I would like to get some information and opinions of how the prestige of your so-called "credentials" helps or hurts in getting an academic job. I'm interested in this with respect to pure mathematics (I state this since I'm not sure if this is the same of all STEM subjects).
I guess that it is universally accepted the most important factor in one's academic career is his research. Of course, there are some more intricate relevant matters like a relationship with faculty of the department you apply to, success in teaching and the intersection of your research interests and faculty members' research interests.
However, the I wonder how pure prestige of one's credentials matter. Let me state it more clearly:
1) The prestige of your graduate school. If we have two candidates for a postdoc/tenure track with relatively equal research experience, but one holds a PhD from Harvard, while another from University of British Columbia, or University of Pennsylvania, or something else good, but not as famous and "shiny" as Harvard (or MIT/Princeton, for that matter), would a Harvard alumni be in a better position?
2) The fame of your doctoral advisor. Again, I advise you to consider how much the pure "fame" of your advisor matters, all other things aside, like the fact that a strong mathematician could help you mature as a mathematician too. Would having a Breakthrough Prize in Math/Fields winner boost on CV in way that math departments would more readily accept you as a postdoctorial researcher/assistant professor? Of course, this question is not only about Fields medalist, it was just an example.
3) The prestige of the institution where you did your postdoc. While points 1) and 2) were also related to getting a postdoc position, this matters only to getting an assistant professorship/tenure track. Usually, the advice for choosing a postdoc is to find a suitable researchers with respect to your interests. But this question again asks you to look on the matter from a different angle: the one of how being a postodoctorial researcher in a high-ranked institution assists in getting a tenure track.
Again, I wish to emphasize that this question is not about how a quality of a graduate school, a strength of a PhD advisor and a research atmosphere in a postdoctorial institution matters to an aspiring mathematician with respect to getting stronger as a researcher and a teacher. This much I know. I wish to hear some opinions on more "petty" stuff like "shiny", "glowing" names of departments or advisors, or relatively less famous ones.