeating "it is not fair," until the people are upon her. 

The Lottery--Shirley Jackson

Tessie Hutchinson was in the center of a cleared space by now, and she held her hands out desperately as the villagers moved in on her. "It isn't fair," she said. A stone hit her on the side of the head. Old Man Warner was saying, "Come on, come on, everyone." Steve Adams was in the front of the crowd of villagers, with Mrs. Graves beside him.

"It isn't fair, it isn't right," Mrs. Hutchinson screamed, and then they were upon her.

From this perspective, it is a patriarchal town where men rule and women follow. 

On another level, if you are looking for another sources of power, you can mention tradition. Why does the town hold a lottery? Tradition. They way things are done is a strong force.