What are two symbols in "Paul's Case" and what do they represent?

When Paul appears before the school faculty to get back into school, he is wearing a red carnation. The staff feels this is too bold for someone trying to show humility and repentance. Paul wears it as a way of expressing his artistic side and for this meeting, as a way of expressing his defiance and contempt for the faculty. For Paul, his daily life at home and at school is dull and gray. His fascination with the theater and the art world is full of color and life. The red carnation stands out for its color. It represents this world that Paul longs to belong to. When Paul runs away to the city, he stays in a fancy hotel. He is pleased with the room except for one missing detail: flowers. Here, it is the fragrance of the flowers that brings him to life. In the end, when Paul decides to end his life, the carnation is symbolically drooping. The short life of flowers parallels Paul's brief happiness in life: 

It was only one splendid breath they had, in spite of their brave mockery at the winter outside the glass; and it was a losing game in the end, it seemed, this revolt against the homilies by which the world is run.

The narrator makes a clear comparison between Paul and the flowers. The flowers mock the winter as Paul mocked his teachers. The flowers revolt against the homilies. Cather uses the same word ("homilies") to describe the lessons of Paul's teachers. Paul symbolically buries one of the blossoms before he takes his life.

The stage entrance of the theater represents the door between Paul's gray world and the wondrous life of the theater. When he enters, he is transformed. This world is marked by artificiality, which is interesting because Paul chooses this "made up" world to the "real world." When he is banned from entering this world, he chooses to run away. The theater itself is an alternative reality for Paul. It represents the world in which he thinks he belongs. Fittingly, it is the place where things are made up, acted out, performed. It is a place where fiction is presented as an alternative to the real world outside.

What is the symbolism of the windows in "Paul's Case?"

Paul is an interesting "case" because while he is like many adolescents who rebel, Paul simply can't stand to go along with the provincial life that he is forced to accept. Consequently, he reinvents himself by making up stories of his connections with the theater and its inhabitants. Paul does not intend to become an actor or a part of the theater. He simply wants to dwell in its atmosphere. Paul's inventions are like the theater's performances: staged, creative, and in many cases so unrealistic as to be escapes from reality (like fairy tales). So, Paul is drawn to the romance of this escape. He flees to New York when this escape is taken from him. 

Paul steals money to live luxuriously if only for a few days. Here we see that Paul's concept of an ideal life becomes conflated with wealth. And although Paul's concept of this wealthy, artificial paradise is flawed because of its superficiality, it is still paradise to Paul. He sees in the theater and in the world of the wealthy something more beautiful than the mundane world of Cordelia Street. When this is taken from him, Paul sees no escape other than death. 

Paul was always looking for an escape. It is the sense of looking that relates to Paul's dream of being, and therefore looking, somewhere other than where he was. It's as if he were always looking through a window to some other place. After the performance, Paul followed the soprano to her hotel, watched her walk in, and began to daydream that he was with her. When he snaps out of his daydream, he is left with a dichotomy: the rain-soaked, dreary world he is standing in and the "orange glow of the windows above him." What dwelt behind that window was another fantasy, the escape Paul longed for. 

There it was, what he wanted--tangibly before him, like the fairy world of a Christmas pantomime--but mocking spirits stood guard at the doors, and, as the rain beat in his face, Paul wondered whether he were destined always to shiver in the black night outside, looking up at it. 

Paul goes home and decides that in order to avoid his father's disapproval, he will wait in the basement. He uses a window to get in the basement. Again, the window is a means of escape and avoiding his reality. 

When Paul is walking around Central Park, he notices the "avenue stages" (window displays). Particularly striking to him are the flowers enclosed in glass, untouched by the cold and snow outside. Throughout this story, flowers are a symbol of the vitality of life Paul strives to seek: 

Here and there on the corners were stands, with whole flower gardens blooming under glass cases, against the sides of which the snowflakes stuck and melted; 

In his sitting room, Paul is actually in the fantasy world. He looks out the window to note his separation from the cold reality outside, that he is protected (like the flowers in the window displays). Windows symbolize an actual and figurative border between reality and fantasy.

We might also consider the front of the theater's stage (fourth wall) as a window between the audience and the world of fiction/performance.