Flowers represent Paul's spirit in this story.
I come to this conclusion for several reasons. First, this line: " His teachers felt this afternoon that his whole attitude was symbolized by his shrug and his flippantly red carnation flower, and they fell upon him without mercy, his English teacher leading the pack."
That indicates directly that those in the story found the flower representative of his attitude. I build from there to reading it as his spirit by this sequence: " The carnations in his coat were drooping with the cold, he noticed, their red glory all over. It occurred to him that all the flowers he had seen in the glass cases that first night must have gone the same way, long before this. 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."
Since he's drooping then, and about to kill himself, and since he (Paul) had rebelled against these homilies, it seems a clear association.