Suppose a student is applying to a graduate school. Who is typically on the admissions committee? Is the chair of the department on the admissions committee?
Since this question has somehow come to the top of the queue: in the U.S., in mathematics, the department chair would rarely participate. The person in charge of graduate admissions would be "Director of Graduate Studies in Math.", or equivalent title. This would be a position that would/should involve genuine PR and recruitment work. The old tradition was that the position would be a backwater/sinecure for otherwise inactive faculty. In modern times, the level of energy required to do the job has increased to the point that most faculty could not cope, and do not want such a job, since it would take away from "refereed publications", the baseline for salary and status improvements.
But, of course, grad admissions and policies around this issue have a tremendous impact on the atmosphere in a department. Somehow the internet has made things more intense, more stressed, so choice of people to be around is all the more critical... ironically?
To some degree, the "grad admissions committee" is supposed to represent "all constituencies", but, in reality, this doesn't make much sense, since in the U.S. the thin-ness of undergrad preparation does not allow students to make competent announcements of their eventual interests.
The "real-politick" of grad admissions, involving understanding of the wildly varying undergrad or Master's level preparation around the world, and how those things translate to functioning in the U.S., is not interesting to most faculty, so the responsibility descends to the shoulders of a relative few who've paid attention to the reality, rather than the PR.
Who this is in a given place, and whether it's anyone at all, depends...