With one of its alumnae, Elena Kagan, poised for confirmation as a justice on the United States Supreme Court, it should be a triumphant season for Hunter College High School, a New York City public school for the intellectually gifted.

But instead, the school is in turmoil, with much of the faculty in an uproar over the resignation of a popular principal, the third in five years. In her departure speech to teachers in late June, the principal cited several reasons for her decision, including tensions over a lack of diversity at the school, which had been the subject of a controversial graduation address the day before by one of the school’s few African-American students.

Hours after the principal’s address, a committee of Hunter High teachers that included Ms. Kagan’s brother, Irving, read aloud a notice of no confidence to the president of Hunter College, who ultimately oversees the high school, one of the most prestigious public schools in the nation.

The events fanned a long-standing disagreement between much of the high school faculty and the administration of Hunter College over the use of a single, teacher-written test for admission to the school, which has grades 7 through 12. Faculty committees have recommended broadening the admissions process to include criteria like interviews, observations or portfolios of student work, in part to increase minority enrollment and blunt the impact of the professional test preparation undertaken by many prospective students.