Reaching out to the hiring manager after applying, a good idea?

I applied for a role on LinkedIn, the HR person in charge of hiring for the position was an alum, and had the same major as mine (small non-target school, small program too) I added them and we connected on LinkedIn because we were in the same program and we had a bunch of mutual connections including professors.

Is it a good idea to cold message this person on LinkedIn and express my interest in the role? something along the lines of "I wanted to connect and possibly learn more about the role" The individuals profile was posted with the job posting, that's how I found them